# On the principal frequency of non-homogeneous membranes

V. Gol’dshtein, V. Pchelintsev<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We obtained estimates for first eigenvalues of the Dirichlet boundary value
problem for elliptic operators in divergence form (i.e. for the principal
frequency of non-homogeneous membranes) in bounded domains
$\Omega\subset\mathbb{C}$ satisfying quasihyperbolic boundary conditions. The
suggested method is based on the quasiconformal composition operators on
Sobolev spaces and their applications to constant estimates in the
corresponding Sobolev-Poincaré inequalities. We also prove a variant of the
Rayleigh-Faber-Khran inequality for a special case of these elliptic
operators.

00footnotetext: Key words and phrases: Elliptic equations, Sobolev spaces,
quasiconformal mappings.00footnotetext: 2010 Mathematics Subject
Classification: 35P15, 46E35, 30C65.

##  1\. Introduction

In this paper we give applications of the theory of quasiconformal mappings to
the Dirichlet eigenvalue problem for two-dimensional elliptic operators in
divergence form

(1.1) $L_{A}f(z)=-\textrm{div}[A(z)\nabla f(z)],\quad z=(x,y)\in\Omega,\quad
f(x,y)=0\,\,\text{on}\,\,\partial\Omega,$

in bounded domains $\Omega\subset\mathbb{C}$ satisfying quasihyperbolic
boundary conditions [19, 20]. We assume that $A\in M^{2\times 2}(\Omega)$,
where $M^{2\times 2}(\Omega)$ is the class of all $2\times 2$ symmetric matrix
functions $A(z)=\left\\{a_{kl}(z)\right\\}$, $\textrm{det}A=1$ a. e. in
$\Omega$, with measurable entries satisfying to the uniform ellipticity
condition

(1.2) $\frac{1}{K}|\xi|^{2}\leq\left\langle A(w)\xi,\xi\right\rangle\leq
K|\xi|^{2}\,\,\,\text{a.\leavevmode\nobreak\ e. in}\,\,\,\Omega,$

for every $\xi\in\mathbb{C}$, where $1\leq K<\infty$. Such elliptic operators
in divergence form arise in various problems of mathematical physics (see, for
example, [3]).

Under these conditions the matrix $A$ induces a quasiconformal homeomorphism
$\varphi_{A}:\Omega\to\widetilde{\Omega}$ which we call the $A$-quasiconformal
mapping [11]. This allow us to reduce (by quasiconformal change of variable)
this elliptic operator in divergence form to the Laplace operator.

A domain $\Omega$ satisfies the $\gamma$-quasihyperbolic boundary condition
with some $\gamma>0$ if the growth condition on the quasihyperbolic metric

$k_{\Omega}(x_{0},x)\leq\frac{1}{\gamma}\log\frac{\operatorname{dist}(x_{0},\partial\Omega)}{\operatorname{dist}(x,\partial\Omega)}+C_{0}$

satisfied for all $x\in\Omega$, where $x_{0}\in\Omega$ is a fixed base point
and $C_{0}=C_{0}(x_{0})<\infty$, [7, 17]. Here

$k_{\Omega}(x_{1},x_{2}):=\inf\int\limits_{\gamma}\frac{ds}{\operatorname{dist}(x,\partial\Omega)},$

where the infimum is taken over all rectifiable curves $\gamma$ joining
$x_{1}$ and $x_{2}$.

This class of domains includes, in particular, domains with Lipschitz
boundary, some domains with Hölder singularities, and domains of snowflakes
type [23].

A function $f\in W_{0}^{1,2}(\Omega,A)$ is a solution to the generalized
spectral problem for the elliptic operator in divergence form $L_{A}f(z)$ with
the Dirichlet boundary condition if

$\iint\limits_{\Omega}\langle A(z)\nabla
f(z),\nabla\overline{g(z)}\rangle\leavevmode\nobreak\
dxdy=\lambda\iint\limits_{\Omega}f(z)\overline{g(z)}\leavevmode\nobreak\
dxdy,\,\,\forall g\in W^{1,2}_{0}(\Omega,A).$

It is known [16, 22] that in a bounded domain $\Omega\subset\mathbb{C}$ the
operator $L_{A}f(z)$ with the Dirichlet boundary condition has discrete
spectrum represented as the non-decreasing sequence

$0<\lambda_{1}(A,\Omega)\leq\lambda_{2}(A,\Omega)\leq\ldots\leq\lambda_{n}(A,\Omega)\leq\ldots,$

where each eigenvalue is repeated as many time as its multiplicity. By the
min-max principle, the first eigenvalue $\lambda_{1}(A,\Omega)$ is defined by

$\lambda_{1}(A,\Omega)=\inf_{f\in
W_{0}^{1,2}(\Omega,A)\setminus\\{0\\}}\frac{\|f\mid
L^{1,2}_{A}(\Omega)\|^{2}}{\|f\mid L^{2}(\Omega)\|^{2}}\,.$

The lower estimates for the first eigenvalues of the Laplace operator with the
Dirichlet boundary condition in a bounded domain are connected by the
Rayleigh-Faber-Khran inequality [6, 21] which means that the first Dirichlet
eigenvalue in a bounded domain $\Omega$ is not less than the corresponding
Dirichlet eigenvalue in the disc of the same area $\Omega^{\ast}$ with
$R_{\ast}$ as its radius, i.e.,

$\lambda_{1}(I,\Omega):=\lambda_{1}(\Omega)\geq\lambda_{1}(\Omega^{\ast})=\frac{{j_{0,1}^{2}}}{R^{2}_{\ast}},$

where $j_{0,1}\approx 2.4048$ is the first positive zero of the Bessel
function $J_{0}$. This inequality was improved by the method based on the
capacity theory [22].

Unfortunately, for the first eigenvalue $\lambda_{1}(A,\Omega)$ Rayleigh-
Faber-Khran type inequality has not been proven. However, lower estimates for
the first eigenvalues the operator $L_{A}f(z)$ with the Dirichlet boundary
condition in bounded domains can be obtained easily by using the Rayleigh-
Faber-Krahn inequality and the uniform ellipticity condition (1.2): Let
$\Omega\subset\mathbb{C}$ be a bounded domain such that
$|\Omega|=|\Omega^{\ast}|$ and $K$ is the ellipticity constant of the matrix
$A$. Then

(1.3)
$\lambda_{1}(A,\Omega)\geq\frac{\lambda_{1}(\Omega)}{K}\geq\frac{\lambda_{1}(\Omega^{\ast})}{K}=\frac{{j_{0,1}^{2}}}{KR^{2}_{\ast}}.$

In this paper we obtain lower estimates for the first Dirichlet eigenvalues of
the divergence form elliptic operators $L_{A}f(z)$ in bounded domains with
some quasihyperbolic boundary condition. We call such domains as
$\beta$-regular domains for some $\beta\in(1,\infty]$. The class of all
$\beta$-regular domains coincides with the class of all domains with
quasihyperbolic boundary conditions.

Our machinery is based on connections between $A$-quasiconformal mappings [3,
4, 15] and composition operators on Sobolev spaces [11].

One of the main results of the article states the following estimate for
$\infty$-regular domains:  If a simply connected bounded domain
$\Omega\subset\mathbb{C}$ satisfies to the quasihyperbolic boundary condition,
then

(1.4)
$\lambda_{1}(A,\Omega)\geq\frac{\lambda_{1}(\widetilde{\Omega})}{\|J_{\varphi_{A}^{-1}}\mid
L^{\infty}(\widetilde{\Omega})\|},$

where $\lambda_{1}(\widetilde{\Omega})$ is the first Dirichlet eigenvalue of
the Laplace operator and $J_{\varphi_{A}^{-1}}$ is a Jacobian of the inverse
mapping to the $A$-quasiconformal mapping
$\varphi_{A}:\Omega\to\widetilde{\Omega}$.

A detailed discussion about $\beta$-regular domains can be found in Section 3.

Note that if $\widetilde{\Omega}=\Omega^{\ast}$ and
$\|J_{\varphi_{A}^{-1}}\mid L^{\infty}(\Omega^{\ast})\|<K$ then estimate (1.4)
is better than estimates (1.3). For example, this condition is satisfied for
measure preserving $A$-quasiconformal mappings
$\varphi_{A}:\Omega\to\Omega^{\ast}$ ($|J(z,\varphi_{A})|=1$ a.e. in
$\Omega$). Some examples can be found at the end of this paper.

Taking into account the domain monotonicity property for the Dirichlet
eigenvalues of the operator $L_{A}f(z)$ (see, for example, [16]) and estimate
(1.4) we obtain estimates for variations of the first Dirichlet eigenvalues of
the operator $L_{A}f(z)$ under quasiconformal deformations of the domain.
Namely: Let $\Omega\subset\mathbb{C}$ be a bounded $\infty$-regular domain. We
assume that $\varphi_{A}(\Omega):=\widetilde{\Omega}\supset\Omega$, then

$\lambda_{1}(A,\Omega)-\lambda_{1}(\widetilde{\Omega})\geq\frac{1-\|J_{\varphi_{A}^{-1}}\mid
L^{\infty}(\widetilde{\Omega})\|}{\|J_{\varphi_{A}^{-1}}\mid
L^{\infty}(\widetilde{\Omega})\|}\lambda_{1}(\widetilde{\Omega}),$

where $\lambda_{1}(\widetilde{\Omega})$ is the first Dirichlet eigenvalue of
the Laplace operator and $J_{\varphi_{A}^{-1}}$ is a Jacobian of the inverse
mapping to the $A$-quasiconformal mapping
$\varphi_{A}:\Omega\to\widetilde{\Omega}$.

