# Partial regularity for $\bm{BV^{\mathcal{B}}}$ minimizers

Federico Franceschini  ETH, Rämistrasse 101, 8092 Zürich, Switzerland
<EMAIL_ADDRESS>
###### Abstract.

We prove an $\varepsilon$-regularity theorem for $BV^{\mathcal{B}}$ minimizers
of strongly $\mathcal{B}$-quasiconvex functionals with linear growth, where
$\mathcal{B}$ is an elliptic operator of the first order. This generalises to
the $BV^{\mathcal{B}}$ setting the analogous result for $BV$ functions by F.
Gmeineder and J. Kristensen [Arch. Rational Mech. Anal. 232 (2019)].

The results of this work cannot be directly derived from the
$\mathcal{B}=\nabla$ case essentially because of Ornstein’s “non-inequality”.

This adaptation requires an abstract local Poincaré inequality and a fine
Fubini-type property to avoid the use of trace theorems, which in general fail
when $\mathcal{B}$ is elliptic.

##  1\. Introduction

###  1.1. Main result

In this work we prove an $\varepsilon$-regularity theorem for
$BV^{\mathcal{B}}$ \- minimizers of strongly $\mathcal{B}$-quasiconvex
functionals with linear growth, where $\mathcal{B}$ is an elliptic operator of
the first order.

Recently (especially after [PR16]), there has been interest to understand
which results available for $BV$ maps extend to the $BV^{\mathcal{B}}$
framework (see [KR22, Rai19, Rai19a, ARPR17, AR+19, AR21, CG22]). This work
falls in this line of research as our main result was proved in [GK18] in the
case $\mathcal{B}=\nabla$. We show that those arguments can be adapted to
general first order elliptic operators.

In order to state precisely our result we introduce briefly some vocabulary,
further details will be given in Section 2.

####  1.1.1. The operator $\mathcal{B}$.

We start fixing $\mathcal{B}$, an elliptic operator with constant
coefficients, homogeneous of order 1 from $\mathbf{R}^{m\times n}$ to
$\mathbf{R}^{N}$. That is to say, for each $v\in
C^{\infty}(\mathbf{R}^{n},\mathbf{R}^{m})$ we set

$\mathcal{B}v:=\sum_{j=1}^{n}B_{j}\partial_{j}v,\text{ for some linear maps
}B_{j}\colon\mathbf{R}^{m}\to\mathbf{R}^{N}.$

By elliptic, we mean that $\ker\widehat{\mathcal{B}}[\xi]=\\{0\\}$ for all
$\xi\in\mathbf{R}^{n}\setminus\\{0\\}$, where the _symbol_ is the linear map
between $\mathbf{R}^{m}$ and $\mathbf{R}^{N}$ defined as

$\widehat{\mathcal{B}}[\xi]:=\sum_{j=1}^{n}\xi_{j}B_{j}\text{ for each
}\xi\in\mathbf{R}^{n},$

so necessarily $m\leq N$. We also define the _wave cone_ of $\mathcal{B}$ as

$\Lambda_{\mathcal{B}}:=\bigcup_{\xi\neq
0}\operatorname{ran}\widehat{\mathcal{B}}[\xi]\ \subset\mathbf{R}^{N}.$

####  1.1.2. Functions with bounded $\mathcal{B}$-variation

This space of functions arises naturally when looking at distributional limits
of sequences $\\{v_{k}\\}\subset W^{1,1}(\mathbf{R}^{n},\mathbf{R}^{m})$
having a bound on $\|\mathcal{B}v_{k}\|_{L^{1}}$.

For an open set $\Omega\subset\mathbf{R}^{n}$, define

$BV^{\mathcal{B}}(\Omega):=\\{v\in
L^{1}(\Omega,\mathbf{R}^{m}):\mathcal{B}v\in\mathcal{M}(\Omega,\mathbf{R}^{N})\\},$

where $\mathcal{M}(\Omega,\mathbf{R}^{n})$ is the space of
$\mathbf{R}^{N}$-valued Borel measures with finite total variation in
$\Omega$. By a famous result of Ornstein (seee [Orn62, KK16]),
$BV^{\mathcal{B}}(\Omega)\subsetneq BV(\Omega,\mathbf{R}^{m})$ unless
$\mathcal{B}=\nabla$111That is to say, (unless) there is a linear map
$p\colon\mathbf{R}^{N}\to\mathbf{R}^{m\times n}$ such that $\nabla
u=p(\mathcal{B}u)$..

####  1.1.3. Functionals defined on measures

We explain the meaning of

$\int_{\Omega}f(\mathcal{B}u)$

when $u\in BV^{\mathcal{B}}$ and $f\colon\mathbf{R}^{N}\to\mathbf{R}$ has
linear growth

(H1) $|f(y)|\leq L\langle y\rangle\text{ for all }y\in\mathbf{R}^{N},$

and

(1.1) $f^{\infty}(y):=\lim_{t\to+\infty,y^{\prime}\to
y}\frac{f(ty^{\prime})}{t}\text{ exists for all
}y\in\operatorname{span}\Lambda_{\mathcal{B}}.$

Here, $L>0$ and $\langle y\rangle:=\sqrt{1+|y|^{2}}$ is the japanese bracket.

For all $v\in W^{1,1}(\Omega,\mathbf{R}^{m})$ consider the functional

(1.2) $\mathcal{F}[v,\Omega]:=\int_{\Omega}f(\mathcal{B}v(x))\,dx,$

which can be extended222This extension is continuous with respect to the area-
strict convergence, see Remark 2.5 to $BV^{\mathcal{B}}$ setting

$\mathcal{F}[v,\Omega]:=\int_{\Omega}f(\mathcal{B}v^{ac}(x))\,dx+\int_{\Omega}f^{\infty}\Big{(}\frac{d\mathcal{B}v^{s}}{d|\mathcal{B}v^{s}|}\Big{)}\,d|\mathcal{B}v^{s}|,$

where we decomposed the measure $\mathcal{B}v$ with respect to the Lebesgue
measure. We will then denote

$\int_{\Omega}f(\mathcal{B}v)=\mathcal{F}[v,\Omega]\text{ for all }v\in
BV^{\mathcal{B}}(\Omega),$

(without the $dx$).

####  1.1.4. $\mathcal{B}$-quasiconvexity

Following [FM99], we say that a continuous function
$f\colon\mathbf{R}^{N}\to\mathbf{R}$ is $\mathcal{B}$-quasiconvex if, for all
$y\in\mathbf{R}^{N}$, we have

$f(y)\leq\int_{Q}f(y+\mathcal{B}\varphi(x))\,dx\text{ for all }\varphi\in
C^{1}_{c}(Q,\mathbf{R}^{m}),$

where $Q\subset\mathbf{R}^{n}$ is the unit cube.

$\mathcal{B}$-quasiconvex functions with linear growth are automatically
Lipschitz and satisfy (1.1), thus we can define $\mathcal{F}[v,\Omega]$ for
$v\in BV^{\mathcal{B}}$. Furthermore, $\mathcal{F}[\cdot,\Omega]$ will be
weakly∗ lower semicontinuous up to boundary terms, see Theorem 2.18 below.

We say that $f$ is strongly $\mathcal{B}$-quasiconvex if there is $\ell>0$
such that

(H2) $f-\ell\langle\cdot\rangle\text{ is $\mathcal{B}$-quasiconvex.}$

Strong quasiconvexity is a natural assumption in the framework of minimization
problems: it is a necessary condition if we want $\mathcal{F}[\cdot,\Omega]$
to be $L^{1}$-coercive, see Remark 1.4.

####  1.1.5. Excess

We will prove regularity of local minimizers of $\mathcal{F}[\cdot,\Omega]$ in
the balls of $\Omega$ where a suitable energy denisity (called the excess
following [DG61]) is smaller than a parameter $\varepsilon$ which does not
depend on the particular solution.

In our situation the right definition of excess in a ball
$B_{R}(x_{0})\Subset\Omega$, is

(1.3)
$\Phi(x_{0},R):=\frac{1}{\omega_{n}R^{n}}\int_{B_{R}(x_{0})}E\big{(}\mathcal{B}u-(\mathcal{B}u)_{B_{R}(x_{0})}\mathscr{L}^{n}\big{)},$

where $E(y):=\sqrt{1+|y|^{2}}-1$ and

$(\mathcal{B}u)_{B_{R}(x_{0})}=\frac{\mathcal{B}u(B_{R}(x_{0}))}{\omega_{n}R^{n}}\in\mathbf{R}^{n}.$

$\Phi$ is a sort of $L^{1}$ oscillation of $\mathcal{B}u$ where we are
replacing the standard norm $|\cdot|$ with $E(\cdot)$, which has the advantage
of being strictly convex.

Under the further regularity assumption on the lagrangian

(H3) $f\in C^{2,1}_{\text{{loc}}}(\mathbf{R}^{N}),$

we are going to show

######  Theorem 1.1.

Let $f$ satisfy (H1), (H2) and (H3), and let $u\in
BV^{\mathcal{B}}(\mathbf{R}^{n})$ be a local minimizer of
$\mathcal{F}[\cdot,\Omega]$, that is

$\int_{\Omega}f(\mathcal{B}u)\leq\int_{\Omega}f(\mathcal{B}u+\mathcal{B}\varphi)\text{
for all }\varphi\in C^{1}_{c}(\Omega,\mathbf{R}^{m}).$

Then for every $\alpha\geq 1$ and $\gamma\in(0,1)$ there is a critical
threshold
$\varepsilon=\varepsilon(\alpha,\gamma,\mathcal{B},f^{\prime\prime},L/\ell)>0$
such that the following implication holds. If

$B_{R}(x_{0})\Subset\Omega,\quad\left|(\mathcal{B}u)_{B_{R}(x_{0})}\right|\leq\alpha,\quad\Phi(x_{0},R)\leq\varepsilon,$

then $u\in C^{2,\gamma}(B_{R/2}(x_{0}))$ and

$[\nabla^{2}u]_{C^{\gamma}(B_{R/2}(x_{0}))}\leq
CR^{-\gamma}\sqrt{\Phi(x_{0},R)}.$

for some $C=C({{\alpha,\gamma,\mathcal{B},f^{\prime\prime},L/\ell}})$.

Some remarks are in order.

######  Remark 1.2.

Non-convex Lagrangians $f$ satisfying (H1), (H2) and (H3) exist, even if they
are not given by explicit formulas, but rather regularizing quasi-convex
envelopes.

######  Remark 1.3.

Local minimizers $u\in BV^{\mathcal{B}}_{{\text{{loc}}}}$ to which Theorem 1.1
applies, can be constructed looking at minimizing sequences of the functional
$\mathcal{F}[v,\Omega]$ in a given Dirichlet class.

######  Remark 1.4.

Assumption (H2) is optimal in the following sense. Assume that $f$ has linear
growth and $\mathcal{F}[\cdot,\Omega]$ is coercive, that is for all sequences
$\\{v_{k}\\}$ with fixed boudnary values on $\partial\Omega$ it holds:

$\int_{\Omega}f(\mathcal{B}v_{k})\text{ bounded implies
}\|\mathcal{B}v_{k}\|_{L^{1}(\Omega)}\text{ bounded.}$

Then necessarily there is some $\ell>0$ such that
$f(\cdot)-\ell\langle\cdot\rangle$ is $\mathcal{B}$-quasiconvex at some
$z\in\mathbf{R}^{N}$. See Section 2.8.

###  1.2. Comparison with the full gradient case

Of course there is (a unique) $F\colon\mathbf{R}^{m\times n}\to\mathbf{R}$
such that $f(\mathcal{B}v)=F(\nabla v)$ and Theorem 1.1 in the case
$\mathcal{B}=\nabla$ has been already proved in [GK18]. Still, our result
cannot be reduced to the $\nabla$ case, indeed it is easily checked that $F$
satisfies (H1) and (H3), but (H2) does not hold for the full gradient
$\nabla$. Thus we only have $L^{1}$ bounds $\mathcal{B}u$, that do not imply
$L^{1}$ bounds on $\nabla u$, because of Ornstein non-inequality.

This fundamental difference can be worked out under the assumption that
$\mathcal{B}$ is elliptic, which a priori was not clear at all.

A posteriori, the main differences with respect to [GK18] are: a fine Fubini-
type argument to bypass the lack of a trace theorem for $BV^{\mathcal{B}}$
functions (cf. Section 2.6), an abstract Poincaré inequality to deal with
$\mathcal{B}$-affine functions, based on the general form of Ehrenpreis
fundamental principle (cf. Section 2.4).

We also remark that this adaptation would be straightforward if we assumed a
much stronger ellipticity condition on $\mathcal{B}$, namely complex
ellipticity. In this case both the trace theorem and the Poincaré inequality
are available by the results in [GR19].

###  1.3. Organization of the paper

In Section 2 we repeat the main definitions, fix the notation and prove the
core results for $BV^{\mathcal{B}}$ functions. In Section 3 we prove Theorem
1.1 following the implant of [GK18].

###  1.4. Acknowledgements

The author gratefully thanks Jan Kristensen for bringing the problem to his
attention and for the continuous guidance. Most of this work has been carried
out while the author was visiting Oxford University. The author would like to
thank the Mathematics Department and Magdalen College for the warm
hospitality.

The author has been also supported by Swiss NSF Ambizione Grant PZ00P2 180042
and by the European Research Council (ERC) under the Grant Agreement No.
948029 and under the Grant Agreement No. 721675.

##  2\. Framework and Preliminaries

We collect some preliminary results.

###  2.1. General notation

We work in $\mathbf{R}^{n}$ with its standard Euclidean structure, denote with
$B_{r}$ the balls centred at the origin of radius $r>0$. $\Omega$ will always
denote an open, bounded set with Lipschitz boundary.

When $f\colon\mathbf{R}^{N}\to\mathbf{R}$ we denote differentiation by apexes

$f^{\prime}(x)[z]:=\frac{d}{dt}\Big{|}_{t=0}f(x+tz),\quad
f^{\prime\prime}(x)[z,z]:=\frac{d^{2}}{dt^{2}}\Big{|}_{t=0}f(x+tz).$

Similarly the action of a bilinear map $Q$ on vectors $z,z^{\prime}$ is
denoted by $Q[z,z^{\prime}]$.

We denote with $\mathscr{D}$ the space of test functions and with
$\mathscr{D}^{\prime}$ the space of distributions.

We denote with $C_{c}$ the space of continuous compactly supported functions
and with $C_{0}$ its closure in the uniform topology.

The space $\mathcal{M}(\Omega,\mathbf{R}^{N})$ of $\mathbf{R}^{N}$-valued
Borel measures on $\Omega$ with finite total variation, will be identified
with

$\mathcal{M}(\Omega,\mathbf{R}^{N})\simeq C_{0}(\Omega,\mathbf{R}^{N})^{*}.$

Similarly we have

$\mathcal{M}_{loc}(\Omega,\mathbf{R}^{N})\simeq
C_{c}(\Omega,\mathbf{R}^{N})^{*}\text{ and
}\mathcal{M}(\overline{}\Omega,\mathbf{R}^{N})\simeq
C(\overline{}\Omega,\mathbf{R}^{N})^{*}.$

We denote with the angular bracket $\langle\cdot,\cdot\rangle$ these
dualities, using the standard scalar product on $\mathbf{R}^{N}$.