In the case of the measure preserving $A$-quasiconformal mappings
$\varphi_{A}:\Omega\to\widetilde{\Omega}$ we prove Rayleigh-Faber-Khran type
inequality for the operator $L_{A}f(z)$: Let $\Omega\subset\mathbb{C}$ be a
simply connected bounded domain such that there exists a measure preserving
A-quasiconformal mapping $\varphi_{A}:\Omega\to\widetilde{\Omega}$.
$|\Omega|=|\Omega^{\ast}|$. Then

$\lambda_{1}(A,\Omega)\geq\lambda_{1}(\Omega^{\ast})=\frac{{j_{0,1}^{2}}}{R^{2}_{\ast}}.$

Here $\Omega^{\ast}$ is the disc of the radius $R_{\ast}$ such that
$|\Omega|=|\Omega^{\ast}|$. and $j_{0,1}\approx 2.4048$ is the first positive
zero of the Bessel function $J_{0}$.

##  2\. Sobolev spaces and $A$-quasiconformal mappings

Let $E\subset\mathbb{C}$ be a measurable set on the complex plane and
$h:E\to\mathbb{R}$ be a positive almost everywhere (a.e.) locally integrable
function, i.e. a weight. The weighted Lebesgue space $L^{p}(E,h)$, $1\leq
p<\infty$, is the space of all locally integrable functions endowed with the
following norm

$\|f\,|\,L^{p}(E,h)\|=\left(\iint\limits_{E}|f(z)|^{p}h(z)\,dxdy\right)^{\frac{1}{p}}<\infty.$

The two-weighted Sobolev space $W^{1,p}(\Omega,h,1)$, $1\leq p<\infty$, is
defined as the normed space of all locally integrable weakly differentiable
functions $f:\Omega\to\mathbb{R}$ endowed with the following norm:

$\|f\mid W^{1,p}(\Omega,h,1)\|=\|f\,|\,L^{p}(\Omega,h)\|+\|\nabla f\mid
L^{p}(\Omega)\|.$

In the case $h=1$ this weighted Sobolev space coincides with the classical
Sobolev space $W^{1,p}(\Omega)$. The seminormed Sobolev space
$L^{1,p}(\Omega)$, $1\leq p<\infty$, is the space of all locally integrable
weakly differentiable functions $f:\Omega\to\mathbb{R}$ endowed with the
following seminorm:

$\|f\mid L^{1,p}(\Omega)\|=\|\nabla f\mid L^{p}(\Omega)\|,\,\,1\leq p<\infty.$

We also need a weighted seminormed Sobolev space $L^{1,2}_{A}(\Omega)$
(associated with the matrix $A$), defined as the space of all locally
integrable weakly differentiable functions $f:\Omega\to\mathbb{R}$ with the
finite seminorm given by:

$\|f\mid L^{1,2}_{A}(\Omega)\|=\left(\iint\limits_{\Omega}\left\langle
A(z)\nabla f(z),\nabla f(z)\right\rangle\,dxdy\right)^{\frac{1}{2}}.$

The corresponding Sobolev space $W^{1,2}(\Omega,A)$ is defined as the normed
space of all locally integrable weakly differentiable functions
$f:\Omega\to\mathbb{R}$ endowed with the following norm:

$\|f\mid W^{1,2}(\Omega,A)\|=\|f\,|\,L^{2}(\Omega)\|+\|f\mid
L^{1,2}_{A}(\Omega)\|.$

The Sobolev space $W^{1,2}_{0}(\Omega,A)$ is the closure in the
$W^{1,2}(\Omega,A)$-norm of the space $C^{\infty}_{0}(\Omega)$.

We consider the Sobolev spaces as Banach spaces of equivalence classes of
functions up to a set of $p$-capacity zero [22].

Recall that a homeomorphism $\varphi:\Omega\to\widetilde{\Omega}$,
$\Omega,\,\widetilde{\Omega}\subset\mathbb{C}$, is called a $Q$-quasiconformal
mapping if $\varphi\in W^{1,2}_{\operatorname{loc}}({\Omega})$ and there
exists a constant $1\leq Q<\infty$ such that

$|D\varphi(z)|^{2}\leq Q|J(z,\varphi)|\,\,\text{for almost
all}\,\,z\in\Omega.$

Note that quasiconformal mappings have a finite distortion and possesses the
Luzin $N$-property (i.e. a image of any set of measure zero has measure zero)
[26].

If $\varphi:\Omega\to\widetilde{\Omega}$ is a $Q$-quasiconformal mapping then
$\varphi$ is differentiable almost everywhere in $\Omega$ and

$|J(z,\varphi)|=J_{\varphi}(z):=\lim\limits_{r\to
0}\frac{|\varphi(B(z,r))|}{|B(z,r)|}\,\,\text{for almost all}\,\,z\in\Omega.$

Now we give a construction of $A$-quasiconformal mappings connected with the
matrix $A$.

Recall that matrix functions $A(z)=\left\\{a_{kl}(z)\right\\}$ with measurable
entries $a_{kl}(z)$ belongs to a class $M^{2\times 2}(\Omega)$ of all $2\times
2$ symmetric matrix functions that satisfy to an additional condition
$\textrm{det}A=1$ a.e. and to the uniform ellipticity condition:

(2.1) $\frac{1}{K}|\xi|^{2}\leq\left\langle A(z)\xi,\xi\right\rangle\leq
K|\xi|^{2}\,\,\,\text{a.e. in}\,\,\,\Omega,$

for every $\xi\in\mathbb{C}$ and for some $1\leq K<\infty$. The basic idea is
that every positive quadratic form

$ds^{2}=a_{11}(x,y)dx^{2}+2a_{12}(x,y)dxdy+a_{22}(x,y)dy^{2}$

defined in a planar domain $\Omega$ can be reduced, by means of a
quasiconformal change of variables, to the canonical form

$ds^{2}=\Lambda(du^{2}+dv^{2}),\,\,\Lambda\neq 0,\,\,\text{a.e.
in}\,\,\widetilde{\Omega},$

given that $a_{11}a_{22}-a^{2}_{12}\geq\kappa_{0}>0$, $a_{11}>0$, almost
everywhere in $\Omega$ [1, 4].

By [4] any matrix $A$ of the type under discussion induces a quasiconformal
homeomorphism as a solution to the corresponding Beltrami equation. The
detailed procedure is described below

Let $\xi(z)=\operatorname{Re}\varphi(z)$ be a real part of a quasiconformal
mapping $\varphi(z)=\xi(z)+i\eta(z)$, which satisfies to the Beltrami
equation:

(2.2) $\varphi_{\overline{z}}(z)=\mu(z)\varphi_{z}(z),\,\,\,\text{a.e.
in}\,\,\,\Omega,$

where

$\varphi_{z}=\frac{1}{2}\left(\frac{\partial\varphi}{\partial
x}-i\frac{\partial\varphi}{\partial
y}\right)\quad\text{and}\quad\varphi_{\overline{z}}=\frac{1}{2}\left(\frac{\partial\varphi}{\partial
x}+i\frac{\partial\varphi}{\partial y}\right),$

with the complex dilatation $\mu(z)$ given by

(2.3) $\mu(z)=\frac{a_{22}(z)-a_{11}(z)-2ia_{12}(z)}{\det(I+A(z))},\quad
I=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}.$

We call this quasiconformal mapping (with the complex dilatation $\mu$ defined
by (2.3)) as an $A$-quasiconformal mapping and we will use the notation
$\varphi_{A}$ for this quasiconformal mapping.

Note that the uniform ellipticity condition (2.1) can be written as

(2.4) $|\mu(z)|\leq\frac{K-1}{K+1},\,\,\,\text{a.e. in}\,\,\,\Omega.$

Conversely from (2.3) (see, for example, [3], p. 412) one can recover the
matrix $A$ :

(2.5)
$A(z)=\begin{pmatrix}\frac{|1-\mu|^{2}}{1-|\mu|^{2}}&\frac{-2\operatorname{Im}\mu}{1-|\mu|^{2}}\\\
\frac{-2\operatorname{Im}\mu}{1-|\mu|^{2}}&\frac{|1+\mu|^{2}}{1-|\mu|^{2}}\end{pmatrix},\,\,\,\text{a.e.
in}\,\,\,\Omega.$

Thus, for any $A\in M^{2\times 2}(\Omega)$ by (2.4) can be produced the
complex dilatation $\mu(z)$, for which, in turn, the Beltrami equation (2.2)
induces an $A$-quasiconformal homeomorphism
$\varphi:\Omega\to\widetilde{\Omega}$ as its solution (by the Riemann
measurable mapping theorem (see, for example, [1])). Let us briefly say that
$A$ and $\varphi_{A}$ are agreed.

Therefore with the given $A$-divergent form elliptic operator defined in a
domain $\Omega\subset\mathbb{C}$ can be assochiated the $A$-quasiconformal
mapping $\varphi_{A}:\Omega\to\widetilde{\Omega}$ with the quasiconformality
coefficient

$Q_{A}=\frac{1+\|\mu\mid L^{\infty}(\Omega)\|}{1-\|\mu\mid
L^{\infty}(\Omega)\|},$

where $\mu$ defined by (2.3).

From the estimate $|\mu(z)|\leq\frac{K-1}{K+1}$ immediately follows that
$Q_{A}\leq K$.

Note that the inverse mapping to the $A$-quasiconformal mapping
$\varphi_{A}:\Omega\to\widetilde{\Omega}$ is the $A^{-1}$-quasiconformal
mapping [11].

In [11] the relationship between composition operators on Sobolev spaces and
$A$-quasiconformal mappings was studied and the following theorem was proved.

######  Theorem 2.1.

Let $\Omega,\widetilde{\Omega}$ be domains in $\mathbb{C}$. Then a
homeomorphism $\varphi_{A}:\Omega\to\widetilde{\Omega}$ is an
$A$-quasiconformal mapping if and only if $\varphi$ induces, by the
composition rule $\varphi^{*}(f)=f\circ\varphi$, an isometry of Sobolev spaces
$L^{1,2}_{A}(\Omega)$ and $L^{1,2}(\widetilde{\Omega})$ i.e.

$\|\varphi_{A}^{*}(f)\,|\,L^{1,2}_{A}(\Omega)\|=\|f\,|\,L^{1,2}(\widetilde{\Omega})\|$

for any $f\in L^{1,2}(\widetilde{\Omega})$.

This theorem generalizes the well known property of conformal mappings
generate the isometry of uniform Sobolev spaces $L^{1}_{2}(\Omega)$ and
$L^{1}_{2}(\widetilde{\Omega})$ (see, for example, [5]). It is also refines
(in the case $n=2$) the functional characterization of quasiconformal mappings
in the terms of isomorphisms of uniform Sobolev spaces [25].

##  3\. Estimate of the constant in Sobolev-Poincaré inequality

In [12] was proved the following weighted Sobolev-Poincaré inequality for a
bounded domain ${\Omega}\subset\mathbb{C}$. We denote by
$h(z)=|J(z,\varphi_{A})|$ the quasihyperbolic weight defined by an
A-quasiconformal mapping $\varphi_{A}:\Omega\to\widetilde{\Omega}$.

######  Theorem 3.1.

Let $A$ belongs to a class $M^{2\times 2}(\Omega)$ and $\Omega$ be a bounded
simply connected planar domain. Then for any function $f\in
W^{1,2}_{0}(\Omega,A)$ the following weighted Sobolev-Poincaré inequality

$\left(\iint\limits_{\Omega}|f(z)|^{r}h(z)dxdy\right)^{\frac{1}{r}}\leq
C_{r,2}(h,A,\Omega)\left(\iint\limits_{\Omega}\left\langle A(z)\nabla
f(z),\nabla f(z)\right\rangle dxdy\right)^{\frac{1}{2}}$

holds for any $r\geq 2$ with the constant
$C_{r,2}(h,A,\Omega)=C_{r,2}(\widetilde{\Omega})$.

Here $C_{r,2}(\widetilde{\Omega})$ is the best constant in the (non-weight)
Sobolev-Poincaré inequality in a bounded domain
$\widetilde{\Omega}\subset\mathbb{C}$ with the upper estimate (see [10]):

(3.1)
$C_{r,2}(\widetilde{\Omega})\leq\inf\limits_{p\in\left(\frac{2r}{r+2},2\right)}\left(\frac{p-1}{2-p}\right)^{\frac{p-1}{p}}\frac{\left(\sqrt{\pi}\cdot\sqrt[p]{2}\right)^{-1}|\widetilde{\Omega}|^{\frac{1}{r}}}{\sqrt{\Gamma(2/p)\Gamma(3-2/p)}}.$

Using Theorem 3.1 we give an upper estimate for the constant in the Sobolev-
Poincaré inequality in domains with the quasihyperbolic boundary condition. As
shown in [2], the Jacobians of quasiconformal mappings
$\psi:\widetilde{\Omega}\to\Omega$ belong to $L^{\beta}(\widetilde{\Omega})$
for some $\beta>1$ if and only if ${\Omega}$ has the $\gamma$-quasihyperbolic
boundary condition for some $\gamma$. We note that $\beta$ depends only on
$\widetilde{\Omega}$ and the quasiconformal coefficient $K(\psi)$.

Since we need the exact value of the integrability exponent $\beta$ for the
Jacobians of quasiconformal mappings, we consider an equivalent description of
domains with quasihyperbolic boundary in terms of the integrability of
Jacobians [9].

A simply connected domain $\Omega\subset\mathbb{C}$ is called an
$A$-quasiconformal $\beta$-regular domain about a simply connected domain
$\widetilde{\Omega}\subset\mathbb{C}$ if

$\iint\limits_{\widetilde{\Omega}}|J(w,\varphi_{A}^{-1})|^{\beta}\leavevmode\nobreak\
dudv<\infty$

for some $\beta>1$, where $\varphi_{A}:\Omega\to\widetilde{\Omega}$ is a
corresponding $A$-quasiconformal mapping.

The property of the quasiconformal $\beta$-regularity implies the
integrability of a Jacobian of quasiconformal mappings and therefore for any
quasiconformal $\beta$-regular domain we have the embedding of weighted
Lebesgue spaces $L^{r}(\Omega,h)$ into non-weighted Lebesgue spaces
$L^{s}(\Omega)$ for $s=\frac{\beta-1}{\beta}r$ [9].

######  Lemma 3.2.

[9] Let $\Omega$ be an $A$-quasiconformal $\beta$-regular domain. Then for any
function $f\in L^{r}(\Omega,h)$, $\beta/(\beta-1)\leq r<\infty$, the
inequality

$\|f\mid
L^{s}(\Omega)\|\leq\left(\iint\limits_{\widetilde{\Omega}}|J(w,\varphi^{-1}_{A})|^{\beta}dudv\right)^{\frac{1}{\beta}\cdot\frac{1}{s}}\|f\mid
L^{r}(\Omega,h)\|$

holds for $s=\frac{\beta-1}{\beta}r$.

We ready to prove the upper estimate for the constant in the Sobolev-Poincaré
inequality in quasiconformal regular domains.

######  Theorem 3.3.

Let $A$ belong to a class $M^{2\times 2}(\Omega)$ and a domain $\Omega$ be
$A$-quasiconformal $\beta$-regular about $\widetilde{\Omega}$. Then:

  1. (1)

for any function $f\in W^{1,2}_{0}(\Omega,A)$ and for any $s\geq 1$, the
Sobolev-Poincaré inequality

$\|f\mid L^{s}(\Omega)\|\leq C_{s,2}(A,\Omega)\|f\mid L^{1,2}_{A}(\Omega)\|$

holds with the constant

$C_{s,2}(A,\Omega)\leq C_{\frac{\beta
s}{\beta-1},2}(\widetilde{\Omega})\|J_{\varphi_{A}^{-1}}\mid
L^{\beta}(\widetilde{\Omega})\|^{\frac{1}{s}},\quad 1<\beta<\infty;$

  2. (2)

if $\beta=\infty$ then for any function $f\in W^{1,2}_{0}(\Omega,A)$, the
Sobolev-Poincaré inequality

$\|f\mid L^{2}(\Omega)\|\leq C_{2,2}(A,\Omega)\|f\mid L^{1,2}_{A}(\Omega)\|$

holds with the constant

$C_{2,2}(A,\Omega)\leq
C_{2,2}(\widetilde{\Omega})\big{\|}J_{\varphi_{A}^{-1}}\mid
L^{\infty}(\widetilde{\Omega})\big{\|}^{\frac{1}{2}}\,.$

Here $J_{\varphi_{A}^{-1}}$ is a Jacobian of the inverse mapping to the
$A$-quasiconformal mapping $\varphi_{A}:\Omega\to\widetilde{\Omega}$.

######  Remark 3.4.

The constant
$C_{2,2}^{2}(\widetilde{\Omega})=1/\lambda_{1}(\widetilde{\Omega})$, where
$\lambda_{1}(\widetilde{\Omega})$ is the first Dirichlet eigenvalue of
Laplacian in a domain $\widetilde{\Omega}\subset\mathbb{C}$.

###### Proof.

1\. Let $\frac{\beta}{\beta-1}\leq r$ and $s=\frac{\beta-1}{\beta}r$. It means
that such $r$ exists for any $s\geq 1$. By Lemma 3.2 and Theorem 3.1 we get

$\|f\mid
L^{s}(\Omega)\|=\left(\iint\limits_{\Omega}|f(z)|^{s}dxdy\right)^{\frac{1}{s}}\\\
\leq\left(\iint\limits_{\widetilde{\Omega}}|J(w,\varphi^{-1}_{A})|^{\beta}dudv\right)^{\frac{1}{\beta}\cdot\frac{1}{s}}\left(\iint\limits_{\Omega}|f(z)|^{r}|J(\varphi_{A},x,y)|dxdy\right)^{\frac{1}{r}}\\\
\leq
C_{r,2}(\widetilde{\Omega})\left(\iint\limits_{\widetilde{\Omega}}|J(w,\varphi^{-1}_{A})|^{\beta}dudv\right)^{\frac{1}{\beta}\cdot\frac{1}{s}}\left(\iint\limits_{\Omega}\left\langle
A(z)\nabla f(z),\nabla f(z)\right\rangle dxdy\right)^{\frac{1}{2}}$

for any $s\geq 1$.