We will use the the trace spaces $W^{s,p}$ defined by the Gagliardo seminorms

$[u]_{W^{s,p}(B_{1})}:=\left(\int_{B_{1}}\int_{B_{1}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\,dx\,dy\right)^{1/p},$

we also need the sphere version

$[u]_{W^{s,p}(\partial B_{1})}:=\left(\int_{\partial B_{1}}\int_{\partial
B_{1}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n-1+sp}}\,d\sigma_{x}\,d\sigma_{y}\right)^{1/p}.$

In estimates we write $X\lesssim_{a,b,c}Y$ meaning that, if one fixes the
parameters $a,b,c$, then the ratio $X/Y$ is bounded.

###  2.2. Functionals defined on measures

We introduce notation to deal with functionals defined on measures. We refer
to [AFP00] for background in measure theory.

######  Definition 2.1.

We say that continuous function
$f\colon\overline{\Omega}\times\mathbf{R}^{N}\to\mathbf{R}$ belongs to
$\mathbf{E}_{1}(\Omega,\mathbf{R}^{N})$ if the limit

$\lim_{t\to+\infty}\frac{f(x,tz)}{t}=:f^{\infty}(x,z)\text{ exists in
}\mathbf{R},\text{ locally uniformly in
}x\in\overline{\Omega},z\in\mathbf{R}^{N}.$

The function $f^{\infty}(\cdot,\cdot)$ is called the “strong recession
function”.

We remark that, by definition,
$f^{\infty}\colon\overline{}\Omega\times\mathbf{R}^{N}\to\mathbf{R}$ is
continuous and positively one-homogeneous in its second argument.

For any $f\in\mathbf{E}_{1}$ and any
$\mu\in\mathcal{M}(\overline{}\Omega,\mathbf{R}^{N})$, we take the
decomposition of $\mu$ with respect to the Lebesgue measure
$\mu=\mu^{ac}(x)\mathscr{L}^{n}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\Omega+\mu^{s}$ and further
decompose the singular part $\mu^{s}$ in terms of its own total variation
$\mu^{s}=\frac{d\mu^{s}}{d|\mu^{s}|}(x)\,|\mu^{s}|$. Then we define for any
Borel set $A\subset\overline{\Omega}$

(2.1)
$\int_{A}f(x,\mu):=\int_{A}f(x,\mu^{ac}(x))\,dx+\int_{A}f^{\infty}\Big{(}x,\frac{d\mu^{s}}{d|\mu^{s}|}(x)\Big{)}\,d|\mu^{s}|(x).$

The same construction can of course be carried out for every Radon measure
$\mu\in C_{0}(\Omega,\mathbf{R}^{N})^{*}$ and Borel set $A\Subset\Omega$.

We can now define a suitable notion of strong convergence of measures. Define
$E:\mathbf{R}^{N}\to\mathbf{R}$ by

(2.2) $E(z):=\sqrt{1+|z|^{2}}-1=\langle z\rangle-1,$

it is easily checked that $E$ belongs to
$\mathbf{E}_{1}(\Omega,\mathbf{R}^{N})$ and it has the nice property of being
strictly convex. Furthermore, simple computations shows that

(2.3)
$\int_{\overline{}\Omega}E(\mu)=|\tilde{\mu}|(\overline{\Omega})\quad\text{
where }\tilde{\mu}:=\left(\mu,\mathscr{L}^{n}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\Omega\right)\in
C(\overline{\Omega},\mathbf{R}^{N}\times\mathbf{R})^{*}.$

######  Definition 2.2 (Area-strict convergence).

Given $\mu$ and $\\{\mu_{j}\\}_{j\in\mathbf{N}}$ in
$\mathcal{M}(\Omega,\mathbf{R}^{N})$ we say that
$\\{\mu_{j}\\}_{j\in\mathbf{N}}$ converges “area-strictly” to $\mu$ in
$\Omega$, and write $\mu_{j}\stackrel{{\scriptstyle
E}}{{\rightharpoondown}}\mu$, as $j\to+\infty$, provided
$\mu_{j}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\mu$ in
$C_{c}(\Omega,\mathbf{R}^{N})^{*}$ and

(2.4) $\int_{\Omega}E(\mu_{j})\to\int_{\Omega}E(\mu)\quad\text{ as
}j\to+\infty.$

Intuitively (2.4) prevents oscillations and loss of mass to $\partial\Omega$,
this ensures continuity of the functional $\mu\mapsto\int_{\Omega}f(x,\mu)$
for all $f\in\mathbf{E}_{1}(\Omega,\mathbf{R}^{N})$ as the following version
of Reshetnyak continuity shows.

######  Theorem 2.3 (Theorem 5 in [KR09]).

For every $f\in\mathbf{E}_{1}(\Omega,\mathbf{R}^{N})$ we have

$\lim_{j\to+\infty}\int_{\overline{}\Omega}f(x,\mu_{j})=\int_{\overline{}\Omega}f(x,\mu),$

provided $\mu_{j}\stackrel{{\scriptstyle E}}{{\rightharpoondown}}\mu$ in
$\Omega$.

######  Remark 2.4.

If we know something more about the limit measure $\mu$, we can relax the
assumptions on $f$. In fact, what is really needed in the proof of Theorem 2.3
is that the “perspective integrand”
$F\colon\overline{}\Omega\times\mathbf{R}^{N}\times\mathbf{R}\to\mathbf{R}$,
defined by

$F(x,z,t):=\begin{cases}|t|\,f(x,z/|t|)&\text{ if }t\neq 0,\\\
f^{\infty}(x,z)&\text{ if }t=0,\end{cases}$

has a $|\tilde{\mu}|$-neglegible set of discontinuity points (see [AFP00,
Proposition 1.62, (b)]), where $\tilde{\mu}$ is as in (2.3).

######  Remark 2.5.

Definition (2.1) can be justified a posteriori by Theorem 2.3. In fact (2.1)
is obtained as extension by area strict continuity of

$\phi\mapsto\ \int_{\Omega}f(x,\phi(x))\,dx,$

where we think $\phi\in
L^{1}(\Omega,\mathbf{R}^{N})\subset\mathcal{M}(\Omega,\mathbf{R}^{N})$.

###  2.3. The operators $\mathcal{A}$ and $\mathcal{B}$

We fix a homogeneous, first-order, elliptic differential operator
$\mathcal{B}$ with constant coefficients over ${\mathbf{R}^{n}}$ from
$\mathbf{R}^{m}$ to $\mathbf{R}^{N}$, as explained in Section 1.1.1.

The Leibniz rule takes the form

(2.5) $\mathcal{B}(\eta
u)=\eta\mathcal{B}u+\widehat{\mathcal{B}}[\nabla\eta]u,\text{ for all }\eta\in
C^{\infty}(\mathbf{R}^{n}),u\in C^{\infty}(\mathbf{R}^{n},\mathbf{R}^{m}).$

Exploiting the ellipticity assumption we find another finite dimensional
vector space $\mathbf{R}^{d}$ and a _homogeneous_ differential operator
$\mathcal{A}$ over ${\mathbf{R}^{n}}$ from $\mathbf{R}^{N}$ to
$\mathbf{R}^{d}$ such that

(symbol exactness)
$\operatorname{ran}\widehat{\mathcal{B}}[\xi]=\ker\widehat{\mathcal{A}}[\xi]\qquad\text{
for all }\xi\in{\mathbf{R}^{n}}\setminus\\{0\\}.$

Notice that $\mathcal{A}$ might have order larger than one. The existence of
such a couple $(d,\mathcal{A})$ is not obvious nor unique, (see [VS11,
Proposition 4.2]). In the elliptic case one can, for example, set $d:=N$ and
define $\mathcal{A}$ via its symbol

$\widehat{\mathcal{A}}:={\det\left(\,\widehat{\mathcal{B}}^{\dagger}\circ\widehat{\mathcal{B}}\,\right)}\cdot\left\\{\text{id}_{\mathbf{R}^{N}}-\widehat{\mathcal{B}}\circ{\left(\,\widehat{\mathcal{B}}^{\dagger}\circ\widehat{\mathcal{B}}\,\right)}^{-1}\circ\widehat{\mathcal{B}}^{\dagger}\right\\},$

homogeneity and symbol exactness are simple to check.

Finally, we remark that
$\mathcal{B}u(x)\in\operatorname{span}\Lambda_{\mathcal{B}}$, so we can (and
do) always restrict ourselves to the case
$\mathbf{R}^{N}=\operatorname{span}\Lambda_{\mathcal{B}}$.

###  2.4. Erhenpreis fundamental principle

We state a very general result concerning the solvability of (possibly
overdetermined) systems of PDEs with constant coefficients.

Consider the set $M_{\mathcal{B}}$ of all constant-coefficients differential
operators

$\alpha\colon C^{\infty}({\mathbf{R}^{n}},\mathbf{R}^{m})\to
C^{\infty}({\mathbf{R}^{n}})$

such that $\alpha\circ\mathcal{B}\equiv 0$. We will call $M_{\mathcal{B}}$ the
“module of compatibility conditions” of $\mathcal{B}$. The following
remarkable result is contained in [Pal70, Theorem 1, Chapter 7].

######  Theorem 2.6.

Let $V\subset{\mathbf{R}^{n}}$ be an open convex set and ${M}_{\mathcal{B}}$
its module of compatibility conditions. If
$f\in\mathscr{D}^{\prime}(V,\mathbf{R}^{m})$ satisfies

(2.6) $\alpha(f)=0\text{ in }\mathscr{D}^{\prime}(V)\text{ for every
}\alpha\in{M}_{\mathcal{B}},$

then there exists $u\in\mathscr{D}^{\prime}(V,X)$ such that $\mathcal{B}u=f$
in $\mathscr{D}^{\prime}(V,\mathbf{R}^{m})$.

######  Remark 2.7.

Then ${M}_{\mathcal{B}}$ has a natural structure of
$\mathbf{R}[\xi_{1},\ldots,\xi_{n}]$-module via the natural action:

$p\cdot\alpha:=p(\partial/\partial x_{1},\ldots,\partial/\partial
x_{n})\circ\alpha,\qquad\text{ for all
}p\in\mathbf{R}[\xi],\alpha\in{M}_{\mathcal{B}}.$

Since ${M}_{\mathcal{B}}$ is a submodule of the Noetherian module
$\mathbf{R}[\xi_{1},\ldots,\xi_{n}]$, it is finitely generated by some
operators $\\{\mathcal{A}_{1},\ldots,\mathcal{A}_{\ell}\\}$. This means that
the compatibility condition (2.6) in this theorem can be checked only for
$\mathcal{A}_{1},\ldots,\mathcal{A}_{\ell}$.

######  Remark 2.8.

This theorem is a far reaching generalization of Poincaré’s Lemma: a closed
form (that is a form that satisfies the compatibility conditions $d\omega=0$)
is in fact exact, provided the domain is simple enough (for example convex
always works). A quicker account of this theory can be found in [Hör90,
Chapter 7].

###  2.5. The space $BV^{\mathcal{B}}$

Let us collect some properties of the space $BV^{\mathcal{B}}$. Recall that

$BV^{\mathcal{B}}(\Omega):=\left\\{u\in
L^{1}(\Omega,\mathbf{R}^{m})\colon\mathcal{B}u\in
C_{0}(\Omega,\mathbf{R}^{N})^{*}\right\\}.$

The space $BV^{\mathcal{B}}_{\text{{loc}}}(\Omega)$ is defined similarly
requiring that $\mathcal{B}u\in C_{c}(\Omega,\mathbf{R}^{N})^{*}$.

Given some sequence $\\{u_{j}\\}_{j\in\mathbf{N}}\subset
BV^{\mathcal{B}}(\Omega)$ and $u\in BV^{\mathcal{B}}(\Omega)$, we say that:

  * •

$\\{u_{j}\\}$ converges weakly∗ to $u$ in
$BV^{\mathcal{B}}_{\text{{loc}}}(\Omega)$ provided

$u_{j}\to u\text{ in }L^{1}_{{\text{{loc}}}}(\Omega)\quad\text{ and
}\quad\mathcal{B}u_{j}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\mathcal{B}u\text{
in }C_{c}(\Omega,\mathbf{R}^{N})^{*};$

  * •

$\\{u_{j}\\}$ converges area-strictly to $u$ in $BV^{\mathcal{B}}(\Omega)$
provided $\lim_{j}\|u-u_{j}\|_{L^{1}(\Omega)}=0$ and

$\int_{\Omega}E(\mathcal{B}u_{j})\to\int_{\Omega}E(\mathcal{B}u),\quad\text{
that is }\mathcal{B}u_{j}\stackrel{{\scriptstyle
E}}{{\rightharpoondown}}\mathcal{B}u\text{ in }\Omega.$

The area-strict closure of $C^{\infty}_{c}(\Omega,\mathbf{R}^{m})$ is denoted
by $BV^{\mathcal{B}}_{0}(\Omega)$.

$BV^{\mathcal{B}}$ functions can be approximated by smooth ones in the
following sense.

######  Lemma 2.9.

Let $u\in BV^{\mathcal{B}}_{{\text{{loc}}}}(\Omega)$ and
$\\{\phi_{\varepsilon}\\}_{\varepsilon>0}$ be a family of standard mollifiers,
extend $u$ to zero outside $\Omega$ and set
$u_{\varepsilon}:=u*\phi_{\varepsilon}$. Then $u_{\varepsilon}$ converges
weakly* to $u$ in $BV^{\mathcal{B}}_{\text{{loc}}}(\Omega)$. Furthermore, for
every $U\Subset\Omega$ such that $\mathscr{L}^{n}(\partial
U)+|\mathcal{B}u|(\partial U)=0$, convergence holds in the area-strict sense
in $BV^{\mathcal{B}}(\omega)$.

The following embedding is crucial for the proof to work in the
$\mathcal{B}\neq\nabla$ case

######  Lemma 2.10.

For all $u\in BV^{\mathcal{B}}(B_{1})$ and $p\in(1,\frac{n+1}{n})$ we have

(2.7)
$\|u\|_{W^{1-1/p,p}(B_{1/2})}\lesssim_{\mathcal{B},p}|\mathcal{B}u|(B_{1})+\|u\|_{L^{1}(B_{1})}.$

###### Proof.

As the inequality is local, we can reduce ourselves, by multiplying by a
cutoff function, to prove (2.7) in the whole $\mathbf{R}^{n}$. Furthermore,
thanks to Lemma 2.9, we can assume $u\in
C^{1}_{c}(\mathbf{R}^{n},\mathbf{R}^{m})$.