2\. Let a function $f\in L^{2}(\Omega)$. Since quasiconformal mappings possess
the Luzin $N$-property, then $|J(z,\varphi_{A})|^{-1}=|J(w,\varphi_{A}^{-1})|$
for almost all $z\in\Omega$ and for almost all
$w=\varphi_{A}(z)\in\widetilde{\Omega}$. Hence

$\left(\iint\limits_{\Omega}|f(z)|^{2}\leavevmode\nobreak\
dxdy\right)^{\frac{1}{2}}=\left(\iint\limits_{\Omega}|f(z)|^{2}|J(z,\varphi_{A})|^{-1}|J(z,\varphi_{A})|\leavevmode\nobreak\
dxdy\right)^{\frac{1}{2}}\\\ \leq\|J_{\varphi_{A}}\mid
L^{\infty}(\Omega)\|^{-\frac{1}{2}}\left(\int\limits_{\Omega}|f(z)|^{2}|J(z,\varphi_{A})|\leavevmode\nobreak\
dxdy\right)^{\frac{1}{2}}.$

Applying the change of variable formula for quasiconformal mappings [26],
(non-weighed) Sobolev-Poincaré inequality [10], and Theorem 2.1 we obtain

$\left(\iint\limits_{\Omega}|f(z)|^{2}\leavevmode\nobreak\
dxdy\right)^{\frac{1}{2}}\\\ \leq\|J_{\varphi_{A}^{-1}}\mid
L^{\infty}(\widetilde{\Omega})\|^{\frac{1}{2}}\left(\iint\limits_{\widetilde{\Omega}}|f\circ\varphi_{A}^{-1}(w)|^{2}\leavevmode\nobreak\
dudv\right)^{\frac{1}{2}}\\\ \leq
C_{2,2}(\widetilde{\Omega})\|J_{\varphi_{A}^{-1}}\mid
L^{\infty}(\widetilde{\Omega})\|^{\frac{1}{2}}\left(\iint\limits_{\widetilde{\Omega}}|\nabla(f\circ\varphi_{A}^{-1}(w))|^{2}\leavevmode\nobreak\
dudv\right)^{\frac{1}{2}}\\\
=C_{2,2}(\widetilde{\Omega})\|J_{\varphi_{A}^{-1}}\mid
L^{\infty}(\widetilde{\Omega})\|^{\frac{1}{2}}\left(\iint\limits_{\Omega}\left\langle
A(z)\nabla f(z),\nabla f(z)\right\rangle dxdy\right)^{\frac{1}{2}}$

for any $f\in W_{0}^{1,2}(\Omega,A)$. ∎

##  4\. Lower estimates for $\lambda_{1}(A,\Omega)$

We consider the generalized formulation of the Dirichlet eigenvalue problem
(1.1):

$\iint\limits_{\Omega}\langle A(z)\nabla
f(z),\nabla\overline{g(z)}\rangle\leavevmode\nobreak\
dxdy=\lambda\iint\limits_{\Omega}f(z)\overline{g(z)}\leavevmode\nobreak\
dxdy,\,\,\forall g\in W^{1,2}_{0}(\Omega,A).$

By the min-max principle (see, for example, [16, 22]) the first Dirichlet
eigenvalue $\lambda_{1}(A,\Omega)$ of the elliptic operator in divergence form
$L_{A}f(z)$ is defined by

$\lambda_{1}(A,\Omega)=\inf_{f\in
W_{0}^{1,2}(\Omega,A)\setminus\\{0\\}}\frac{\|f\mid
L^{1,2}_{A}(\Omega)\|^{2}}{\|f\mid L^{2}(\Omega)\|^{2}}\,.$

In other words, $\lambda_{1}^{-\frac{1}{2}}(A,\Omega)$ is the exact constant
$C_{2,2}(A,\Omega)$ in the Sobolev-Poincaré inequality

$\|f\mid L^{2}(\Omega)\|\leq C_{2,2}(A,\Omega)\|f\mid
L^{1,2}_{A}(\Omega)\|,\,\,f\in W^{1,2}_{0}(\Omega,A).$

######  Theorem 4.1.

Let $A$ belong to the class $M^{2\times 2}(\Omega)$ and $\Omega$ be an
$A$-quasiconformal $\beta$-regular domain about a domain $\widetilde{\Omega}$.
Then

$\frac{1}{\lambda_{1}(A,\Omega)}\leq
C^{2}_{\frac{2\beta}{\beta-1},2}(\widetilde{\Omega})\|J_{\varphi_{A}^{-1}}\mid
L^{\beta}(\widetilde{\Omega})\|,$

where $J_{\varphi_{A}^{-1}}$ is a Jacobian of the inverse mapping to the
$A$-quasiconformal mapping $\varphi_{A}:\Omega\to\widetilde{\Omega}$ and

$C_{\frac{2\beta}{\beta-1},2}(\widetilde{\Omega})\leq\inf\limits_{p\in\left(\frac{2\beta}{2\beta-1},2\right)}\left(\frac{p-1}{2-p}\right)\frac{\left(\sqrt{\pi}\cdot\sqrt[p]{2}\right)^{-1}|\widetilde{\Omega}|^{\frac{\beta-1}{2\beta}}}{\sqrt{\Gamma(2/p)\Gamma(3-2/p)}}.$

###### Proof.

By the min-max principle and Theorem 3.3 in the case $s=2$, we have

$\iint\limits_{\Omega}|f(z)|^{2}\leavevmode\nobreak\ dxdy\leq
C^{2}_{2,2}(A,\Omega)\iint\limits_{\Omega}\langle A(z)\nabla f(z),\nabla
f(z)\rangle\leavevmode\nobreak\ dxdy,$

where

$C_{2,2}(A,\Omega)\leq
C_{\frac{2\beta}{\beta-1},2}(\widetilde{\Omega})\left(\iint\limits_{\widetilde{\Omega}}|J(w,\varphi_{A}^{-1})|^{\beta}\leavevmode\nobreak\
dudv\right)^{\frac{1}{2\beta}}.$

Thus

$\frac{1}{\lambda_{1}(A,\Omega)}\leq
C^{2}_{\frac{2\beta}{\beta-1},2}(\widetilde{\Omega})\left(\iint\limits_{\widetilde{\Omega}}|J(w,\varphi_{A}^{-1})|^{\beta}\leavevmode\nobreak\
dudv\right)^{\frac{1}{\beta}}.$

∎

In the limit case $\beta=\infty$ we have the following assertion:

######  Theorem 4.2.

Let $A$ belong to a class $M^{2\times 2}(\Omega)$ and $\Omega$ be an
$A$-quasiconformal $\infty$-regular domain about a domain
$\widetilde{\Omega}$. Then

(4.1) $\frac{1}{\lambda_{1}(A,\Omega)}\leq
C^{2}_{2,2}(\widetilde{\Omega})\|J_{\varphi_{A}^{-1}}\mid
L^{\infty}(\widetilde{\Omega})\|=\frac{\|J_{\varphi_{A}^{-1}}\mid
L^{\infty}(\widetilde{\Omega})\|}{\lambda_{1}(\widetilde{\Omega})},$

where $J_{\varphi_{A}^{-1}}$ is a Jacobian of the inverse mapping to the
$A$-quasiconformal mapping $\varphi_{A}:\Omega\to\widetilde{\Omega}$.

As an application of Theorem 4.2 we consider several examples.

######  Example 4.3.

The homeomorphism

$\varphi(z)=\frac{a}{a^{2}-b^{2}}z-\frac{b}{a^{2}-b^{2}}\overline{z},\quad
z=x+iy,\quad a>b\geq 0,$

is an $A$-quasiconformal and maps the right triangle with arbitrary angels

$\Omega=\left\\{(x,y)\in\mathbb{R}^{2}:0\leq x\leq a+b,\,\,0\leq
y\leq-\frac{a-b}{a+b}x+(a-b)\right\\}$

onto the $45^{\circ}$ right triangle

$\widetilde{\Omega}=\left\\{(u,v)\in\mathbb{R}^{2}:0\leq u\leq 1,\,\,0\leq
v\leq 1-u\right\\}.$

The mapping $\varphi$ satisfies the Beltrami equation with

$\mu(z)=\frac{\varphi_{\overline{z}}}{\varphi_{z}}=-\frac{b}{a}$

and the Jacobian
$J(z,\varphi)=|\varphi_{z}|^{2}-|\varphi_{\overline{z}}|^{2}=1/(a^{2}-b^{2})$.
It is easy to verify that $\mu$ induces, by formula (2.5), the matrix function
$A(z)$ form

$A(z)=\begin{pmatrix}\frac{a+b}{a-b}&0\\\ 0&\frac{a-b}{a+b}\end{pmatrix}.$

Given that $|J(w,\varphi^{-1})|=|J(z,\varphi)|^{-1}=a^{2}-b^{2}$ and
$\lambda_{1}(\widetilde{\Omega})=5\pi^{2}$ (see, for example, [14]). Then by
Theorem 4.2 we have

$\lambda_{1}(A,\Omega)\geq\frac{\lambda_{1}(\widetilde{\Omega})}{\|J_{\varphi^{-1}}\mid
L^{\infty}(\widetilde{\Omega})\|}=\frac{5\pi^{2}}{a^{2}-b^{2}}.$

######  Example 4.4.

The homeomorphism

$\varphi(z)=\frac{2\cdot
z^{\frac{3}{8}}}{\overline{z}^{\frac{1}{8}}}-1,\,\,\varphi(0)=-1,\quad
z=x+iy,$

is an $A$-quasiconformal and maps the interior of the non-convex domain

$\Omega:=\left\\{(\rho,\theta)\in\mathbb{R}^{2}:\rho=\cos^{4}\left(\frac{\theta}{2}\right),\quad-\pi\leq\theta\leq\pi\right\\}$

onto the unit disc $\mathbb{D}$. The mapping $\varphi$ satisfies the Beltrami
equation with

$\mu(z)=\frac{\varphi_{\overline{z}}}{\varphi_{z}}=-\frac{1}{3}\frac{z}{\overline{z}}$

and the Jacobian

$J(z,\varphi)=|\varphi_{z}|^{2}-|\varphi_{\overline{z}}|^{2}=\frac{1}{2\cdot|z|^{\frac{3}{2}}}.$

We see that $\mu$ induces, by formula (2.5), the matrix function $A(z)$ form

$A(z)=\begin{pmatrix}\frac{|3\overline{z}+z|^{2}}{8|\overline{z}|^{2}}&\frac{3}{4}\operatorname{Im}\frac{z}{\overline{z}}\\\
\frac{3}{4}\operatorname{Im}\frac{z}{\overline{z}}&\frac{|3\overline{z}-z|^{2}}{8|\overline{z}|^{2}}\end{pmatrix}.$

Given that $|J(w,\varphi^{-1})|=|J(z,\varphi)|^{-1}=2\cdot|z|^{\frac{3}{2}}$
and $\lambda_{1}(\mathbb{D})=(j_{0,1})^{2}$. Then by Theorem 4.2 we have

$\lambda_{1}(A,\Omega)\geq\frac{\lambda_{1}(\mathbb{D})}{\|J_{\varphi^{-1}}\mid
L^{\infty}(\mathbb{D})\|}\geq\frac{(j_{0,1})^{2}}{2}.$

Taking into account Theorem 4.2 and the domain monotonicity property of the
Dirichlet eigenvalues for the elliptic operator in divergence form
$L_{A}f(z)$, we obtain the following result for the special case
$\|J_{\varphi_{A}^{-1}}\mid L^{\infty}(\widetilde{\Omega})\|<1$.

######  Proposition 4.5.

Let $\Omega$ be an $A$-quasiconformal $\infty$-regular domain about
$\widetilde{\Omega}$. We assume that $\widetilde{\Omega}\supset\Omega$, then

$\lambda_{1}(A,\Omega)-\lambda_{1}(\widetilde{\Omega})\geq\frac{1-\|J_{\varphi_{A}^{-1}}\mid
L^{\infty}(\widetilde{\Omega})\|}{\|J_{\varphi_{A}^{-1}}\mid
L^{\infty}(\widetilde{\Omega})\|}\lambda_{1}(\widetilde{\Omega}).$

###### Proof.

Since $\widetilde{\Omega}\supset\Omega$, we have
$\lambda_{1}(A,\Omega)\geq\lambda_{1}(\widetilde{\Omega})$. Taking into
account the inequality (4.1) in Theorem 4.2 and making elementary calculation,
we get

$\lambda_{1}(A,\Omega)-\lambda_{1}(\widetilde{\Omega})\geq\frac{1-\|J_{\varphi_{A}^{-1}}\mid
L^{\infty}(\widetilde{\Omega})\|}{\|J_{\varphi_{A}^{-1}}\mid
L^{\infty}(\widetilde{\Omega})\|}\lambda_{1}(\widetilde{\Omega}).$

∎

###  4.1. The Rayleigh-Faber-Krahn type inequality

The theory of composition operators [11] allows us to reduce the spectral
problem for the divergence form elliptic operator (1.1) defined in simply
connected bounded domain $\Omega\subset\mathbb{C}$ to a weighted spectral
problem for the Laplace operator in a simply connected bounded domain
$\widetilde{\Omega}\subset\mathbb{C}$. By the chain rule applied to a function
$f(z)=g\circ\varphi_{A}(z)$ [15], we have

$-\textrm{div}[A(z)\nabla f(z)]=-\textrm{div}[A(z)\nabla
g\circ\varphi_{A}(z)]=-h(w)\Delta g(w),$

where the weight $h(w)=|J(w,\varphi_{A}^{-1})|^{-1}$ is a Jacobian of the
inverse mapping to the $A$-quasiconformal mapping
$\varphi_{A}:\Omega\to\widetilde{\Omega}$.

From here we can point out that

$\lambda_{1}(A,\Omega)=\inf_{f\in
W_{0}^{1,2}(\Omega,A)\setminus\\{0\\}}\frac{\iint\limits_{\Omega}\left\langle
A(z)\nabla f,\nabla f\right\rangle dxdy}{\iint\limits_{\Omega}|f|^{2}dxdy}\\\
=\inf_{g\in
W_{0}^{1,2}(\widetilde{\Omega},h,1)\setminus\\{0\\}}\frac{\iint\limits_{\widetilde{\Omega}}|\nabla
g|^{2}dudv}{\iint\limits_{\widetilde{\Omega}}|g|^{2}h(w)dudv}=\lambda_{1}(h,\widetilde{\Omega}).$

Let $\varphi_{A}:\Omega\to\widetilde{\Omega}$ be $A$-quasiconformal mappings
for which $|J(z,\varphi_{A})|=1$ for almost all $z\in\Omega$. In this case
quasiconformal weights $h(w)=|J(w,\varphi_{A}^{-1})|=1$ for almost all
$w\in\widetilde{\Omega}$. Hence we get that
$\lambda_{1}(A,\Omega)=\lambda_{1}(\widetilde{\Omega})$.

Using this equality and the classical Rayleigh-Faber-Krahn inequality we
immediately obtain a version of this inequality for elliptic operators in
divergence form. Namely:

######  Theorem 4.6.

Let $\Omega\subset\mathbb{C}$ be a simply connected bounded domain such that
there exists a measure preserving A-quasiconformal mapping
$\varphi_{A}:\Omega\to\widetilde{\Omega}$. Then

$\lambda_{1}(A,\Omega)\geq\lambda_{1}(\Omega^{\ast})=\frac{{j_{0,1}^{2}}}{R^{2}_{\ast}},$

Here $\Omega^{\ast}$ is the disc of the radius $R_{\ast}$ such that
$|\Omega|=|\Omega^{\ast}|$. and $j_{0,1}\approx 2.4048$ is the first positive
zero of the Bessel function $J_{0}$.

###### Proof.

By assumptions, the mapping $\varphi_{A}:\Omega\to\widetilde{\Omega}$ is
measure preserving A-quasiconformal, i.e., $|J(z,\varphi_{A})|=1$ almost
everywhere in $\Omega$. From the equality
$|J(z,\varphi_{A})|=|J(z,\varphi^{-1}_{A})|=1$ for almost all $z\in\Omega$ and
almost all $w=\varphi_{A}(z)\in\widetilde{\Omega}$ we obtain the equality
$\lambda_{1}(A,\Omega)=\lambda_{1}(\widetilde{\Omega})$. Now using the
classical Rayleigh-Faber-Krahn inequality,

$\lambda_{1}(\widetilde{\Omega})\geq\lambda_{1}(\Omega^{\ast})$

we obtain

$\lambda_{1}(A,\Omega)\geq\lambda_{1}(\Omega^{\ast}).$

The theorem is proved. ∎

A description of the class of measure preserving $A$-quasiconformal mappings
and/or corresponding divergence form elliptic equations is an open problem.
Let us give simple examples of such mappings.

######  Example 4.7.

Let $\varphi(z)=(ax,\frac{1}{a}y)$, $z=x+iy$ and $a>1$. Then $J(z,\varphi)=1$,
the quasiconformality coefficient $Q_{\varphi}=a^{2}$ and
$\mu_{\varphi}=\frac{a^{2}-1}{a^{2}+1}$. The matrix can be easily recontracted

$A(z)=\begin{pmatrix}\frac{1}{a^{2}}&0\\\ 0&a^{2}\end{pmatrix}.$

A little bit more complicated example.

######  Example 4.8.

Let $\varphi(z)=(ax+by,\frac{1}{a}y)$, $z=x+iy$ and $a>1$. Then
$J(z,\varphi)=1$, the quasiconformality coefficient $Q_{\varphi}=a^{2}$.
Calculation of $\mu_{\varphi}$ is more complicated. We use the Beltrami
equation, i.e

$\varphi_{z}=\frac{1}{2}(a+\frac{1}{a})-i\frac{b}{2},\quad\varphi_{\overline{z}}=\frac{1}{2}(a-\frac{1}{a})+i\frac{b}{2}.$

Hence

$\mu_{\varphi}=\frac{(a^{2}-1)(a^{2}+1)-a^{2}b^{2}}{(a^{2}+1)^{2}+a^{2}b^{2}}+i\frac{2a^{3}b}{(a^{2}+1)^{2}+a^{2}b^{2}}.$

The matrix $A$ can be easily reconstracted by elementary calculations.

######  Example 4.9.

Let $f\in L^{1}_{\infty}(\mathbb{R})$. Then $\varphi(z)=(x+f(y),y)$, $z=x+iy$,
is a quasiconformal mapping with $|J(z,\varphi)|=1$ (see, [11]). A basic
calculation implies

$\varphi_{z}=1-i\frac{f^{\prime}(y)}{2},\quad\varphi_{\overline{z}}=i\frac{f^{\prime}(y)}{2}.$

Hence

$\mu_{\varphi}=-\frac{(f^{\prime}(y))^{2}}{4+(f^{\prime}(y))^{2}}+i\frac{2f^{\prime}(y)}{4+(f^{\prime}(y))^{2}}.$

The matrix can be easily recontracted

$A(z)=\begin{pmatrix}1&-f^{\prime}(y)\\\
-f^{\prime}(y)&1+(f^{\prime}(y))^{2}\end{pmatrix}.$

This algorithm can be used for more complicated examples.

##  5\. Appendix

Few remarks about measure preserving and quasi-preserving (bi-Lipschitz)
quasiconformal mappings and$A$-quasiconformal mappings.

A quasiconformal mapping $\varphi$ is a solution of the corresponding Beltrami
equation

(5.1) $\varphi_{\overline{z}}(z)=\mu(z)\varphi_{z}(z).$

Because $J(z,\varphi)=|\varphi_{z}^{2}|(z)-|\varphi_{\overline{z}}^{2}|(z)$,
the condition $J(z,\varphi)=1$ can be written as

(5.2) $|\varphi_{z}|^{2}(z)(1-|\mu(z)|^{2})=1.$

Recall that for $A$-quasiconformal mappings we have

$\mu(z)=\frac{a_{22}(z)-a_{11}(z)-2ia_{12}(z)}{\det(I+A(z))}.$

By elementary calculations it is easy to verify that any measure preserving
quasiconformal mapping $\varphi:\Omega\to\widetilde{\Omega}$ is a bi-Lipschitz
mapping in the following sense: $D(\varphi)\in L^{1}_{\infty}(\Omega)$ and
$D(\varphi^{-1})\in L^{1}_{\infty}(\widetilde{\Omega})$ (or $D\varphi\in
L^{\infty}(\Omega)$ and $D\varphi^{-1}\in L^{\infty}(\widetilde{\Omega})$). An
equivalent geometric description is the following. The mapping $\varphi$ and
its inverse are locally Lipschitz homeomorphisms with uniformly bounded
Lipschitz constants.