Let $s<0$ to be fixed later. First, recall that by Sobolev embedding it holds

$H^{-s,p^{\prime}}(\mathbf{R}^{n})\subset C_{0}(\mathbf{R}^{n})\text{ provided
}-sp^{\prime}>n,$

where $p^{\prime}$ is the Hölder conjugate. Taking duals we find
$\mathcal{M}(\mathbf{R}^{n})\subset H^{s,p}(\mathbf{R}^{n})$. Mihlin
multiplier theorem $(p>1)$ and ellipticity of $\mathcal{B}$ then give

$\|\nabla u\|_{H^{s,p}}\lesssim_{n,s,p}\|\mathcal{B}u\|_{H^{s,p}}.$

Furthermore, in this range of $p$333Since $\mathcal{B}^{-1}$ can be
represented as a convolution kernel which is $1-n$ homogeneous, by classical
boundedness of convolution operators we have
$BV^{\mathcal{B}}(\mathbf{R}^{n})\subset L^{n/(n-1)}_{weak}(\mathbf{R}^{n})$.
Since $p<n/(n-1)$, we deduce $BV^{\mathcal{B}}(\mathbf{R}^{n})\subset
L^{p}(\mathbf{R}^{n})$. we have

$\|u\|_{L^{p}(\mathbf{R}^{n})}\lesssim_{n,p}\|\mathcal{B}u\|_{L^{1}(\mathbf{R}^{n})}+\|u\|_{L^{1}(\mathbf{R}^{n})}.$

Thus, we proved that $BV^{\mathcal{B}}(\mathbf{R}^{n})\subset H^{1+s,p}$,
provided $sp^{\prime}>n$.

Now we conclude using the relationship between Hardy and Besov spaces, we
refer to [BL76] for background.

Since $p\in(1,2]$ we have

$H^{1+s,p}(\mathbf{R}^{n})\subset B^{1+s}_{p2}(\mathbf{R}^{n}),$

see for example [BL76, Theorem 6.4.4]. Then, for all $\delta>0$, we use that
(see for example [Tri73, Theorem 5.2.1])

$B^{1+s}_{p2}(\mathbf{R}^{n})\subset B^{1+s-\delta}_{pp}(\mathbf{R}^{n}).$

Since $B^{1-1/p}_{pp}(\mathbf{R}^{n})=W^{1-1/p,p}(\mathbf{R}^{n})$, we are
finished if we find $s,\delta$ such that

$\delta>0,\quad s<0,\quad 1+s-\delta=1/p^{\prime},\quad-sp^{\prime}>n.$

Such choice is possible if and only if $1/p>n/p^{\prime}$, which rewrites as
$p<(n+1)/n$. ∎

######  Remark 2.11.

Recall that $W^{1-1/p,p}(\mathbf{R}^{n-1})$ is the trace space of
$W^{1,p}(\mathbf{R}^{n})$, for all $p>1$, thanks to Gagliardo’s trace theorem
[Gag57].

We define the set of $\mathcal{B}$-affine maps in an open set
$U\subset\mathbf{R}^{n}$ as

$\operatorname{Aff}(\mathcal{B},U)=\\{u\in\mathscr{D}^{\prime}(U,\mathbf{R}^{m}):\mathcal{B}u\equiv\text{cost.}\\}=\\{u\in
C^{\infty}(U,\mathbf{R}^{m}):\mathcal{B}u\equiv\text{cost.}\\},$

and the kernel

$\ker(\mathcal{B},U)=\\{u\in\mathscr{D}^{\prime}(U,\mathbf{R}^{m}):\mathcal{B}u=0\\}=\\{u\in
C^{\infty}(U,\mathbf{R}^{m}):\mathcal{B}u=0\\},$

where we use the local regularity of constant coefficients elliptic operators.
These maps in general depend on $U$.

The closed graph theorem then entails the local Poincaré inequality.

######  Proposition 2.12 (Poincaré).

For all $u\in BV^{\mathcal{B}}(B_{1})$ we have

(2.8)
$\inf_{h\in\ker(\mathcal{B},B_{1})}\|u-h\|_{L^{1}(B_{1/2})}\lesssim_{\mathcal{B}}|\mathcal{B}u|(B_{1}).$

And, for all $p\in(1,\frac{n+1}{n})$ and $R>0$

(2.9)
$\inf_{h\in\ker(\mathcal{B},B_{R})}[u-h]_{W^{1-1/p,p}(B_{R/2})}\lesssim_{\mathcal{B},p}R^{\frac{n+1}{p}-n}|\mathcal{B}u|(B_{R}).$

###### Proof.

We start reformulating (2.8) abstractly. Consider the Frechét spaces

$X:=BV^{\mathcal{B}}_{{\text{{loc}}}}(B_{1})\text{ and
}Y:=\mathcal{M}_{{\text{{loc}}}}(B_{1},\mathbf{R}^{N}).$

The structure of Frechét spaces is induced by the seminorms ($k\geq 0$)

$p^{X}_{k}(u):=|\mathcal{B}u|(B_{1-2^{-k}})+\int_{B_{1-2^{-k}}}|u|\quad\text{
and }\quad p^{Y}_{k}(\mu)=|\mu|(B_{1-2^{-k}}).$

If we consider the continuous map $\mathcal{B}\colon X\to Y$ sending $u$ to
$\mathcal{B}u$, (2.8) is equivalent to show that the inverse of the map

$\widetilde{\mathcal{B}}\colon
X/\ker(\mathcal{B},B_{1})\to\operatorname{ran}\mathcal{B}$

is continuous. Thus, by the open mapping theorem, everything boils down to
prove that $\operatorname{ran}\mathcal{B}$ is closed in $Y$. Consider a
sequence $u_{j}\in X$ such that $\mathcal{B}u_{j}\to\mu$ in $Y$ for some
measure $\mu$. Then for all $\alpha\in M_{\mathcal{B}}$ (see Section 2.4) we
have

$\alpha(\mu)=\mathscr{D}^{\prime}-\lim_{j}\alpha(\mathcal{B}u_{j})=0.$

Then Theorem 2.6 applies and gives
$v\in\mathscr{D}^{\prime}(B_{1},\mathbf{R}^{m})$ such that $\mathcal{B}v=\mu$.
Finally, since $\mathcal{B}$ is elliptic,
$\mathcal{B}u\in\mathcal{M}_{{\text{{loc}}}}$ forces $v\in
L^{q}_{{\text{{loc}}}}$ for all $q\in[1,n/n-1)$, thus $v\in X$ and
$\mu\in\operatorname{ran}\mathcal{B}$.

We turn to the proof of (2.9), by scaling we can take $R=1$. Take a smooth
cutoff function $\bm{1}_{B_{1/2}}\leq\varrho\leq\bm{1}_{B_{2/3}}$ and any
$h\in\ker(\mathcal{B},B_{1})$, then using Lemma 2.10 we have

$\displaystyle[u-h]_{W^{1-1/p,p}(B_{1/2})}$
$\displaystyle\leq[\varrho(u-h)]_{W^{1-1/p,p}(\mathbf{R}^{n})}$
$\displaystyle\lesssim_{n,s,\mathcal{B}}\left\|\mathcal{B}(\varrho(u-h))\right\|_{\mathcal{M}(\mathbf{R}^{n})}+\|\varrho(u-h)\|_{L^{1}(\mathbf{R}^{n})}$
$\displaystyle\leq\left\|\varrho\mathcal{B}u\right\|_{\mathcal{M}(\mathbf{R}^{n})}+\left\|\widehat{\mathcal{B}}[\nabla\varrho](u-h)\right\|_{L^{1}(\mathbf{R}^{n})}+\|\varrho(u-h)\|_{L^{1}(\mathbf{R}^{n})}$
$\displaystyle\lesssim_{\mathcal{B}}|\mathcal{B}u|(B_{1})+\|u-h\|_{L^{1}(B_{3/2})}.$

Now thanks to (2.8) we can choose $h$ so that
$\|u-h\|_{L^{1}(B_{2/3})}\lesssim|\mathcal{B}u|(B_{1})$ and we are done. ∎

For $u\in BV^{\mathcal{B}}_{{\text{{loc}}}}(\Omega)$ consider the Lebesgue
decomposition of the measure $\mathcal{B}u$

$\mathcal{B}u=(\mathcal{B}u)^{ac}\,\mathscr{L}^{n}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\Omega+\frac{d(\mathcal{B}u)^{s}}{d|\mathcal{B}u|^{s}}\,|\mathcal{B}u|^{s},$

where the Borel function
$\frac{d(\mathcal{B}u)^{s}}{d|\mathcal{B}u|^{s}}\colon\Omega\to\mathbf{R}^{N}$
is defined only $|\mathcal{B}u|^{s}$-almost everywhere. Then the following
generalization of Alberti’s rank-one Theorem ([Alb93, PR16]) holds

(2.10) $\frac{d(\mathcal{B}u)^{s}}{d|\mathcal{B}u|^{s}}(x)\text{ belongs to
}\Lambda_{\mathcal{B}}\text{ for $|\mathcal{B}u|^{s}$-a.e. $x\in\Omega$}.$

###  2.6. Fubini property

We will also use a Fubini-type property for maps $f\in
W^{s,p}(B_{1},\mathbf{R}^{m})$.

######  Lemma 2.13.

Let $s\in(0,1),p>1$ and let $\phi$ be a standard mollifier. Let $f\in
W^{s,p}(B_{1},\mathbf{R}^{m})$ and denote by $B_{1}\setminus S_{f}$ the set of
$L^{p}$-Lebesgue points of $f$ and by $\tilde{f}\colon B_{1}\setminus
S_{f}\to\mathbf{R}^{m}$ the precise representative.

For all $0<r<R<1$ with $R\leq 100r$, there is a set of “good radii”
$G\subset(r,R)$ such that $\mathscr{L}^{1}(G)>0$ and for all $t\in G$ the
following holds

  * (a)

$\mathcal{H}^{n-1}(S_{f}\cap\partial B_{t})=0$;

  * (b)

for $\phi_{\varepsilon}:=\varepsilon^{-n}\phi(\cdot/\varepsilon)$ it holds

$\|\tilde{f}-f*\phi_{\varepsilon}\|_{L^{p}(\partial B_{t})}\to 0\text{ as
}\varepsilon\downarrow 0;$

  * (c)

we have the bound

$[\tilde{f}]_{W^{s,p}(\partial
B_{t})}\lesssim_{n,s,p,}(R-r)^{-1/p}[f]_{W^{s,p}(B_{R})}.$

###### Proof.

Assume that $[f]_{W^{s,p}(B_{1})}\leq 1$, the proof is based on the Fubini
inequality

(2.11) $\int_{r}^{R}[f]_{W^{s,p}(\partial B_{t})}dt\lesssim 1.$

While known (see [Gme17, Proposition 8.25]), let us sketch its proof. Recall
the following norm which is equivalent 444This is based on the nice fact that
for subadditive functions $0\leq g(x+x^{\prime})\leq g(x)+g(x^{\prime})$ we
have $\sup_{B_{1}}g\lesssim\int_{B_{1}}g(x)/|x|^{n}$. to the $W^{s,p}$ one
(see for example [Leo17, Proposition 17.21]):

$|f|^{p}_{W^{s,p}(\mathbf{R}^{n})}:=\int_{0}^{\infty}\sup_{\xi\in\mathbf{R}^{n},|\xi|\leq
t}\|f-f(\cdot-\xi)\|^{p}_{L^{p}(\mathbf{R}^{n})}\frac{dt}{t^{1+sp}}.$

Now for $f\in C^{1}_{c}(\mathbf{R}^{n},\mathbf{R}^{m})$ we have

$\displaystyle\int_{0}^{\infty}[f(\cdot,s)]^{p}_{W^{s,p}(\mathbf{R}^{n-1})}ds$
$\displaystyle=\int_{0}^{\infty}ds\int_{\mathbf{R}^{n-1}}dx^{\prime}\int_{\mathbf{R}^{n-1}}dy^{\prime}\frac{|f(x^{\prime},s)-f(y^{\prime},s)|^{p}}{|x^{\prime}-y^{\prime}|^{n-1+sp}}$
$\displaystyle(\tau^{\prime}:=x^{\prime}-y^{\prime})\quad$
$\displaystyle=\int_{\mathbf{R}^{n-1}}\frac{d\tau^{\prime}}{|\tau^{\prime}|^{n-1+sp}}\|f(\cdot-(\tau^{\prime},0))-f\|^{p}_{L^{p}(\mathbf{R}^{n-1}\times(0,\infty))}$
$\displaystyle\leq\int_{\mathbf{R}^{n-1}}\frac{d\tau^{\prime}}{|\tau^{\prime}|^{n-1+sp}}\sup_{\xi\in\mathbf{R}^{n},|\xi|\leq|\tau^{\prime}|}\|f(\cdot-\xi)-f\|^{p}_{L^{p}(\mathbf{R}^{n})}$
$\displaystyle(\text{polar})\quad$
$\displaystyle=\int_{0}^{\infty}\frac{dt}{t^{1+sp}}\sup_{\xi\in\mathbf{R}^{n},|\xi|\leq
t}\|f(\cdot-\xi)-f\|^{p}_{L^{p}(\mathbf{R}^{n})}$
$\displaystyle=|f|^{p}_{W^{s,p}(\mathbf{R}^{n})}\leq
C(n,s,p)[f]^{p}_{W^{s,p}(\mathbf{R}^{n})}.$

This computation works also slicing with spheres, at least as long as $r/R$ is
bounded below.

Now (a) is just a consequence of Fubini’s theorem in polar coordinates and
Lebesgue differentiation Theorem, indeed we have $\mathscr{L}^{1}(I)=R-r$
where

$I:=\\{t\in[r,R]:\mathcal{H}^{n-1}(S_{f}\cap\partial B_{t})=0\\}.$

In order to prove (b), we apply (2.11) to
$f_{\varepsilon}=f*\phi_{\varepsilon}$, and employ Fatou’s Lemma to find
$\mathscr{L}^{1}(J)=R-r$ where

$J:=\\{t\in[r,R]:\liminf_{\varepsilon}\|f_{\varepsilon}\|_{W^{s,p}(\partial
B_{t})}<\infty\\}.$

So, by Rellich’s Theorem, for each $t\in J$ we have that
$\\{f_{\varepsilon}|_{\partial B_{t}}\\}_{\varepsilon}$ is pre-compact in
$L^{p}(\partial B_{t})$. If $t\in I\cap J$, then necessarily

$f_{\varepsilon}\to\tilde{f}\text{ in }L^{p}(\partial B_{t}),$

by uniqueness of the $\mathcal{H}^{n-1}$-a.e. limit.

Finally (c) follows by (2.11) and the mean value inequality. ∎

###  2.7. $\mathcal{B}$-quasiconvexity and lower semicontinuity

Let us first repeat the definition given in Section 1.1.4:

######  Definition 2.14 ($\mathcal{B}$-quasiconvexity).

A locally bounded Borel function $f\colon\mathbf{R}^{N}\to\mathbf{R}$ is said
to be $\mathcal{B}$-quasiconvex if, for all $y\in\mathbf{R}^{N}$, it holds

(2.12)
$f(y)=\inf\left\\{\fint_{Q}f(y+\mathcal{B}\varphi(x))\,dx:\varphi\in\mathscr{D}(Q,\mathbf{R}^{m})\right\\},$

where $Q$ is the unit cube in $\mathbf{R}^{n}$.

######  Remark 2.15.

This is equivalent to ask that $f$ is $\mathcal{A}$-quasiconvex in the sense
of [FM99], see for example [GR19, Corollary 6, Lemma 5].

It is immediate to check that (2.12) is equivalent to require that
$f\circ\beta\colon\mathbf{R}^{m}\otimes\mathbf{R}^{n}\to\mathbf{R}$ is quasi-
convex in the sense of Morrey ([Mor52]), where
$\beta\colon\mathbf{R}^{m}\otimes\mathbf{R}^{n}\to\mathbf{R}^{N}$ is the
linear map such that $\mathcal{B}u(x)=\beta(\nabla u(x))$, which is given by

$\beta(v\otimes\xi):=\widehat{\mathcal{B}}[\xi]v\text{ for
}\xi\in\mathbf{R}^{n},v\in\mathbf{R}^{m}.$

Since $\mathbf{R}^{N}=\operatorname{span}\Lambda^{\mathcal{B}}$, $\beta$ is
surjective, but in general not injective; nevertheless
$\Lambda^{\mathcal{B}}=\beta(\Lambda^{\nabla})$ since
$\Lambda^{\nabla}=\\{\text{rank-one matrices}\\}$. This observation grants
that many properties of $f$ are immediately deduced from the corresponding
properties of Morrey’s quasiconvex functions, for example

######  Lemma 2.16.

If $f$ is $\mathcal{B}$-quasiconvex and has linear growth, then it is globally
Lipschitz and $\Lambda_{\mathcal{B}}$-convex, meaning that for all
$t\in[0,1],y,y^{\prime}\in\mathbf{R}^{N}$ it holds

$f(ty+(1-t)y^{\prime})\leq tf(y)+(1-t)f(y^{\prime}),\text{ provided
}y-y^{\prime}\in\Lambda_{\mathcal{B}}.$

This Lemma entails the following

######  Proposition 2.17.

Assume $f\colon\mathbf{R}^{N}\to\mathbf{R}$ is $\mathcal{B}$-quasiconvex and
has linear growth. Consider the upper and lower recession functions

$f^{\\#}(y)=\limsup_{y^{\prime}\to y,t\uparrow\infty}f(ty^{\prime})/t,\quad
f_{\\#}(y)=\liminf_{y^{\prime}\to y,t\uparrow\infty}f(ty^{\prime})/t,$

which are real valued and positively 1-homogeneous. Then
$\Lambda_{\mathcal{B}}\subset\\{f^{\\#}=f_{\\#}\\}$ so that the functional
$u\mapsto\int_{\Omega}f(\mathcal{B}u)$ is well defined on
$BV^{\mathcal{B}}(\Omega)$ and area-strict continuous (cf. Remark 2.4 and
(2.10)).

Now that we known how to give a meaning to $\int_{\Omega}f(\mathcal{B}u)$ it
is natural to hope that this functional is l.s.c. with respect the weak*
convergence (this is not used in the proof of Theorem 1.1). The only issue is
the possible concentration of mass on $\partial\Omega$, which is a problem
since no trace operator is available for a general elliptic operator in the
$L^{1}$ setting 555This would no be an issue if $\mathcal{B}$ was complex
elliptic.. We report the following

######  Theorem 2.18.

Let $u,\\{u_{j}\\}_{j\in\mathbf{N}}$ in $BV^{\mathcal{B}}(\Omega)$ and assume
$u_{j}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}u$. Then there exist a
measure $\lambda\in\mathcal{M}(\overline{}\Omega)^{+}$ such that

$\liminf_{j}\int_{\omega}f(\mathcal{B}u_{j})\geq\int_{\omega}f(\mathcal{B}u)$

whenever $\omega\subset\Omega$ is an open set such that
$\lambda(\partial\omega)=0$.

###  2.8. On the strong $\mathcal{B}$-quasiconvexity assumption

The following striking result fully justifies assumption equation H2, it
follows with notational changes from the case $\mathcal{B}=\nabla$, see [GK18,
Proposition 3.1] and [CK17] for a more detailed treatment.

######  Proposition 2.19.

Assume $f\colon\mathbf{R}^{N}\to\mathbf{R}$ is a continuous integrand of
linear growth, let $\Omega\subset\mathbf{R}^{n}$ be a bounded Lipschitz domain
and $g\in W^{1,1}(\mathbf{R}^{n},\mathbf{R}^{N})$. Then minimizing sequences
for the variational problem

$\inf_{u\in
W^{1,1}(\mathbf{R}^{n},\mathbf{R}^{N}),\operatorname{spt}(u-g)\subset\Omega}\int_{\Omega}f(\mathcal{B}u(x))\,dx$

are all bounded in $BV^{\mathcal{B}}(\Omega)$ if and only if $f-\ell E$ is
$\mathcal{B}$-quasiconvex at some point $y_{0}\in\mathbf{R}^{N}$, for some
$\ell>0$.

###  2.9. $L^{p}$ regularity for Legendre-Hadamard elliptic systems

A symmetric bilinear form $Q\colon\mathbf{R}^{N}\times\mathbf{R}^{N}\to R$ is
called $\mathcal{B}$-Legendre–Hadamard elliptic if there is $\lambda>0$

(2.13)
$Q(\widehat{\mathcal{B}}[\xi]v,\widehat{\mathcal{B}}[\xi]v)\geq\lambda|\xi|^{2}|v|^{2}\text{
for all }\xi\in\mathbf{R}^{n},v\in\mathbf{R}^{m},$

the positive constant $\lambda$ is called the ellipticity constant. This is
equivalent to ask that $\widetilde{Q}=Q(\beta(\cdot),\beta(\cdot))$ is
Legendre–Hadamard elliptic on $\mathbf{R}^{m}\otimes\mathbf{R}^{n}$. When
$f-\ell E$ is $C^{2}$ and $\mathcal{B}$-quasiconvex, using that $f\circ\beta$
is rank-one convex we immediately find that $f^{\prime\prime}(y)$ is
$\mathcal{B}$-Legendre–Hadamard elliptic for each $y$ for some
$\lambda=\lambda(\ell,\mathcal{B},y)$.

######  Theorem 2.20.

Let $Q\colon\mathbf{R}^{N}\times\mathbf{R}^{N}\to\mathbf{R}$ be a symmetric,
$\mathcal{B}$-Legendre–Hadamard elliptic, bilinear form with ellipticity
constant $\lambda$ and $|Q|\leq\Lambda$. Given some ball
$B=B_{R}(x_{0})\subset{\mathbf{R}^{n}}$ and some exponents $p\in(1,+\infty)$
and $q\geq 2$, the following holds.

  * (a)

For each $g\in W^{1-1/p,p}(\partial B,\mathbf{R}^{m})$ there exist a unique
solution $h\in W^{1,p}(B,\mathbf{R}^{m})$ to the system

(2.14) $\begin{cases}-\mathcal{B}^{*}\left(Q.\mathcal{B}h\right)=0&\text{ in
}B,\\\ \operatorname{Tr}_{\partial B}(h)=g&\text{ on }\partial B,\end{cases}$

where the first equation is intended in the distribution sense, and

$\|\nabla
h\|_{L^{p}(B,\mathbf{R}^{m}\otimes{\mathbf{R}^{n}})}\lesssim_{n,p,\mathcal{B},\Lambda/\lambda}[g]_{W^{1-1/p,p}(\partial
B,\mathbf{R}^{m})}.$

Furthermore, $h\in C^{\infty}(B,\mathbf{R}^{m})$ and for every $0<r<R$ and
$z\in\mathbf{R}^{m}\otimes{\mathbf{R}^{n}}$:

$\sup_{B_{r/2}(x_{0})}|\nabla
h-z|+r\,\sup_{B_{r/2}(x_{0})}|\nabla^{2}h|\lesssim_{n,\mathcal{B},\Lambda/\lambda}\fint_{B_{r}(x_{0})}|\nabla
h(x)-z|\,dx.$

  * (b)

For each $f\in L^{q}(B,\mathbf{R}^{m})$ there exist a unique solution $w\in
W^{2,q}(B,\mathbf{R}^{m})$ to the system

(2.15) $\begin{cases}-\mathcal{B}^{*}\left(Q.\mathcal{B}w\right)=f&\text{ in
}B,\\\ \operatorname{Tr}_{\partial B}(w)=0&\text{ on }\partial B,\end{cases}$

where the first equation is intended in the distribution sense, and

$\|w\|_{W^{2,q}(B,\mathbf{R}^{m}\otimes{\mathbf{R}^{n}})}\lesssim_{n,q,\mathcal{B},\Lambda/\lambda,R}\|f\|_{L^{q}(B,\mathbf{R}^{m})}.$

###### Proof.

We just show how to reduce ourselves to the case $\mathcal{B}=\nabla$, then
the theorem is essentially the $L^{p}$ regularity of Legendre-Hadamard
elliptic systems, see for example [GK18, Proposition 2.11] and the references
therein.

Suppose you have a solution $h\in W^{1,p}(B,\mathbf{R}^{m})$ of the system
(2.14), then

$0=\int_{B}Q[\mathcal{B}h(x),\mathcal{B}\varphi(x)]\,dx=\int_{B}\widetilde{Q}[\nabla
h(x),\nabla\varphi(x)]\,dx.$

Conversely, the same formula shows that if we are able to solve the system

$\begin{cases}-\operatorname{div}\big{(}\widetilde{Q}.\nabla h\big{)}=0&\text{
in }B,\\\ \operatorname{Tr}_{\partial B}(h)=g&\text{ on }\partial
B,\end{cases}$

for any Legendre-Hadamard elliptic form $\tilde{Q}$ with ellipticity constants
$\sim\lambda,\sim\Lambda$, then we have in fact found a solution of system
(2.14). An identical reasoning applies to system (2.15). ∎

###  2.10. Some auxiliary estimates for $\bm{E}$ and $\bm{f}$

In this section we collect some auxiliary estimates, we start with the area
function $E$.

######  Lemma 2.21.

For every $y,y^{\prime}\in\mathbf{R}^{N}$ and $\alpha\geq 1$:

(2.16) $\displaystyle(\sqrt{2}-1)\,\min\left\\{|y|^{2},|y|\right\\}\leq
E(y)\leq\min\left\\{|y|^{2},|y|\right\\},$ (2.17) $\displaystyle E(\alpha
y)\leq\alpha^{2}\,E(y),$ (2.18) $\displaystyle E(y+y^{\prime})\leq
2\left(E(y)+E(y^{\prime})\right).$

######  Lemma 2.22 (Lemma 2.8 in [GK18]).

For every $\mu\in C_{0}(\omega,\mathbf{R}^{N})^{*}$ we have

(2.19)
$\inf_{y\in\mathbf{R}^{N}}\int_{\omega}E(\mu-y)\leq\int_{\omega}E(\mu-(\mu)_{\omega})\leq
4\,\inf_{y\in\mathbf{R}^{N}}\int_{\omega}E(\mu-y),$

where $(\mu)_{\omega}:=\frac{\mu(\omega)}{\mathscr{L}^{n}(\omega)}$ is the
mean value of $\mu$ and $\omega\subset\mathbf{R}^{n}$ is an open set.

######  Lemma 2.23 (Lemma 2.9 in [GK18]).

For every $\mu\in C_{0}(\omega,\mathbf{R}^{N})^{*}$ we have

(2.20)
$\frac{1}{\mathscr{L}^{n}(\omega)}\int_{\omega}|\mu|\leq\sqrt{\Phi^{2}+2\Phi}\quad\text{
with }\Phi:=\frac{1}{\mathscr{L}^{n}(\omega)}\int_{\omega}E(\mu).$

In particular for $\Phi\leq 1$ we have $\fint_{\omega}|\mu|\leq\sqrt{3\Phi}$.

It’s clear that $E$ is strictly convex, the following bounds explicitly show
that the ellipticity constants are bounded below on every compact set.

######  Lemma 2.24 (Lemma 4.1 in [GK18]).

For all $y_{0},y\in\mathbf{R}^{N}$ we have

$\displaystyle
E^{\prime\prime}(y_{0})[y,y]=\Big{\\{}1+|y_{0}|^{2}-|y_{0}|^{2}\big{(}\frac{y_{0}}{|y_{0}|}\cdot\frac{y}{|y|}\big{)}\Big{\\}}\langle
y_{0}\rangle^{-3}|y|^{2};$ $\displaystyle
E(y+y_{0})-E(y_{0})-E^{\prime}(y_{0})y\geq 2^{-4}\langle
y_{0}\rangle^{-3}\,E(y).$

We turn to similar convexity properties of $f$, assuming that it satisfies
assumptions (H1), (H2) and (H3).

Given $y_{0}\in\mathbf{R}^{N}$ we define the linearized functions $E_{y_{0}}$
and $f_{y_{0}}$ by the formula:

$\displaystyle f_{y_{0}}(y)$
$\displaystyle:=f(y_{0}+y)-f(y_{0})-f^{\prime}(y_{0}).[y]$
$\displaystyle=\int_{0}^{1}(1-t)f^{\prime\prime}(y_{0}+ty)[y,y]\,dt.$

######  Lemma 2.25 (Lemma 4.2 in [GK18]).

For all $y,y_{0}\in\mathbf{R}^{N}$ with $|y_{0}|\leq\alpha$,
$v\in\mathbf{R}^{m}$ and $\xi\in\mathbf{R}^{n}$ we have

(2.21) $\displaystyle|f_{y_{0}}(y)|\lesssim_{\alpha}L\,E(y),$ (2.22)
$\displaystyle|f^{\prime}_{y_{0}}(y)|\lesssim_{\alpha}L\,\min\\{|y|,1\\},$
(2.23)
$\displaystyle|f^{\prime\prime}_{y_{0}}(0)y-f^{\prime}_{y_{0}}(y)|_{\mathbf{R}^{N}}\lesssim_{\alpha}L\,E(y),$
(2.24) $\displaystyle
f^{\prime\prime}_{y_{0}}(0)[\widehat{\mathcal{B}}[\xi]v,\widehat{\mathcal{B}}[\xi]v]\gtrsim_{\alpha}\ell|v|^{2}|\xi|^{2}.$

Furthermore, for all $\varphi\in C^{1}_{c}({\mathbf{R}^{n}},\mathbf{R}^{m})$
it holds

(2.25)
$\int_{\mathbf{R}^{n}}f_{y_{0}}(\mathcal{B}\varphi(x))\,dx\gtrsim_{\alpha}\ell\int_{\mathbf{R}^{n}}E(\mathcal{B}\varphi(x))\,dx.$

In particular the last inequality is the $\mathcal{B}$-Legendre-Hadamard
ellipticity condition introduced in (2.13).

##  3\. Proof of Theorem 1.1

We fix $u\in BV^{\mathcal{B}}(\Omega)$ satisfying the local minimality
condition

(3.1)
$\int_{\Omega}f(\mathcal{B}u)\leq\int_{\Omega}f(\mathcal{B}(u+\varphi))\text{
for all }\varphi\in BV^{\mathcal{B}}_{0}(\Omega),$

where the lagrangian $f\colon\mathbf{R}^{N}\to\mathbf{R}$ satisfies (H1),(H2)
and (H3).