The condition (5.2) can be soften in the spirit of quasiconformality up to the
inequality

$\ \frac{1}{C}<J(z,\varphi)<C$

for some positive constant $C$. We call such homeomorphism as quasi-preserving
measure homeomorphisms.

For quasiconformal quasi-preserving measure homeomorphisms this condition can
be written as

(5.3) $\frac{1}{C}<|\varphi_{z}^{2}|(z)(1-|\mu(z)|^{2})<C.$

The class of quasiconformal quasi-preserving measure homeomorphisms coincide
with the class of bi-Lipschitz homeomorphisms. The constant $C$ can be easily
recalculated in terms of uniform Lipschitz constants for the corresponding
homeomorphism and its inverse.

We do not have any geometric description of such homeomorphisms i.e to
solution of systems ((5.1),(5.2)) or ((5.1),(5.3)). Let us look for some
simplified cases.

Suppose the matrix $A$ is a diagonal matrix. Then the quasiconformality
coefficient $\mu(z)$ is a real number. If $a_{11}(z)>0$ then $0<\mu(z)<1$ and
the condition (5.2) can be simplified:

(5.4) $|\varphi_{z}|^{2}(z)(1-\mu(z)^{2})=1.$

By simple calculations

$\mu(z)=\frac{a_{11}(z)+a_{11}^{-1}(z)}{2+a_{11}(z)+a_{11}^{-1}(z)}.$

Let us give an example of such homeomorphisms.

######  Example 5.1.

Suppose $\Omega:=(0,1)\times(0,1)$ and $\varphi(x,y)=a(x)+ib(y)$ where $a(x)$
and $b(y)$ belong to $C^{1}(\Omega)$. We also suppose that

$\inf\limits_{x\in(0,1)}(da/dx)>\sup\limits_{y\in(0,1)}(db/dy)$

and $I_{1}:=\inf\limits_{y\in(0,1)}(db/dy)>0$.

Any such mapping is a quasi-preserving measure one, because

$I_{1}\leq
J(z,\varphi)\leq\left(\left|da/dx\right|_{C^{1}(\Omega)}\times\left|db/dy\right|_{C^{1}(\Omega)}\right)$

and quasiconformal one because

$Q\leq\frac{\sup\limits_{x\in(0,1)}(da/dx)}{\inf\limits_{y\in(0,1)}(db/dy)}.$

The corresponding matrix $A$ can be easily reconstructed

(5.5) $A(z)=\begin{pmatrix}\frac{1-\mu(z)}{1+\mu(z)}&0\\\
0&\frac{1+\mu(z)}{1-\mu(z)}\end{pmatrix},\,\,\,\text{a.e. in}\,\,\,\Omega,$

where for $z=x+iy$

$\mu(z)=\frac{da/dx(x)-db/dy(y)}{da/dx(x)+db/dy(y)}\,.$

Acknowledgements.

The second author was supported by the Ministry of Science and Higher
Education of Russia (agreement No. 075-02-2022-884).

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Department of Mathematics, Ben-Gurion University of the Negev, P.O.Box 653,
Beer Sheva, 8410501, Israel

_E-mail address:_

Regional Scientific and Educational Mathematical Center, Tomsk State
University, 634050 Tomsk, Lenin Ave. 36, Russia

_E-mail address:_


# DeSIQ: Towards an Unbiased, Challenging Benchmark for Social Intelligence
Understanding

Xiao-Yu Guo  Yuan-Fang Li  Gholamreza Haffari

###### Abstract

Social intelligence is essential for understanding and reasoning about human
expressions, intents and interactions. One representative benchmark for its
study is Social Intelligence Queries (Social-IQ), a dataset of multiple-choice
questions on videos of complex social interactions. We define a comprehensive
methodology to study the soundness of Social-IQ, as the soundness of such
benchmark datasets is crucial to the investigation of the underlying research
problem. Our analysis reveals that Social-IQ contains substantial biases,
which can be exploited by a moderately strong language model to learn spurious
correlations to achieve perfect performance without being given the context or
even the question. We introduce DeSIQ, a new challenging dataset, constructed
by applying simple perturbations to Social-IQ. Our empirical analysis shows
DeSIQ significantly reduces the biases in the original Social-IQ dataset.
Furthermore, we examine and shed light on the effect of model size, model
style, learning settings, commonsense knowledge, and multi-modality on the new
benchmark performance. Our new dataset, observations and findings open up
important research questions for the study of social intelligence.

##  1 Introduction

Social intelligence is a long-standing research area in social science and
psychology Thorndike (1920); Andreou (2006); Goleman (2007). It is the
capacity to understand and navigate complex social situations. Social
intelligence is more than the perception of objects and human actions, as it
requires a deeper understanding of human intents and interactions behind these
actions or words.

The study of social intelligence is an emerging area in both the NLP and
computer vision communities. One representative work, Social-IQ (Social
Intelligence Queries) Zadeh et al. (2019), is a benchmark dataset measuring
social intelligence of current AI systems. It is a multiple choice question
answering dataset with multi-modal inputs, including questions, answer
options, videos, etc; see an example in Figure 1. Although Social-IQ contains
rigorously human-annotated data, surprisingly, we find even small models like
T5-small Raffel et al. (2020) could easily achieve 100% answer option accuracy
(Table 3).

The perfect performance of such an underpowered model prompted us to conduct
further investigation to identify its source. Through employing different
models and perturbation methods on the answer options, we identify significant
biases in the Social-IQ dataset, in which the representations of correct and
incorrect options are easily separable, regardless of the questions (Figure
3). Thus, the models are able to exploit such a _shortcut_ Jia and Liang
(2017); Jiang and Bansal (2019) to answer questions with a high accuracy,
without necessarily understanding social intelligence.

To debias the Social-IQ dataset, we propose a simple yet effective debiasing
approach and present a new unbiased benchmark DeSIQ, by substituting all the
incorrect answer options with correct answer options from randomly selected
other questions.

We establish a performance baseline on DeSIQ with T5-small and Delphi Jiang et
al. (2021), a language model pretrained with commonsense and social norms
knowledge. Given answer options only or question-answers, both T5-small and
Delphi obtain close to random accuracy. By making use of multi-modal inputs,
both T5-small and Delphi achieve an accuracy of up to 77%. These results
demonstrate that DeSIQ is unbiased and challenging. Interestingly, both models
also outperform GPT-3 and ChatGPT, further indicating the challenging nature
of the social intelligence understanding problem.

Figure 1: One example in the Social-IQ and DeSIQ benchmark. For Social-IQ, q,
a, i stand for question, correct and incorrect answer respectively, while a’
with yellow background color is the unbiased incorrect answer we substitute in
DeSIQ. Different colors represent different persons, including the facial
expressions and oral speaking words. The transcripts are in three black
squares related to certain video clips in the above.

Our contributions are:

  * •

We propose six formally defined methods to identify the bias in Social-IQ.
From the answer perturbation experiments, we find that the bias mainly exists
in the answer options.

  * •

We propose DeSIQ, an unbiased, and more challenging multi-modal question
answering benchmark, designed to better measure social intelligence for
machine learning models.

  * •

We propose two effective models that outperform the baseline and GPT-3/ChatGPT
on our new benchmark. We also make detailed analysis and comparison on the
performance of these models.

##  2 Identifying Biases in Social-IQ

###  2.1 The Social Intelligence Datasets

Social-IQ Zadeh et al. (2019) is an unconstrained multi-modal, multiple-choice
question answering (MCQA) dataset designed to evaluate the social intelligence
of machine learning models. It contains videos about social interactions,
questions and multiple-choice answer options, in which the questions and
answer options were crowdsourced. For each video, the context for all
questions and answer options includes not only the original video, but also
the corresponding extracted audio and automatically generated transcripts111We
don’t have access to the raw transcript, video and audio data so we use
extracted features downloaded from https://github.com/matsuolab/CMU-
MultimodalSDK.. Detailed dataset statistics are shown in Table 2.

Number | Training | Development | Total
---|---|---|---
Video | 888 | 127 | 1,015
Question | 5,328 | 762 | 6,090
Correct | 21,312 | 3,048 | 24,360
Incorrect | 15,984 | 2,286 | 18,270

Table 1: Statistics of the Social-IQ dataset. On average, each video has 6
questions; for each question, there are 4 correct answer options and 3
incorrect answer options.

Number | Training | Development | Total
---|---|---|---
Video | 987 | 145 | 1,132
Question | 6,159 | 943 | 7,102
Correct | 6,159 | 943 | 7,102
Incorrect | 18,477 | 2,829 | 21,306

Table 2: Statistics of the Social-IQ-2.0 dataset. For each question, there is
only 1 correct answer option.

Social-IQ provides two configurations: A2 (2-way, i.e. one correct answer
option and one incorrect option for each question) and A4 (2-way, i.e. one
correct option and 3 incorrect options for each question) for training and
evaluation, in which model performance is measured using binary and 4-way
accuracy respectively.

Most recently, Social-IQ-2.0 was released online222https://cmu-multicomp-
lab.github.io/social-iq-2.0/ with the A4 configuration. Though nearly half of
the videos overlap with Social-IQ, almost all questions and answers were newly
annotated. Moreover, raw video and audio files have been provided instead of
only features in the original Social-IQ dataset. The detailed statistics are
shown in Table 2. For simplicity, v1 and v2 represent Social-IQ and Social-
IQ-2.0 respectively, which are used interchangeably.