###  3.1. Euler-Lagrange equation

We start with a

######  Lemma 3.1.

For all $\lambda,\lambda^{\prime}\in\Lambda$ we have
$f^{\infty}(\lambda+\lambda^{\prime})\leq
f^{\infty}(\lambda)+f^{\infty}(\lambda^{\prime}).$

###### Proof.

We exploit that $f$ is $\Lambda$-convex. Fix any $t>1$ and write the two-slope
inequality along the line $\\{\lambda+s\lambda^{\prime}:s\in\mathbf{R}\\}$

$\frac{f(t\lambda+t^{2}\lambda^{\prime})-f(t\lambda)}{t^{2}}\geq\frac{f(t(\lambda+\lambda^{\prime}))-f(t\lambda)}{t},$

we conclude sending $t\to+\infty$, and using the existence of the strong
recession function at points in $\Lambda$. ∎

The following Proposition is inspired by [GK18, Lemma 2.15].

######  Proposition 3.2 (Euler-Lagrange equation).

For every $\varphi\in BV^{\mathcal{B}}_{0}(\Omega)$ we have

(3.2)
$-\int_{\Omega}f^{\infty}(\mathcal{B}\varphi^{s})\leq\int_{\Omega}f^{\prime}\left({\mathcal{B}u^{ac}(x)}\right).\left[\mathcal{B}\varphi^{ac}(x)\right]\,dx\leq\int_{\Omega}f^{\infty}(-\mathcal{B}\varphi^{s}).$

In particular, using smooth variations we find that in $\Omega$

$\mathcal{B}^{*}\left[f^{\prime}\left({\mathcal{B}u^{ac}}\right)\right]=0\quad\text{
in the sense of distributions.}$

###### Proof.

Let $\varepsilon>0$, notice that by uniqueness of the Lebesgue decomposition
of measures we have

$(\mathcal{B}(u+\varepsilon\varphi))^{s}=(\mathcal{B}u)^{s}+\varepsilon(\mathcal{B}\varphi)^{s},$

thus we use the singular measure
$\tau:=|\mathcal{B}^{s}u|+|\mathcal{B}^{s}\varphi|$. By Besicovitch
differentiation and (2.10) we have

$\frac{d(\mathcal{B}u)^{s}}{d\tau}(x)=\frac{d(\mathcal{B}u)^{s}}{d|(\mathcal{B}u)^{s}|}(x)\cdot\frac{d|(\mathcal{B}u)^{s}|}{d\tau}(x)\in\Lambda\quad\text{
for }\tau\text{-a.e. }x\in\Omega,$

so that, since $f^{\infty}$ is positively 1-homogeneous

$\int_{\Omega}f(\mathcal{B}u)=\int_{\Omega}f((\mathcal{B}u)^{ac}(x))\,dx+\int_{\Omega}f^{\infty}\Big{(}\frac{d(\mathcal{B}u)^{s}}{d\tau}\Big{)}\,d\tau.$

The same holds for $\varphi$ and $u+\varepsilon\varphi$, so by Lemma 3.1 we
find

$\displaystyle\int_{\Omega}f^{\infty}\left(\mathcal{B}(u+\varepsilon\varphi)^{s}\right)-f^{\infty}\left(\mathcal{B}u^{s}\right)=\int_{\Omega}f^{\infty}\left(\frac{d\mathcal{B}(u+\varepsilon\varphi)^{s}}{d\tau}\right)-f^{\infty}\left(\frac{d\mathcal{B}u^{s}}{d\tau}\right)\,d\tau$
$\displaystyle\leq\int_{\Omega}f^{\infty}\left(\varepsilon\frac{d\mathcal{B}\varphi^{s}}{d\tau}\right)\,d\tau=\varepsilon\int_{\Omega}f^{\infty}\left(\mathcal{B}\varphi^{s}\right).$

Combining this inequality with the local minimality condition (3.1) we find

$\displaystyle 0$
$\displaystyle\leq\int_{\Omega}f(\mathcal{B}(u+\varepsilon\varphi))-\int_{\Omega}f(\mathcal{B}u)$
$\displaystyle=\int_{\Omega}\int_{0}^{1}f^{\prime}\left(\mathcal{B}u^{ac}+t\varepsilon\mathcal{B}\varphi^{ac}\right)[\varepsilon\mathcal{B}\varphi^{ac}]\,dt\,dx+\int_{\Omega}f^{\infty}\left(\mathcal{B}u^{s}+\varepsilon\mathcal{B}\varphi^{s}\right)-f^{\infty}\left(\mathcal{B}u^{s}\right)$
$\displaystyle\leq\varepsilon\int_{\Omega}\Big{(}\int_{0}^{1}f^{\prime}\left(\mathcal{B}u^{ac}+t\varepsilon\mathcal{B}\varphi^{ac}\right)\,dt\Big{)}[\mathcal{B}\varphi^{ac}]\,dx+\varepsilon\int_{\Omega}f^{\infty}\left(\mathcal{B}\varphi^{s}\right).$

Since $f$ is globally Lipschitz, sending $\varepsilon\to 0^{+}$ and using the
dominated convergence theorem we find

$\int_{\Omega}f^{\prime}\left({\mathcal{B}u^{ac}}\right).\left[\mathcal{B}\varphi^{ac}\right]\,\geq-\int_{\Omega}f^{\infty}\left(\mathcal{B}\varphi^{s}\right).$

Using as test function $-\varphi$ we get the opposite inequality and thus
(3.2). ∎

###  3.2. Caccioppoli inequality

The next step consists in establishing a nonlinear Caccioppoli inequality,
combining the local minimality condition, the strong
$\mathcal{B}$-quasiconvexity, and Widman’s “hole filling” trick. This
technique goes back to Evans [Eva86].

######  Proposition 3.3 (Caccioppoli inequality).

Fix a threshold $\alpha>0$. For every
$a\in\operatorname{Aff}(\mathcal{B},B_{R}(x_{0}))$ with
$B_{R}(x_{0})\Subset\Omega$ and $|\mathcal{B}a|\leq\alpha$, we have

(3.3)
$\int_{B_{R/2}(x_{0})}E(\mathcal{B}(u-a))\lesssim_{\alpha,n,\mathcal{B},L/\ell}\int_{B_{R}(x_{0})}E\left(\frac{u-a}{R}\right).$

###### Proof.

Assume by simplicity $x_{0}=0$. Set $y_{0}:=\mathcal{B}a$, $\tilde{u}:=u-a$
and $f_{y_{0}}(y):=f(y+y_{0})-f(y_{0})-f^{\prime}(y_{0}).y$.

Step 1. For every $\varphi\in BV^{\mathcal{B}}_{c}(\Omega)$ there holds

(3.4)
$\int_{\Omega}f_{y_{0}}(\mathcal{B}\tilde{u}+\mathcal{B}\varphi)\geq\int_{\Omega}f_{y_{0}}(\mathcal{B}\tilde{u}).$

Just subtract to the local minimality condition

$\int_{\Omega}f(y_{0}+\mathcal{B}\tilde{u}+\mathcal{B}\varphi)\geq\int_{\Omega}f(y_{0}+\mathcal{B}\tilde{u}),$

the identity

$\int_{\Omega}f(y_{0})+\int_{\Omega}f^{\prime}(y_{0}).[\mathcal{B}\varphi+\mathcal{B}\tilde{u}]=\int_{\Omega}f(y_{0})+\int_{\Omega}f^{\prime}(y_{0})[\mathcal{B}\tilde{u}].$

Since $y_{0}\in\mathbf{R}^{N}$ lives in the ball of radius $\alpha$ the bounds
of Lemmas 2.24 and 2.25 are available.

Just as in the classical Caccioppoli inequality we use as test function the
solution itself. We choose two balls $B_{s}\subset B_{t}$, with radii
$R/2<s<t<R$, and a cutoff function $\bm{1}_{B_{s}}\leq\eta\leq\bm{1}_{B_{t}}$
smooth and satisfying $|\nabla\eta|\leq 2/(t-s)$.

Step 2. We use (3.4) with $\varphi:=-\eta\tilde{u}$ and (2.21),

(3.5)
$\displaystyle\int_{B_{t}}f_{y_{0}}(\mathcal{B}\tilde{u})\leq\int_{B_{t}}f_{y_{0}}(\mathcal{B}\tilde{u}+\mathcal{B}\varphi)=\int_{B_{t}\setminus
B_{s}}f_{y_{0}}(\mathcal{B}\tilde{u}+\mathcal{B}\varphi)\lesssim_{\alpha}L\int_{B_{t}\setminus
B_{s}}E(\mathcal{B}v)$

where we set $v:=(1-\eta)\tilde{u}=\tilde{u}+\varphi$.

Step 3. Let $\varrho_{\varepsilon}$ be a family of standard smooth mollifiers.
Since $f_{y_{0}}$ is strongly $\mathcal{B}$-quasiconvexity at $y=0$ (remember
that $f_{y_{0}}(0)=E(0)=0$) we find

$0\leq\int_{B_{t}}f_{y_{0}}(\mathcal{B}(\eta(\tilde{u}*\varrho_{\varepsilon})))-\ell
E(\mathcal{B}(\eta(\tilde{u}*\varrho_{\varepsilon}))).$

Since $\operatorname{spt}\eta\subset B_{t}$, Lemma 2.9 implies that
$\mathcal{B}(\eta(\tilde{u}*\varrho_{\varepsilon}))\stackrel{{\scriptstyle
E}}{{\rightharpoondown}}-\mathcal{B}\varphi$ in $B_{t}$, as $\varepsilon\to
0^{+}$. Thus by area-strict continuity we find

(3.6)
$\ell\int_{B_{s}}E(\mathcal{B}\tilde{u})\leq\ell\int_{B_{t}}E(\mathcal{B}\varphi)\leq\int_{B_{t}}f_{y_{0}}(\mathcal{B}\tilde{u}-\mathcal{B}v).$

Step 4. There exist a constant
$\theta=\theta({\alpha,n,\mathcal{B},L/\ell})\in(0,1)$ such that

(3.7)
$\int_{B_{s}}E(\mathcal{B}\tilde{u})\leq\theta\int_{B_{t}}E(\mathcal{B}\tilde{u})+\theta\int_{B_{R}}E\left(\frac{\tilde{u}}{t-s}\right).$

In order to prove this we link (3.5) and (3.6), in the following way

(by 3.6) $\displaystyle\ell\int_{B_{s}}E(\mathcal{B}\tilde{u})$
$\displaystyle\leq\int_{B_{t}}f_{y_{0}}(\mathcal{B}\tilde{u}-\mathcal{B}v)$
(by 2.22)
$\displaystyle\lesssim_{\alpha}\int_{B_{t}}f_{y_{0}}(\mathcal{B}\tilde{u})+L\int_{B_{t}}E(\mathcal{B}v)$
(by 3.5) $\displaystyle\lesssim_{\alpha}L\int_{B_{t}\setminus
B_{s}}E(\mathcal{B}v)+L\int_{B_{t}\setminus B_{s}}E(\mathcal{B}v)$ (by
Leibnitz) $\displaystyle\approx L\int_{B_{t}\setminus
B_{s}}E\left((1-\eta)\mathcal{B}\tilde{u}+\widehat{\mathcal{B}}[\nabla\eta]\tilde{u}\right)$
(by 2.17 and 2.18) $\displaystyle\lesssim_{\mathcal{B}}L\int_{B_{t}\setminus
B_{s}}E(\mathcal{B}\tilde{u})+L\int_{B_{t}\setminus
B_{s}}E\left(\frac{\tilde{u}}{t-s}\right).$

In follows that there exist $c=c({\alpha,L/\ell,\mathcal{B}})$ such that

$\int_{B_{s}}E(\mathcal{B}\tilde{u})\leq c\,\int_{B_{t}\setminus
B_{s}}E(\mathcal{B}\tilde{u})+c\,\int_{B_{R}}E\left(\frac{\tilde{u}}{t-s}\right),$

next we fill the hole adding to both sides the term
$c\,\int_{B_{s}}E(\mathcal{B}\tilde{u})$. Setting $\theta:=\frac{c}{c+1}$ we
find 3.7.

Step 5. The conclusion follows iterating Lemma 3.4 below with

$\varPhi(r):=\int_{B_{r}}E(\mathcal{B}\tilde{u})\text{ and
}\varPsi(t):=\int_{B_{R}}E\left({\tilde{u}/t}\right),$

once we notice that with this choice (3.7) is assumption (3.8) and
$\varPsi(h/2)\leq 4\varPsi(h)$ because of (2.17). ∎

######  Lemma 3.4.

Let $\varPhi,\varPsi:(0,R]\to\mathbf{R}^{+}$ such that $\varPhi$ is
increasing, $\varPsi$ is decreasing, $\varPsi(h/2)\leq 4\ \varPsi(h)$ for
every $h>0$ and

(3.8) $\varPhi(s)\leq\theta\,\varPhi(t)+\theta\,\varPsi(t-s)\text{ for all
}R/2\leq s<t\leq R.$

Then we have $\varPhi(R/2)\lesssim_{\theta}\varPsi(R)$.

###  3.3. Linearisation

We fix from now on an exponent $p\in(1,(n+1)/n)$, say

(3.9) $p:=\frac{2n+1}{2n}.$

Then we have the following harmonic replacement Lemma for good spheres.

######  Proposition 3.5 (Linearisation).

Fix $\alpha>0$ and $1<q<n/(n-1)$. Then

  * •

for every $B_{R}(x_{0})\Subset\Omega$ such that $|\mathcal{B}u|(\partial
B_{R})=0$ and $\partial B_{R}$ is a good sphere for $u$ in the sense of Lemma
2.13,

  * •

for every $a\in\operatorname{Aff}(\mathcal{B},U)$ such that
$|\mathcal{B}a|\leq\alpha$, where ${B}_{R}\Subset U\subset\Omega$,

there exist a unique $h\in W^{1,p}(B_{R},\mathbf{R}^{m})$ which solves the
system

(3.10)
$\begin{cases}\mathcal{B}^{*}\left(f^{\prime\prime}(\mathcal{B}a).\mathcal{B}h\right)=0&\text{
in the sense of distributions,}\\\ \operatorname{Tr}_{\partial
B_{R}(x_{0})}h=\tilde{u}&\mathcal{H}^{n-1}\text{-a.e. on }\partial
B_{R}(x_{0}),\end{cases}$

where $\tilde{u}$ denotes the precise representative of $u$. Furthermore, $h$
satisfies

(3.11) $\displaystyle\|\nabla h-\nabla a\|_{L^{p}(B_{R},\mathbf{R}^{m})}$
$\displaystyle\lesssim_{{\alpha,p,\mathcal{B},L/\ell}}\,\left[{\tilde{u}-a}\right]_{W^{1-1/p,p}(\partial
B_{R},\mathbf{R}^{m})},$ (3.12)
$\displaystyle\int_{B_{R}}E\Big{(}\frac{u-h}{R}\Big{)}\,dx$
$\displaystyle\lesssim_{\alpha,p,q,\mathcal{B},L/\ell}R^{n(1-q)}\left(\int_{B_{R}}E(\mathcal{B}(u-a))\right)^{q}.$

###### Proof.

We set for brevity $B:=B_{R}(x_{0})$. Set $y_{0}:=\mathcal{B}a$ and recall
that by Lemma 2.25

$|f^{\prime\prime}(0).y-f^{\prime}(y)|\lesssim_{\alpha}L\,E(y)\text{ for all
}y\in\mathbf{R}^{N}.$

Step 1. Set $\tilde{u}:=u-a\in BV^{\mathcal{B}}_{loc}(U)$ and notice that, as
in the first step of Proposition 3.3, it satisfies the local minimality
condition

$\int_{U}f_{y_{0}}(\mathcal{B}\tilde{u})\leq\int_{U}f_{y_{0}}(\mathcal{B}\tilde{u}+\mathcal{B}\varphi)\text{
for all }\varphi\in BV^{\mathcal{B}}_{c}(U),$

so applying Proposition 3.2 to $\tilde{u}$ we find

(3.13)
$\mathcal{B}^{*}\left(f^{\prime}_{y_{0}}(\mathcal{B}^{ac}\tilde{u}(x))\right)=0\qquad\text{
weakly in }U.$

Step 2. We have

(3.14)
$\int_{B}f^{\prime\prime}_{y_{0}}(0)[\mathcal{B}\tilde{u},\mathcal{B}\varphi(x)]\lesssim_{\alpha,L}\int_{B}E(\mathcal{B}\tilde{u})\,|\mathcal{B}\varphi(x)|\text{
for all }\varphi\in C^{\infty}_{c}(B,\mathbf{R}^{m}).$

Indeed, using 3.13:

$\displaystyle\int_{B}f^{\prime\prime}_{y_{0}}(0)[\mathcal{B}\tilde{u},\mathcal{B}\varphi(x)]$
$\displaystyle=\int_{B}f^{\prime\prime}_{y_{0}}(0)[\mathcal{B}^{s}\tilde{u},\mathcal{B}\varphi(x)]+\int_{B}f^{\prime\prime}_{y_{0}}(0)[\mathcal{B}^{ac}\tilde{u}(x),\mathcal{B}\varphi(x)]$
$\displaystyle=\int_{B}f^{\prime\prime}_{y_{0}}(0)[\mathcal{B}^{s}\tilde{u},\mathcal{B}\varphi(x)]$
$\displaystyle\qquad\qquad+\int_{B}\left(f^{\prime\prime}_{y_{0}}(0).\mathcal{B}^{ac}\tilde{u}(x)-f^{\prime}_{y_{0}}(\mathcal{B}^{ac}\tilde{u}(x))\right)\cdot\mathcal{B}\varphi(x)\,dx$
(by 2.23)
$\displaystyle\lesssim_{\alpha,L}\int_{B}|\mathcal{B}^{s}\tilde{u}||\mathcal{B}\varphi|+\int_{B}E(\mathcal{B}^{ac}\tilde{u})|\mathcal{B}\varphi|\,dx$
$\displaystyle=\int_{B}E(\mathcal{B}\tilde{u})|\mathcal{B}\varphi|$

Step 3. Let $\tilde{h}\in W^{1,p}(B,\mathbf{R}^{m})$ be the solution
of666Existence and uniqueness are assured by Theorem 2.20, part $(a)$ and the
fact that $\tilde{u}\in W^{1-1/p,p}(\partial B,\mathbf{R}^{m})$ since $a\in
C^{\infty}(U)$.

$\begin{cases}\mathcal{B}^{*}\big{(}f^{\prime\prime}_{y_{0}}(0).\mathcal{B}\tilde{h}\big{)}=0&\text{
in }B,\\\ \tilde{h}=\tilde{u}\raisebox{-1.29167pt}{$|$}_{\partial B}&\text{ on
}\partial B.\end{cases}$

In particular we have

(3.15)
$\int_{B}f^{\prime\prime}_{y_{0}}(0)[\mathcal{B}\tilde{h},\mathcal{B}\varphi]\,dx=0\text{
for all }\varphi\in\mathscr{D}(B).$

By part (b) of Theorem 2.20 it also holds

$\|\nabla\tilde{h}\|_{L^{p}(B,\mathbf{R}^{m})}\lesssim_{\alpha,\mathcal{B}}[\tilde{u}]_{W^{1-1/p,p}(\partial
B,\mathbf{R}^{m})}.$

$W^{1,p}$-extend $\tilde{h}$ to $U$ and set $h:=\tilde{h}+a$, the last
inequality proves (3.11).

Step 4. We now want to study the size of the error
$v:=\tilde{u}-\tilde{h}=u-h\in BV^{\mathcal{B}}_{\text{{loc}}}(U)$.
Subtracting equations 3.14 and 3.15 we get that $v$ satisfies

(3.16)
$\int_{B}f^{\prime\prime}_{y_{0}}(0)[\mathcal{B}v,\mathcal{B}\varphi]\lesssim_{\alpha,L}\int_{B}E(\mathcal{B}\tilde{u})|\mathcal{B}\varphi|\text{
for all }\varphi\in C^{\infty}_{c}(B).$

Actually, for all $\beta>0$ (3.16) holds for every $\varphi\in
W^{1,\infty}_{0}\cap C^{1,\beta}(B,\mathbf{R}^{m})$. In fact both sides of
(3.16) are continuous under the convergence

$``\mathcal{B}\varphi_{k}(x)\to\mathcal{B}\varphi(x)\text{ and
}|\mathcal{B}\varphi_{k}(x)|\leq M,\quad\text{ for every }x\in B",$

and we can approximate (under this convergence) any such $\varphi$ with smooth
functions by extending it to zero outside $B$, mollifying and shrinking back
the support inside the ball.

Step 5. In this step we provide a calibration field that will quickly give the
last estimate 3.12. We shift everything back to the unit ball $\mathbf{B}$,
put for $x\in\mathbf{B}$

$V(x):=\frac{1}{R}v(x_{0}+Rx),\quad\Phi(x):=\frac{1}{R}\varphi(x_{0}+Rx),\quad\tilde{U}(x):=\frac{1}{R}\tilde{u}(x_{0}+Rx).$

Then 3.16 becomes

(3.17)
$\int_{\mathbf{B}}f^{\prime\prime}_{y_{0}}(0)[\mathcal{B}V,\mathcal{B}\Phi]\lesssim_{\alpha,L}\int_{\mathbf{B}}E(\mathcal{B}\tilde{U})|\mathcal{B}\Phi|\text{
for all }\Phi\in W^{1,\infty}_{0}\cap C^{1,\beta}(\mathbf{B},\mathbf{R}^{m}).$

We choose $\Phi=\Phi_{0}$ which is the solution of

(3.18)
$\begin{cases}-\mathcal{B}^{*}\left(f^{\prime\prime}_{y_{0}}(0).\mathcal{B}\Phi_{0}\right)=T(V)&\text{
in }\mathbf{B},\\\ \Phi_{0}=0&\text{ on }\partial\mathbf{B},\end{cases}$

here $T(v)=v$ for $|v|\leq 1$ and $T(v)=v/|v|$ for $|v|\geq 1$, is the
vectorial truncation map. Since $T(V)\in L^{\infty}(\mathbf{B})$, Theorem 2.20
(b) and Morrey’s embedding, give

(3.19)
$\|\nabla\Phi_{0}\|_{C^{\beta}(\mathbf{B})}\lesssim_{q}\|\Phi_{0}\|_{W^{2,q^{\prime}}(\mathbf{B})}\lesssim_{q,\mathcal{B}}\|T(V)\|_{L^{q^{\prime}}}\leq\left(\int_{\mathbf{B}}E(V)\right)^{1/q^{\prime}},$

for some $\beta>0$ small. We used that $q^{\prime}>n$ and that
$(\nabla\Phi_{0})_{\mathbf{B}}=0$. In particular $\nabla\Phi_{0}$ has a trace
on $\partial\mathbf{B}$, it follows that we can use as test map in 3.18 any
$\Psi\in C^{\infty}(\overline{\mathbf{B}},\mathbf{R}^{m})$ and integrate by
parts:

$\displaystyle\int_{\mathbf{B}}T(V)\cdot\Psi\,dx$
$\displaystyle=-\int_{\mathbf{B}}\mathcal{B}^{*}\left(f^{\prime\prime}_{y_{0}}(0).\mathcal{B}\Phi_{0}\right)\cdot\Psi\,dx=$
$\displaystyle=\int_{\mathbf{B}}f^{\prime\prime}_{y_{0}}(0)[\mathcal{B}\Phi_{0},\mathcal{B}\Psi]\,dx+\int_{\partial\mathbf{B}}\left(f^{\prime\prime}_{y_{0}}(0)\mathcal{B}\Phi_{0}\right)\cdot\left(\hat{\mathcal{B}}[x]\Psi\right)\,d\mathcal{H}^{n-1}(x).$

Step 6. There holds

(3.20)
$\int_{\mathbf{B}}\min\left\\{|V|,|V|^{2}\right\\}\,dx=\int_{\mathbf{B}}\tilde{f}^{\prime\prime}_{y_{0}}(0)[\mathcal{B}\Phi_{0},\mathcal{B}V].$

The idea is to put formally $\Psi=V$, but we need some care. We set
$\Psi_{\varepsilon}(x):=\frac{1}{R}(v*\varrho_{\varepsilon})(x_{0}+Rx)$ and
for $\varepsilon$ small the convolution is well-defined on $\mathbf{B}$. Now,

  * (i)

$(h*\varrho_{\varepsilon})\raisebox{-1.29167pt}{$|$}_{\partial
B}\to\operatorname{Tr}_{\partial{B}}h=u\raisebox{-1.29167pt}{$|$}_{\partial{B}}$
strongly in $L^{p}(\partial{B})$, because of the trace Theorem in $W^{1,p}$;

  * (ii)

the sequence $\|u*\varrho_{\varepsilon}\|_{W^{s,p}(\partial
B,\mathbf{R}^{m})}$ is bounded because of the definition of good sphere and
the choice of $\varrho$;

  * (iii)

$\tilde{u}*\varrho_{\varepsilon}(x)\to\tilde{u}(x)$ for $\sigma$-a.e.
$x\in\partial B$, because a good sphere is made of Lebesgue points;

  * (iv)

by the previous two points and the compact embedding $W^{s,p}(\partial
B,\mathbf{R}^{m})\hookrightarrow L^{p^{-}}(\partial B,\mathbf{R}^{m})$ we get
$\tilde{u}*\varrho_{\varepsilon}\to\tilde{u}$ in $L^{p^{-}}(\partial B)$,

  * (v)

by points $(i)$ and $(iv)$ we get
$\Psi_{\varepsilon}\raisebox{-1.29167pt}{$|$}_{\partial\mathbf{B}}\to 0$ in
$L^{p^{-}}(\partial\mathbf{B})$;

  * (vi)

$\mathcal{B}\Psi_{\varepsilon}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\mathcal{B}V$
in $C_{c}((U^{\prime}-x_{0})/R,\mathbf{R}^{N})^{*}$ and
$|\mathcal{B}V|(\partial\mathbf{B})=|\mathcal{B}^{s}u|(\partial B)=0$ by
assumption, so
$\mathcal{B}\Psi_{\varepsilon}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\mathcal{B}V$
in $C(\overline{\mathbf{B}},\mathbf{R}^{N})^{*}$,

  * (vii)

$\Psi_{\varepsilon}\to V$ in $L^{p}(\mathbf{B})$ because of general properties
of convolution.

This observation allows us to pass to the limit

$\int_{\mathbf{B}}T(V)\cdot\Psi_{\varepsilon}\,dx=\int_{\mathbf{B}}f^{\prime\prime}_{y_{0}}(0)[\mathcal{B}\Phi_{0},\mathcal{B}\Psi_{\varepsilon}]\,dx+\int_{\partial\mathbf{B}}\left(f^{\prime\prime}_{y_{0}}(0)\mathcal{B}\Phi_{0}\right)\cdot\left(\hat{\mathcal{B}}[\nu]\Psi_{\varepsilon}\right)\,d\sigma,$

obtaining

(by (vii))
$\displaystyle\lim_{\varepsilon}\int_{\mathbf{B}}T(V)\cdot\Psi_{\varepsilon}\,dx=\int_{\mathbf{B}}T(V)\cdot
V\,dx,$ (by (vi))
$\displaystyle\lim_{\varepsilon}\int_{\mathbf{B}}f^{\prime\prime}_{y_{0}}(0)[\mathcal{B}\Phi_{0},\mathcal{B}\Psi_{\varepsilon}]\,dx=\int_{\mathbf{B}}\tilde{f}^{\prime\prime}_{y_{0}}(0)[\mathcal{B}\Phi_{0},\mathcal{B}V],$
(by (v))
$\displaystyle\lim_{\varepsilon}\int_{\partial\mathbf{B}}\left(f^{\prime\prime}_{y_{0}}(0)\mathcal{B}\Phi_{0}\right)\cdot\left(\hat{\mathcal{B}}[\nu]\Psi_{\varepsilon}\right)\,d\sigma=0,$

so we get 3.20 using $T(v)\cdot v=\min\\{|v|,|v|^{2}\\}$.