###  2.2 Methodology

In this section, we propose several experimental settings to identify biases
in a MCQA dataset. Let $q$ and $q^{\prime}$ denote two different questions,
$a$ and $i$ denote the correct and an incorrect answer option of $q$
respectively, and $a^{\prime}$ and $i^{\prime}$ denote the correct and an
incorrect answer option of $q^{\prime}$ respectively. We define six methods to
identify biases:

No context and question (NCAQ):

the contexts and questions for all answer options are removed. I.e., the model
is only given all answer options.

An MCQA dataset should be sufficiently challenging that no model can predict a
correct answer when neither the input context nor the question is not
provided.

More Powerful Model (MPM):

the model is substituted by a larger, more capable model.

It is plausible to induce a performance increase on the dataset when a
stronger model (e.g. with more trainable parameters and/or one that is fine-
tuned on relevant data) is employed. However, a sufficiently hard dataset
should not induce a perfect model performance (i.e. near 100% accuracy score).
This can be tested with models of different sizes and thus capabilities.

RIWI:

Replace $i$ with $i^{\prime}$, $(q,a,i)\rightarrow(q,a,i^{\prime})$

RIWA:

Replace $i$ with $a^{\prime}$, $(q,a,i)\rightarrow(q,a,a^{\prime})$

RAWI:

Replace $a$ with $i^{\prime}$, $(q,a,i)\rightarrow(q,i^{\prime},i)$

RAWA:

Replace $a$ with $a^{\prime}$, $(q,a,i)\rightarrow(q,a^{\prime},i)$

With the above perturbations, we expect the dataset to induce the following
robustness behaviours. With RIWI or RIWA applied to the dev/test set, we
should expect that a model’s performance should not significantly deviate from
the original dataset. With RAWI or RAWA, the model should perform
significantly worse.

Figure 2: Overall architecture of the LSTM baseline Zadeh et al. (2019).
$q,a,i,t,v$ denote question, correct answer, incorrect answer, transcript and
video features. $r_{q},r_{a},r_{i},r_{t},r_{v}$ are corresponding features
extracted using different LSTMs. Dashed squares represent optional input
features. Two multi-layer perception (MLP) are parameter-shared. The output
will be two scores $s_{1},s_{2}$ respectively of the correct and incorrect
answer options.

###  2.3 Biases in Social-IQ

We evaluate the A2 (binary choice) configuration and A4 (multiple choice)
configurations of Social-IQ, and A4 configuration of Social-IQ-2.0 in the
experimental settings discussed above, and surprisingly, we observe that they
are both biased. Below, we describe our detailed analysis and show that
Social-IQ contains substantial biases that can be exploited by moderately
strong language models. Table 3 summarises the experimental results. In the
fully supervised setting, we evaluate the performance of the LSTM-based model
in the original Social-IQ paper Zadeh et al. (2019) (Figure 2) as well as the
more capable T5-small Raffel et al. (2020), which we use as the encoder to
replace the LSTM in Figure 2.

Data | Model | Settings | A2 | A4
---|---|---|---|---
| Random | none | 50 | 25
v1 | LSTM | NCAQ | 59.45 | 34.84
T5-small (MPM) | NCAQ | 100 | 100
NCAQ+RIWI | 97.37 | 99.97
NCAQ+RIWA | 50.21 | 25.03
NCAQ+RAWI | 49.93 | 23.76
NCAQ+RAWA | 97.25 | 100
v2 |  T5-small (MPM) | NCAQ | - | 63.35
NCAQ+RIWI | - | 59.66
NCAQ+RIWA | - | 24.72
NCAQ+RAWI | - | 23.72
NCAQ+RAWA | - | 62.36
Table 3: Model performance on A2 (binary choice) and A4 (multiple choice)
under different experimental settings, in which only answer options are given
as model inputs (but not questions nor context). ‘-’ represents the results
are inapplicable.

Evidence of Dataset Bias. We start from the NCAQ settings, i.e., only the
answer options ($a$ and $i$) are given as model input, without the question
and video, for both training and evaluation. Under this setting, we also
compare models’ performance with different perturbations on the answer
options. Table 3 shows that the basic LSTM model outperforms the random guess
by 9.45% on v1 (i.e. Social-IQ). With the unreasonable inputs (with no context
nor question), these accuracy scores show that the Social-IQ dataset is
biased.

We postulate that while a stronger model (i.e. MPM) should obtain better
performance than LSTM, without being given sufficient information, even the
stronger model should not perform unreasonably well. Thus, we experiment with
T5-small, a modestly-sized yet more capable model. As it can be seen in Table
3, T5-small outperforms LSTM by a large margin on v1. Surprisingly, it also
achieves a perfect 100% accuracy score on v1 and 63.35% on v2 without being
given the context nor the question. These results provide strong evidence of
the biases in these datasets.

Finally, we study the other four perturbation settings by applying them to the
dev sets. Below we analyse the performance on v1 in detail, followed by a
discussion on v2.

  * •

RIWI. Similar to the performance on the original dataset, T5-small achieves an
unreasonable performance of 97.37% on A2 and 99.97% in A4. It indicates that
the model can easily distinguish the correct answer from the incorrect
options.

  * •

RIWA. It leads to a large performance degradation: A2 from 100% to 50.21%, A4
from 100% to 25.03%, similar to random guess (i.e., 50% and 25%). This shows
that T5-small is unable to distinguish the correct answer options, regardless
of the question it is used for.

  * •

RAWI. This produces a dataset containing only incorrect answer options. We
consider the incorrect answer option that replaces the correct answer option
as the correct answer.

Intuitively, it should lead a model to randomly guess, as none of the options
is correct. In Table 3, we can observe that RAWI leads to a large performance
drop: A2 from 100% to 49.93%, A4 from 100% to 23.76%, indicating that T5-small
cannot distinguish incorrect answers from each other, confirming our
intuition.

  * •

RAWA. It should lead to A2 with 50% and A4 with 25% performance since the
correct answer option is replaced with an irrelevant correct answer of another
question.

Contrary to our intuition, RAWA leads to a near-perfect performance of 97.25%
on A2 and even better 100% on A4.

These unexpectedly high scores indicate that the model can easily distinguish
the correct answer options from the incorrect ones of the original dataset,
regardless of the question they are used for, consistent with the results of
RIWI.

Figure 3 shows the T-distributed Stochastic Neighbor Embedding (T-SNE)
visualization of the embeddings of all answer options in the Social-IQ dev
set. We can observe a clear boundary between correct and incorrect answer
options. The above results provide compelling evidence of the unwanted bias in
Social-IQ, manifested in T5-small’s strong capability in distinguishing the
correct and incorrect answer options.

Similar evidence can be found in Social-IQ-2.0, as can be seen in the v2 rows
in Table 3.

Figure 3: T-SNE visualization of correct and incorrect answer options. Red
dots are incorrect answer options and blue dots are correct answer options.

##  3 DeSIQ: Debiased Social-IQ

In this section, we first describe our approach to debias Social-IQ . We then
study the effectiveness of our debiasing approach and the resultant DeSIQ
datasets, by comparing the performance of both LSTM and T5-small on DeSIQ in
different settings.

###  3.1 Constructing DeSIQ

We propose the following perturbation-based approach to debias Social-IQ and
construct a more meaningful and challenging dataset on social intelligence.
Specifically, we apply the RIWA perturbation on both the training and
development sets of Social-IQ, ie substituting the incorrect answer options
with correct answer options from the other questions. We construct two
debiased datasets333Below we describe the dataset construction process for the
A2 configuration. Similar perturbations are applied to the A4 configuration of
Social-IQ.:

  * •

DeSIQd. Given an original triplet $(q,a,i)$, we randomly sample another
triplet $(q^{\prime},a^{\prime},i^{\prime})$ from _another video_. Thus, for
each original triplet $(q,a,i)$, we form a new triplet $(q,a,a^{\prime})$.

  * •

DeSIQs. We sample $(q^{\prime},a^{\prime},i^{\prime})$ from the _same video_
for each $(q,a,i)$. Similarly, we replace the incorrect answer option $i$ with
$a^{\prime}$. Since $q$ and $q^{\prime}$ are from the same video, their
answers can have a higher chance of referring to the same entity that appears
in the video. Thus, DeSIQs is a more challenging dataset of
$(q,a,a^{\prime})$.

An example video and some associated questions and answer options for both
Social-IQ and DeSIQs can be seen in Figure 1. For Social-IQ-2.0, we do the
same approach to obtain DeSIQd-2.0.

###  3.2 Effectiveness of the Debiasing Approach

We set up a number of models in both fully supervised and zero/few-shot
learning settings to show the effectiveness of our debiasing approach,

#### Supervised Learning.

We train the LSTM and T5-small on Social-IQ, DeSIQd and DeSIQs in the same
architecture (Figure 2), and train T5-small on Social-IQ-2.0. Table 5, Table 5
and Table 6 show the results, where the relevant results are shaded in gray.
The second column “Input” represents the input used in both the training and
evaluation procedures, where “a”, “q”, “t”, and “v” represent answer options,
the question, the transcript and the video, respectively. The third column
“Concat” represents different model architectures. The symbol “✗” denotes that
all inputs are separately encoded as in Figure 2, which is the focus of this
subsection. The symbol “✓” denotes that all inputs are concatenated and
encoded as one sequence as in Figure 4, which will be discussed in the next
section.

Figure 4: Our model structure to address the new benchmark DeSIQ. The A2
configuration is shown here for illustration purposes. $q,a,i,t,v,s$ are
question, correct answer, incorrect answer, transcript, video and audio
features; $m_{q},m_{a},m_{i},m_{t},m_{v},m_{s}$ are the corresponding features
after projection to the same dimension; and
$r_{q},r_{a},r_{i},r_{t},r_{v},r_{s}$ are the corresponding embeddings after
going through the same backbone model. $[\circ]$ denotes the concatenation
operation and dashed boxes denote optional input features. Three multi-layer
perceptrons (MLP${}_{\text{t}}$, MLP${}_{\text{v}}$ and MLP${}_{\text{s}}$) in
yellow above are projectors mapping textual, video and audio features to the
same space. The output will be the two scores $s_{1},s_{2}$ (four scores for
multiple-choice), representing the correct and incorrect answer option
respectively.