Step 7. We put everything together and conclude

(by 2.16) $\displaystyle\int_{\mathbf{B}}E(V)\,dx$
$\displaystyle\lesssim\int_{\mathbf{B}}\min\left\\{|V|,|V|^{2}\right\\}\,dx$
(by 3.20)
$\displaystyle=\int_{\mathbf{B}}f^{\prime\prime}_{y_{0}}(0)[\mathcal{B}\Phi_{0},\mathcal{B}V]$
(by 3.17)
$\displaystyle\lesssim_{\alpha,L}\int_{\mathbf{B}}E(\mathcal{B}\tilde{U})|\mathcal{B}\Phi_{0}|$
$\displaystyle\lesssim_{\mathcal{B}}\|\nabla\Phi_{0}\|_{L^{\infty}(\mathbf{B})}\int_{\mathbf{B}}E(\mathcal{B}\tilde{U})$
(by 3.19)
$\displaystyle\lesssim_{q,\mathcal{B}}\left(\int_{\mathbf{B}}E(V)\right)^{1/q^{\prime}}\int_{\mathbf{B}}E(\mathcal{B}\tilde{U}),$

dividing we get

$\int_{\mathbf{B}}E(V)\,dx\lesssim_{\alpha,q,\mathcal{B},L}\left(\int_{\mathbf{B}}E(\mathcal{B}\tilde{U})\right)^{q},$

changing variables back to the ball $B_{R}(x_{0})$ we finally find 3.12. ∎

###  3.4. Excess decay and iteration

In the previous section we derived two key inequalities for a minimizer $u$ of
$\mathcal{F}$, with the previous notations they roughly look like

$\displaystyle\fint_{B_{R/2}}E(\mathcal{B}u)\stackrel{{\scriptstyle\text{Caccioppoli}}}{{\lesssim}}$
$\displaystyle\fint_{B_{R}}E\left(\frac{u}{R}\right)\,dx,$
$\displaystyle\fint_{B_{R}}E\left(\frac{u-h}{R}\right)\stackrel{{\scriptstyle\text{Linearized
Euler}}}{{\lesssim}}$
$\displaystyle\left(\fint_{B_{R}}E(\mathcal{B}u)\right)^{q},$

where $h$ is the $\mathcal{B}$-affine replacement of $u$ in $B_{R}$ and the
averages are always taken with respect to the Lebesgue measure. The idea is to
link these inequalities to obtain some nonlinear estimate that can be iterated
on smaller balls. As in other regularity results we wish to control a suitable
“excess” function: in our set-up the right definition is

(3.21)
$\mathcal{E}(x_{0},R):=\int_{B_{R}(x_{0})}E\left(\mathcal{B}u-(\mathcal{B}u)_{B_{R}(x_{0})}\right),$

where we recall that for every non-neglegible and bounded Borel set $U$ we use
the notation

$\left(\mathcal{B}u\right)_{U}:=\frac{\mathcal{B}u(U)}{\mathscr{L}^{n}(U)}.$

When the center $x_{0}$ is fixed we will also use the shorthand
$\mathcal{E}(x_{0},R)=\mathcal{E}(R)$. Then we have

######  Proposition 3.6 (Preliminary Decay).

Let $u\in BV^{\mathcal{B}}({\mathbf{R}^{n}})$ be a minimizer of
$\overline{\mathcal{F}}$. Fix a positive threshold $\alpha>0$, any exponent
$1<q<n/(n-1)$ and any ball $B_{R}(x_{0})\Subset\Omega$ such that

(3.22) $\left|\left(\mathcal{B}u\right)_{B_{R}(x_{0})}\right|\leq\alpha\ \
\text{ and }\ \
\left|\mathcal{B}u-\left(\mathcal{B}u\right)_{B_{R}(x_{0})}\right|(B_{R}(x_{0}))\leq\omega_{n}R^{n}.$

Then there is a large constant
$c_{\text{\emph{dec}}}=c_{\text{\emph{dec}}}(\alpha,q,\mathcal{B},L,\ell)$
such that

(3.23) $\mathcal{E}(\sigma R)\leq
c_{\text{\emph{dec}}}\left(\sigma^{n+2}+\frac{1}{\sigma^{2}}{\left(\frac{\mathcal{E}(R)}{\omega_{n}R^{n}}\right)}^{q-1}\right)\,\mathcal{E}(R),$

for any $\sigma\in(0,1/10)$.

###### Proof.

In this proof we shall denote by $C$ a generic constant depending only on
$\alpha,q,L/\ell,\mathcal{B},n$. In particular, we will keep track of the
dependence of the constants from $R$ and $\sigma$. Since we work at a fixed
center we will forget about $x_{0}$.

Step 1. We fix any linear map $A\in\mathbf{R}^{m}\otimes\mathbf{R}^{n}$ such
that $\mathcal{B}(Ax)=(\mathcal{B}u)_{B_{R}}$. This is possible because
$(\mathcal{B}u)_{R}\in\Lambda$.

Step 2. Choose any $a_{0}\in\ker(\mathcal{B},B_{R})\subset C^{\infty}(B_{R})$
such that

$\big{[}u-Ax-
a_{0}\big{]}_{W^{1-1/p,p}(B_{R/2})}\lesssim_{\mathcal{B}}R^{\frac{n+1}{p}-n}|\mathcal{B}u-(\mathcal{B}u)_{B_{R}}|\left(B_{R}\right),$

this is possible thanks to the local Poincaré inequality Proposition 2.12
applied to the function $u(x)-Ax$ in the ball $B_{R}$.

Step 3. Using Lemma 2.13 on $u-Ax-a_{0}$ we find a radius
$r^{*}\in\left(\frac{4}{10}R,\frac{5}{10}R\right)$ such that $\partial
B_{r^{*}}$ is a good sphere for $u-Ax-a_{0}$ (and thus for $u$),
$|\mathcal{B}u|(\partial B_{r^{*}})=0$ and

$\displaystyle\big{[}\ u-Ax-a_{0}\big{]}_{W^{1-1/p,p}(\partial B_{r^{*}})}$
$\displaystyle\lesssim_{n}{R^{-1/p}}\big{[}u-Ax-
a_{0}\big{]}_{W^{1-1/p,p}(B_{R/2})}$
$\displaystyle\lesssim_{\mathcal{B}}R^{-n(1-1/p)}|\mathcal{B}u-(\mathcal{B}u)_{R}|\left(B_{R}\right).$

Setting $\tilde{u}(x):=u(x)-a_{0}(x)-Ax\in BV^{\mathcal{B}}_{loc}(B_{R})$ we
proved

$[{\tilde{u}}]_{W^{1-1/p,p}\partial
B_{r^{*}}}\lesssim_{\mathcal{B}}R^{-n(1-1/p)}|\mathcal{B}\tilde{u}|(B_{R}).$

Step 4. Set $a(x):=a_{0}(x)+Ax\in\operatorname{Aff}(\mathcal{B},B_{R})$ and
notice that

$|\mathcal{B}a|=|\mathcal{B}(a_{0}+Ax)|=|0+(\mathcal{B}u)_{B_{R}}|\leq\alpha.$

We apply now the linearization procedure (Proposition 3.5) in the ball
$B_{r*}$ using $A$ as $\mathcal{B}$-affine map: the choice of $r^{*}$ ensures
assumption $(i)$, the last estimate ensures $(ii)$. Thus we find a map $h\in
C^{\infty}\cap W^{1,p}(B_{r*},\mathbf{R}^{m})$ which solves the system

$\begin{cases}\mathcal{B}^{*}\left(f^{\prime\prime}(\mathcal{B}a).\mathcal{B}h\right)=0&\text{
in }B_{r^{*}},\\\ \operatorname{Tr}_{\partial
B_{r^{*}}}h=u\raisebox{-1.29167pt}{$|$}_{\partial B_{r^{*}}}&\text{ on
}\partial B_{r^{*}},\end{cases}$

and satisfies

(3.24)
$\displaystyle\|\nabla\tilde{h}\|_{L^{p}(B_{r*})}\lesssim_{\alpha,\mathcal{B},L}[{\tilde{u}}]_{W^{1-1/p,p}(\partial
B_{r^{*}})}\lesssim_{\mathcal{B}}R^{-n(1-1/p)}|\mathcal{B}\tilde{u}|(B_{R}),$
(3.25)
$\displaystyle\fint_{B_{r^{*}}}E\Big{(}\frac{\tilde{u}-\tilde{h}}{r^{*}}\Big{)}\,dx\lesssim_{\alpha,\mathcal{B},L}\Big{(}\fint_{B_{r^{*}}}E(\mathcal{B}\tilde{u})\Big{)}^{q}=\Big{(}\frac{\mathcal{E}(r^{*})}{\mathscr{L}^{n}(B_{r^{*}})}\Big{)}^{q},$

where we set for brevity $\tilde{h}:=h-a$.

Step 6. Consider the affine map
$H(x):=\tilde{h}(x_{0})+\nabla\tilde{h}(x_{0})(x-x_{0})$. Then

$|\mathcal{B}(a+H)(x)|\leq\alpha+C_{0}\ \ \ \text{ for all }x\in B_{R},$

for some constant $C_{0}=C_{0}(\mathcal{B})$. We have

$\displaystyle|\mathcal{B}(a+H)|$
$\displaystyle\leq|\mathcal{B}a|+|\mathcal{B}H|\lesssim_{\mathcal{B}}|(\mathcal{B}u)_{R}|+|\nabla\tilde{h}(x_{0})|\leq\alpha+\sup_{x^{\prime}\in
B_{r^{*}/2}}|\nabla\tilde{h}(x^{\prime})|,$

but then Theorem 2.20 and (3.24) give

$\displaystyle\sup_{x^{\prime}\in B_{r^{*}/2}}|\nabla\tilde{h}(x^{\prime})|$
$\displaystyle\lesssim_{\alpha,\mathcal{B},\Lambda/\lambda}\Big{(}\fint_{B_{r^{*}}}|\nabla\tilde{h}|^{p}\,dx\Big{)}^{1/p}$
$\displaystyle\lesssim_{\alpha,\mathcal{B},\Lambda/\lambda}R^{-n/p}[u]_{W^{1-1/p,p}(\partial
B_{r^{*}})}\lesssim_{\mathcal{B}}R^{-n}|\mathcal{B}\tilde{u}|(B_{R})\lesssim
1.$

Step 7. Fix any $\sigma\in(0,1/10)$. We will prove the decay bounding
$\mathcal{E}(\sigma R)$ in terms of $\mathcal{E}(R)$, linking the Caccioppoli
inequality and the harmonic approximation.

Exploiting the quasi-minimality property of the mean (Lemma 2.22) we get

$\mathcal{E}(\sigma R)=\int_{B_{\sigma
R}}E(\mathcal{B}u-(\mathcal{B}u)_{\sigma R})\leq 4\int_{B_{\sigma
R}}E(\mathcal{B}u-\mathcal{B}(a+H))=4\int_{B_{\sigma
R}}E(\mathcal{B}(\tilde{u}-H)).$

Now we link this estimate with the Caccioppoli inequality (applied on the ball
$B_{\sigma R}$ with map $a+H\in\operatorname{Aff}_{\mathcal{B}}(B_{R})$ and
threshold $\alpha+C_{0}$) and the triangular inequality:

$\displaystyle\mathcal{E}(\sigma R)$ $\displaystyle\leq 4\int_{B_{\sigma
R}}E(\mathcal{B}(\tilde{u}-H))\lesssim_{\alpha}\int_{B_{2\sigma
R}}E\Big{(}\frac{\tilde{u}-H}{2\sigma R}\Big{)}\,dx$
$\displaystyle\lesssim_{\alpha}\underbrace{\int_{B_{2\sigma
R}}E\Big{(}\frac{\tilde{u}-\tilde{h}}{2\sigma
R}\Big{)}\,dx}_{:=\bm{I}}+\underbrace{\int_{B_{2\sigma
R}}E\Big{(}\frac{\tilde{h}-H}{2\sigma R}\Big{)}\,dx}_{:=\bm{II}}$

Where we used that $\tilde{h}$ is well defined on $B_{2\sigma R}\subset
B_{r^{*}}$. We estimate the two integrals, let us deal with the first using
Lemma 2.21 and (3.25)

$\displaystyle\bm{I}$ $\displaystyle=\int_{B_{2\sigma
R}}E\left(\frac{\tilde{u}-\tilde{h}}{r^{*}}\frac{r^{*}}{2\sigma
R}\right)\,dx\leq\left(\frac{r^{*}}{2\sigma R}\right)^{2}\int_{B_{2\sigma
R}}E\left(\frac{\tilde{u}-\tilde{h}}{r^{*}}\right)\,dx$
$\displaystyle\lesssim\frac{1}{\sigma^{2}}\int_{B_{r^{*}}}E\left(\frac{\tilde{u}-\tilde{h}}{r^{*}}\right)\,dx\lesssim_{\alpha,\mathcal{B}}\frac{{r^{*}}^{n}}{\sigma^{2}}\,{\left(\frac{\mathcal{E}(r^{*})}{\mathscr{L}^{n}(B_{r^{*}})}\right)}^{q}\lesssim\frac{\mathcal{E}(R)}{\sigma^{2}}\,{\left(\frac{\mathcal{E}(R)}{R^{n}}\right)}^{q-1}.$

For the second term we use that $\tilde{h}$ behaves like an harmonic function.
Using Taylor’s theorem and Theorem 2.20 $(b)$ we have

$\sup_{B_{2\sigma
R}}{|\tilde{h}-H|}\lesssim_{\alpha,\mathcal{B}}\sigma^{2}R^{2}\sup_{B_{r^{*}/2}}|\nabla^{2}\tilde{h}|\lesssim_{\alpha,\mathcal{B},\Lambda/\lambda}\sigma^{2}R\Big{(}\fint_{B_{r^{*}}}|\nabla\tilde{h}|^{p}\,dx\Big{)}^{1/p},$

then using (3.24) as in Step 6 we find

$\Big{(}\fint_{B_{r^{*}}}|\nabla\tilde{h}|^{p}\,dx\Big{)}^{1/p}\lesssim_{\alpha,\mathcal{B},\Lambda/\lambda}R^{-n}|\mathcal{B}\tilde{u}|(B_{R}).$

Integrating over $B_{2\sigma R}$ we get exploiting Lemma 2.21 and Jensen’s
inequality

$\displaystyle\bm{II}$ $\displaystyle=\int_{B_{2\sigma
R}}E\Big{(}\frac{\tilde{h}-H}{2\sigma
R}\Big{)}\,dx\leq\sigma^{n}R^{n}E\Big{(}C\sigma\frac{|\mathcal{B}\tilde{u}|(B_{R})}{\mathscr{L}^{n}(B_{R})}\Big{)}$
$\displaystyle\leq
C^{2}\sigma^{n+2}R^{n}\left(\frac{|\mathcal{B}\tilde{u}|(B_{R})}{\mathscr{L}^{n}(B_{R})}\right)^{2}\lesssim\sigma^{n+2}R^{n}E\Big{(}\frac{|\mathcal{B}\tilde{u}|(B_{R})}{\mathscr{L}^{n}(B_{R})}\Big{)}$
$\displaystyle\leq\sigma^{n+2}R^{n}\fint_{B_{R}}E(\mathcal{B}\tilde{u})\lesssim\sigma^{n+2}\mathcal{E}(R).$

Combining these two estimates we have (3.23). ∎

The next step is to iterate this decay to prove that there is a “critical
threshold $\varepsilon_{0}$” such that, if
$\mathcal{E}(x_{0},R)\leq\varepsilon_{0}R^{n}$, then
$\mathcal{E}(x_{0},r)\lesssim r^{n+\gamma}$ when $r\to 0^{+}$. Since this
estimate will hold also for $x$ near $x_{0}$ too, we will be able to employ
Campanato’s integral characterization of Hölder continuity. We prove a decay
of the normalized excess of a ball $B_{R}\Subset\Omega$:

$\Phi(x_{0},R):=\frac{\mathcal{E}(x_{0},R)}{\mathscr{L}^{n}(B_{R}(x_{0}))}=\fint_{B_{R}(x_{0})}E\left(\mathcal{B}u-(\mathcal{B}u)_{B_{R}(x_{0})}\right).$

######  Proposition 3.7 (Excess decay).

Let $u\in BV^{\mathcal{B}}({\mathbf{R}^{n}})$ be a minimizer of
$\overline{\mathcal{F}}$ and $\gamma\in(0,1)$ a fixed exponent. Then for every
$\alpha>0$ there exists a critical threshold
$\varepsilon_{\text{\emph{crit}}}=\varepsilon_{\text{\emph{crit}}}(\alpha,\gamma,\mathcal{B},L,\ell)>0$
such that the following implication holds: if

(3.26)
$B_{R}(x_{0})\Subset\Omega,\quad\left|(\mathcal{B}u)_{B_{R}(x_{0})}\right|\leq\alpha,\quad\Phi(x_{0},R)\leq\varepsilon_{\text{crit}},$

then

(3.27)
$\Phi(x_{0},r)\lesssim_{{\alpha,\gamma,\mathcal{B},L,\ell}}\left(\frac{r}{R}\right)^{2\gamma}\Phi(x_{0},R)\quad\text{
for every }r\in(0,R).$

###### Proof.

Fix $u,q,\alpha$ and $\gamma$ as in the hypothesis. Let us denote with

$C:=c_{\text{decay}}(\alpha+1,q:=1+1/n,\mathcal{B},L,\ell)$

the constant given by Proposition 3.6 relative to $\alpha+1$, this ensures

$\begin{cases}B_{R}(x_{0})\Subset\Omega,\left|(\mathcal{B}u)_{B_{R}(x_{0})}\right|\leq\alpha+1,\\\
\fint_{B_{R}(x_{0})}\left|\mathcal{B}u-(\mathcal{B}u)_{B_{R}(x_{0})}\right|\leq
1\end{cases}\Rightarrow\Phi(x_{0},\sigma R)\leq
C\,\left(\sigma^{n+2}+\sigma^{-2}\Phi(x_{0},R)^{1/n}\right)\Phi(x_{0},R),$

for every $0<\sigma<1/10$.
Step 1. We now fix two values $\sigma_{0}$ and $\varepsilon_{0}$, both
depending only on $\\{\alpha+1,q,\gamma,\mathcal{B},L,\ell\\}$ such that the
following holds:

(3.28)
$\begin{cases}B_{R}(x_{0})\Subset\Omega,\left|(\mathcal{B}u)_{B_{R}(x_{0})}\right|\leq\alpha+1,\\\
\Phi(x_{0},R)\leq\varepsilon_{0}\end{cases}\Rightarrow\Phi(x_{0},\sigma_{0}R)\leq\sigma_{0}^{2\gamma}\Phi(x_{0},R).$

The choices are

$\sigma_{0}:=\min\left\\{\frac{1}{20},3C^{-\frac{1}{n-2(1-\gamma)}}\right\\},\quad\varepsilon_{0}:=\min\left\\{\frac{1}{3},\left(\frac{\sigma_{0}^{2(1+\gamma)}}{3C}\right)^{n}\right\\},$

in order to reduce 3.28 to Proposition 3.6 just use Lemma 2.23:

$\Phi(x_{0},R)\leq\varepsilon_{0}\leq
1/3\Rightarrow\fint_{B_{R}(x_{0})}\left|\mathcal{B}u-(\mathcal{B}u)_{B_{R}(x_{0})}\right|\leq\sqrt{3\Phi(x_{0},R)}\leq
1.$

We remark that $\varepsilon_{\text{crit}}$ is yet to be chosen and differs
from $\varepsilon_{0}$.
Step 2. We inspect how the hypothesis of 3.28 behaves when passing from
$B_{R}$ to $B_{\sigma_{0}R}$, that is to say we notice that

(3.29)
$\begin{cases}B_{R}(x_{0})\Subset\Omega,\Phi(x_{0},R)\leq\varepsilon_{0},\\\
\left|(\mathcal{B}u)_{B_{R}(x_{0})}\right|\leq\alpha<\alpha+1\end{cases}\Rightarrow\begin{cases}B_{\sigma_{0}R}(x_{0})\Subset\Omega,\Phi(x_{0},\sigma_{0}R)\leq\sigma_{0}^{2\gamma}\Phi(x_{0},R)\leq\varepsilon_{0},\\\
\left|(\mathcal{B}u)_{B_{\sigma_{0}R}(x_{0})}\right|\leq\alpha+\sigma^{-n}_{0}\sqrt{3\Phi(x_{0},R)},\end{cases}$

the only nontrivial verification being

(triangular)
$\displaystyle\left|(\mathcal{B}u)_{B_{\sigma_{0}R}(x_{0})}\right|$
$\displaystyle\leq\left|(\mathcal{B}u)_{B_{R}(x_{0})}\right|+\left|(\mathcal{B}u)_{B_{\sigma_{0}R}(x_{0})}-(\mathcal{B}u)_{B_{R}(x_{0})}\right|$
(monotonicity)
$\displaystyle\leq\alpha+\sigma^{-n}_{0}\fint_{B_{R}(x_{0})}\left|\mathcal{B}u-(\mathcal{B}u)_{B_{R}(x_{0})}\right|$
( by Lemma 2.23)
$\displaystyle\leq\alpha+\sigma^{-n}_{0}\sqrt{3\Phi(x_{0},R)}.$

In order to apply again 3.28 on $B_{\sigma_{0}R}(x_{0})$ we need an additional
smallness condition on the initial excess

(3.30) $\sigma^{-n}_{0}\sqrt{3\Phi(x_{0},R)}\leq 1.$

This is of course possible replacing $\varepsilon_{0}$ with some smaller
$\varepsilon_{1}\leq\varepsilon_{0}$. Nevertheless we want to iterate this
argument infinite times so we must ensure that $\varepsilon_{k}$ does not go
down to $0$. Luckily this can be done, we think it is more clear to provide
the next step of the iteration instead of a formal induction.
Step 3. Suppose that 3.30 is true, then we apply 3.28 on
$B_{\sigma_{0}R}(x_{0})$ and get

$\displaystyle\begin{cases}B_{\sigma_{0}R}(x_{0})\Subset\Omega,\Phi(x_{0},\sigma_{0}R)\leq\sigma_{0}^{2\gamma}\Phi(x_{0},R)\leq\varepsilon_{0},\\\
\left|(\mathcal{B}u)_{B_{\sigma_{0}R}(x_{0})}\right|\leq\alpha+\sigma^{-n}_{0}\sqrt{3\Phi(x_{0},R)}\leq\alpha+1,\end{cases}$
$\displaystyle\Rightarrow$
$\displaystyle\begin{cases}B_{\sigma_{0}^{2}R}(x_{0})\Subset\Omega,\Phi(x_{0},\sigma_{0}^{2}R)\leq\sigma_{0}^{2\gamma}\Phi(x_{0},\sigma_{0}R),\\\
\left|(\mathcal{B}u)_{B_{\sigma_{0}^{2}R}(x_{0})}\right|\leq\alpha+\sigma^{-n}_{0}\sqrt{3\Phi(x_{0},R)}+\sigma^{-n}_{0}\sqrt{3\Phi(x_{0},\sigma_{0}R)},\end{cases}$
$\displaystyle\Rightarrow$
$\displaystyle\begin{cases}B_{\sigma_{0}^{2}R}(x_{0})\Subset\Omega,\Phi(x_{0},\sigma_{0}^{2}R)\leq\sigma_{0}^{4\gamma}\Phi(x_{0},R),\\\
\left|(\mathcal{B}u)_{B_{\sigma_{0}^{2}R}(x_{0})}\right|\leq\alpha+(1+\sigma_{0}^{\gamma})\sigma^{-n}_{0}\sqrt{3\Phi(x_{0},R)},\end{cases}$

In order to apply 3.28 on $B_{\sigma_{0}^{2}R}(x_{0})$ we need a further
smallness assumption on the excess

$(1+\sigma_{0}^{\gamma})\sigma^{-n}_{0}\sqrt{3\Phi(x_{0},R)}\leq 1,$

this requirement is slightly stronger than 3.30. Going on in this fashion one
easily devise the pattern: the condition

$\left(\sum_{k\in\mathbf{N}}\sigma_{0}^{k\gamma}\right)\sigma^{-n}_{0}\sqrt{3\Phi(x_{0},R)}\leq
1,$

is enough for infinitely many steps and, since the series converges, we can
set

$\varepsilon_{\text{crit}}:=\min\left\\{\varepsilon_{0};\frac{1}{3}\sigma_{0}^{2n}\left(\sum_{k\in\mathbf{N}}\sigma_{0}^{k\gamma}\right)^{-2}\right\\}.$

With this choice we have

(3.31)
$\begin{cases}B_{R}(x_{0})\Subset\Omega,\left|(\mathcal{B}u)_{B_{R}(x_{0})}\right|\leq\alpha,\\\
\Phi(x_{0},R)\leq\varepsilon_{\text{crit}}\end{cases}\Rightarrow\Phi(x_{0},\sigma_{0}^{k}R)\leq\sigma_{0}^{2\gamma
k}\Phi(x_{0},R),\quad k\in\mathbf{N},$

and this gives the Hölder estimate 3.27 by discrete interpolation. ∎

We can finally prove

######  Theorem 3.8 (Main Theorem).

Let $u\in BV^{\mathcal{B}}({\mathbf{R}^{n}})$ be a minimizer of
$\overline{\mathcal{F}}$ and $\gamma\in(0,1)$ some fixed exponent. Then for
every $\alpha>0$ there exists a critical threshold
$\varepsilon=\varepsilon(\alpha,\gamma,\mathcal{B},L,\ell)>0$ such that the
following implication holds: if

(3.32)
$B_{R}(x_{0})\Subset\Omega,\quad\left|(\mathcal{B}u)_{B_{R}(x_{0})}\right|\leq\alpha,\quad\Phi(x_{0},R)\leq\varepsilon,$

then $\mathcal{B}u\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}B_{R/2}(x_{0})\ll\mathscr{L}^{n}$,
$\mathcal{B}u\in C^{0,\gamma}(B_{R/2}(x_{0}))$ and

(3.33)
$\left[\mathcal{B}u\right]_{C^{0,\gamma}(B_{R/2}(x_{0}))}\lesssim_{\alpha,\mathcal{B},L,\ell}R^{-\gamma}\sqrt{\Phi(x_{0},R)}.$

###### Proof.

We start with two simple estimates that relates the oscillation of nested
balls, whose centers do not necessarily agree. Given
$B_{R}(x_{0})\Subset\Omega$ there holds

$\displaystyle\Phi(x,R/2)\leq 2^{n+2}\,\Phi(x_{0},R)$ $\displaystyle\text{ for
every }x\in B_{R/2}(x_{0}),$
$\displaystyle\left|(\mathcal{B}u)_{B_{R/2}(x)}\right|\leq\left|(\mathcal{B}u)_{B_{R}(x_{0})}\right|+2^{n}\sqrt{3\Phi(x_{0},R)}$
$\displaystyle\text{ for every }x\in B_{R/2}(x_{0}).$

These are simply proven by Lemmas 2.22, 2.23 and the quasi-triangular
inequality. So defining $\varepsilon$ as

(3.34) $\varepsilon:=\min\left\\{\frac{1}{3\cdot
2^{n}};\frac{\varepsilon_{\text{crit}}(\alpha+1,\gamma,\mathcal{B},L,\ell)}{2^{n+2}}\right\\},$

these two estimates gives us

$\begin{cases}B_{R}(x_{0})\Subset\Omega,\Phi(x_{0},R)\leq\varepsilon,\\\
\left|(\mathcal{B}u)_{B_{R}(x_{0})}\right|\leq\alpha\end{cases}\Rightarrow\begin{cases}B_{R/2}(x)\Subset\Omega,\Phi(x,R/2)\leq\varepsilon_{\text{crit}}(\alpha+1,\gamma,\mathcal{B},L,\ell),\\\
\left|(\mathcal{B}u)_{B_{R/2}(x)}\right|\leq\alpha+1,\end{cases}$

so that we can apply Proposition 3.7 on the ball $B_{R/2}(x)$ with exponent
$\gamma$ and threshold $\alpha+1$ and find:

$\Phi(x,r)\lesssim_{{\alpha,\gamma,\mathcal{B},L,\ell}}\left(\frac{r}{R}\right)^{2\gamma}\,\Phi(x_{0},R)\quad\text{
for every }r\in(0,R/2),$

but the constant involved does not depend on the center $x\in B_{R/2}(x_{0})$
so there holds

$\Phi(x,r)\lesssim_{{\alpha,\gamma,\mathcal{B},L,\ell}}\left(\frac{r}{R}\right)^{2\gamma}\,\Phi(x_{0},R)\quad\text{
for every }x\in B_{R/2}(x_{0}),r\in(0,R/2).$

This in particular gives us that for every $x\in B_{R/2}(x_{0})$ we have

$\limsup_{r\to
0^{+}}\frac{|\mathcal{B}^{s}u|(\overline{B}_{r}(x))}{r^{n}}\lesssim\limsup_{r\to
0^{+}}\left(\frac{r}{R}\right)^{2\gamma}\,\Phi(x_{0},R)=0,$

so by standard results about upper densities of measures (see Theorem 2.56 in
[AFP00]) we get $\mathcal{B}u\ll\mathscr{L}^{n}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}B_{R/2}(x_{0})$, so we can write
$\mathcal{B}u=\mathcal{B}u(x)\in L^{1}(B_{R/2}(x_{0}),\mathbf{R}^{N})$. We now
show that $\mathcal{B}u$ belongs to the Campanato space
$\mathcal{L}^{n+\gamma}(B_{R/2}(x_{0}))$, in fact by Lemma 2.23 we have

$\displaystyle\left(\fint_{B_{r}(x)}\left|\mathcal{B}u(x^{\prime})-(\mathcal{B}u)_{B_{r}(x)}\right|\,dx^{\prime}\right)^{2}$
$\displaystyle\leq\Phi(x,r)^{2}+2\Phi(x,r)$
$\displaystyle\lesssim_{{\alpha,\gamma,\mathcal{B},L,\ell}}\left(\frac{r}{R}\right)^{4\gamma}\Phi(x_{0},R)^{2}+\left(\frac{r}{R}\right)^{2\gamma}\Phi(x_{0},R)$
$\displaystyle\lesssim\left(\frac{r}{R}\right)^{2\gamma}\Phi(x_{0},R).$

We conclude that

$\sup_{x\in
B_{R/2}(x_{0}),0<r<R/2}r^{-n-\gamma}\int_{B_{r}(x)}\left|\mathcal{B}u(x^{\prime})-(\mathcal{B}u)_{B_{r}(x)}\right|\,dx^{\prime}\lesssim_{{\alpha,\gamma,\mathcal{B},L,\ell}}\frac{\sqrt{\Phi(x_{0},R)}}{R^{\gamma}},$

so by the integral characterization of Hölder continuity we get 3.33:

$\left[\mathcal{B}u\right]_{C^{0,\gamma}(B_{R/2}(x_{0}))}\sim_{n,\gamma}\left[\mathcal{B}u\right]_{\mathcal{L}^{1,n+\gamma}(B_{R/2}(x_{0}))}\lesssim_{\alpha,\mathcal{B},L,\ell}R^{-\gamma}\sqrt{\Phi(x_{0},R)}.$

∎

This is the key result, using the boundedness of Calderón-Zygmund kernels
between Hölder spaces one easily gets that $u\in
C^{1,\gamma}(B_{R/4}(x_{0}),\mathbf{R}^{m})$ for every $\gamma\in(0,1)$. Then
exploiting the finite-difference method one can prove full $C^{2,\gamma}$
regularity, we do not enter in too much detail here, a quick account is given
in Theorem 4.9 in [GK18].

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