As seen in Table 5, DeSIQd largely reduces the bias in Social-IQ, effectively
reducing the performance of both LSTM and T5-small close to random guess. For
LSTM, when given only answer options, we observe a performance drop of 59.45%
$\rightarrow$ 48.52% on A2 and 34.84% $\rightarrow$ 27.23% on A4. For
T5-small, it suffers a larger performance drop on both A2 and A4 of DeSIQd (to
50.16% and 34.15% respectively), although 100% A2 on Social-IQ.

The results in Table 5 show that DeSIQs effectively reduces the bias in
Social-IQ. For LSTM, DeSIQs leads to a performance drop of 59.45%
$\rightarrow$ 48.24% on A2 and 34.84% $\rightarrow$ 27.06%. For T5-small, we
can observe a performance drop of 100% $\rightarrow$ 48.73% on A2 and of 100%
$\rightarrow$ 33.53% on A4.

Comparing results in Tables 5 and 5, we see DeSIQs is generally more
challenging than DeSIQd, i.e. compare T5-small’s performance under the same
settings across the two datasets. For instance, with the question feature
added (“q+a” as inputs), T5-small achieves on 60.55% on DeSIQd and 49.17% on
DeSIQs in A2, and 27.57% on DeSIQd and 24.22% on DeSIQs in A4.

In Table 6, DeSIQd-2.0 also reduces the bias in Social-IQ-2.0, effectively
reducing the performance of T5-small close to random guess. For T5-small, it
suffers a larger performance drop on A4 of DeSIQd-2.0 (63.35% to 28.07%).

Dataset | Input | Concat | A2 | A4
---|---|---|---|---
LSTM | T5-small | T5-smallDelphi | LSTM | T5-small | T5-smallDelphi
Social-IQ | a | ✗ | 59.45 | 100 | 100 | 34.84 | 100 | 100
q+a | ✗ | 59.78 | 100 | 100 | 38.55 | 100 | 100
q+a+t | ✗ | 60.00 | 100 | 100 | 43.84 | 100 | 100
q+a+v | ✗ | 64.38 | 100 | 100 | 46.08 | 100 | 100
DeSIQd | a | ✗ | 48.52 | 50.16 | 50.33 | 27.23 | 34.15 | 28.97
q+a | ✗ | 58.58 | 60.55 | 50.19 | 26.05 | 27.57 | 25.78
q+a+t | ✗ | 60.46 | 50.16 | 50.40 | 27.59 | 28.84 | 27.15
q+a+v | ✗ | 61.05 | 50.59 | 50.70 | 25.91 | 27.60 | 25.55
DeSIQd | a | ✓ | 49.20 | 49.52 | 50.33 | 34.85 | 36.30 | 38.18
q+a | ✓ | 61.41 | 73.47 | 75.69 | 34.91 | 62.43 | 72.91
q+a+t | ✓ | 13.17 | 74.69 | 76.77 | 29.22 | 70.80 | 74.51
q+a+v | ✓ | 56.67 | 76.72 | 74.99 | 41.77 | 72.69 | 73.24

Table 4: Accuracy on the Social-IQ and DeSIQd development sets. Results shaded
in gray are relevant to Sec. 3.

Dataset | Input | Concat | A2 | A4
---|---|---|---|---
LSTM | T5-small | T5-smallDelphi | LSTM | T5-small | T5-smallDelphi
Social-IQ | a | ✗ | 59.45 | 100 | 100 | 34.84 | 100 | 100
q+a | ✗ | 59.78 | 100 | 100 | 38.55 | 100 | 100
q+a+t | ✗ | 60 | 100 | 100 | 43.84 | 100 | 100
q+a+v | ✗ | 64.38 | 100 | 100 | 46.08 | 100 | 100
DeSIQs | a | ✗ | 48.24 | 48.73 | 48.96 | 27.06 | 33.53 | 28.12
q+a | ✗ | 59.97 | 49.17 | 59.59 | 26.16 | 24.22 | 25.20
q+a+t | ✗ | 60.02 | 58.89 | 56.83 | 27.31 | 26.79 | 23.83
q+a+v | ✗ | 61.00 | 59.19 | 56.42 | 26.99 | 25.00 | 24.22
DeSIQs | a | ✓ | 48.24 | 48.73 | 48.71 | 29.67 | 29.17 | 32.32
q+a | ✓ | 59.35 | 63.08 | 63.42 | 34.73 | 62.47 | 60.58
q+a+t | ✓ | 11.52 | 65.41 | 67.70 | 22.33 | 65.23 | 51.69
q+a+v | ✓ | 51.04 | 65.96 | 65.02 | 32.56 | 56.61 | 55.05

Table 5: Accuracy on the Social-IQ and DeSIQs development sets. Results shaded in gray are relevant to Sec. 3. Dataset | Input | Concat | A4
---|---|---|---
Social-IQ-2.0 | a | ✗ | 63.35
q+a | ✗ | 64.63
q+a+t | ✗ | 64.06
q+a+v | ✗ | 62.28
DeSIQd-2.0 | a | ✗ | 28.07
q+a | ✗ | 28.45
q+a+t | ✗ | 22.17
q+a+v | ✗ | 24.13
| q+a+s | ✗ | 25.87
DeSIQd-2.0 | a | ✓ | 28.07
q+a | ✓ | 57.23
q+a+t | ✓ | 52.02
q+a+v | ✓ | 68.93
q+a+s | ✓ | 74.13
q+a+t+v+s | ✓ | 37.72
Table 6: Accuracy on the Social-IQ-2.0 and DeSIQd-2.0 development sets.
Results shaded in gray are relevant to Sec. 3.

Zero-shot and Few-shot Learning. We employ the strong GPT-3 model Brown et al.
(2020) and ChatGPT OpenAI (2022) with both zero-shot and few-shot learning to
show the strength of our debiased datasets. Social-IQ experiments are
performed in the A2 configuration using GPT-3, while Social-IQ-2.0 experiments
in the A4 using ChatGPT. For zero-shot evaluation, we concatenate the question
with correct and incorrect answer options (i.e. “q+a”) to form the prompt444We
observe that when only given answer options but not the question, GPT-3 tends
to select the first option., where the order of the two answer options is
randomly shuffled. The zero-shot prompt is constructed as follows:

> “Choose the correct answer option corresponding to the question: ” + $q$ \+
> “ A: ” + $a$ \+ “ B: ” + $i$

For the few-shot evaluation, we use the question similarity to find exemplars
for in-context learning. For each question in the development set, we choose
the top-3 most similar questions from the training set. We measure the
semantic distance between questions based on their embeddings obtained from
Sentence-Transformers Reimers and Gurevych (2019). The few-shot prompts
leverage the same format as in the zero-shot evaluation, with the correct
option appended to each exemplar:

> “Choose the correct answer option corresponding to the question: ” +
> $(q^{\prime}$ \+ “ A: ” + $a^{\prime}$ \+ “ B: ” + $i^{\prime}$ \+ “A or
> B”$)$*3 + zero-shot prompt

Table 7 shows the results. For Social-IQ, under the zero-shot setting, GPT-3
can obtain 58.26% with “q+a” and 64.63% with “q+a+t” on Social-IQ. In
comparison, under either zero-shot or few-shot setting, both the DeSIQd and
DeSIQs dataset lead to a performance drop of more than $4\%$. Under the few-
shot setting for “q+a”, GPT-3 does not seem to learn _shortcuts_ , as the
performance is unchanged compared to the zero-shot setting555We could not
perform few-shot experiments with “q+a+t” due to GPT-3’s limit of prompt
length.. These results show that DeSIQd and DeSIQs are less biased and more
challenging than Social-IQ. For Social-IQ-2.0, the performance does not change
that much when leveraging ChatGPT under both zero-shot and few-shot learning
settings, which also proves it is less biased than Social-IQ.

Dataset | Input | A2 (GPT-3)/A4 (ChatGPT)
---|---|---
Zero-shot | Few-shot
Social-IQ | q+a | 58.26 | 56.22
q+a+t | 64.63 | -
DeSIQd | q+a | 54.78 | 54.13
q+a+t | 59.79 | -
DeSIQs | q+a | 54.39 | 53.29
q+a+t | 60.13 | -
DeSIQd-2.0 | q+a | 59.61 | 58.02
q+a+t | 59.24 | -
Table 7: GPT-3 performance on the A2 of Social-IQ and DeSIQ, ChatGPT
performance on the A4 of DeSIQ-2.0.

##  4 Setting Baseline Performance on DeSIQ

For our more challenging DeSIQ benchmark, we introduce a new baseline model to
better handle multi-modal inputs. Its architecture is shown in Figure 4.
Compared with the model in Figure 2, we add three more projection layers
(three yellow MLPs) to map the original feature representations into the same
dimensions. We then concatenate all the resulting representations as the
inputs to a backbone MPM. For DeSIQd-2.0 containing raw data, we employ Vision
Transformer (ViT) Dosovitskiy et al. (2021) and Wav2Vec 2.0 Baevski et al.
(2020) to obtain the video and audio representations respectively. We note
again that raw video and audio files are not available for v1, thus we develop
the above architecture to uniformly handle both datasets, and leave how to
best use multi-modal inputs in DeSIQ-2.0 for future work.

As social intelligence usually requires commonsense knowledge, we posit that
injecting commonsense knowledge into the backbone language model in our
architecture would improve the model’s performance. Therefore, inspired by
Jiang et al. (2021), we distill commonsense social knowledge from the
following datasets into T5-small: Social Chemistry 101 Forbes et al. (2020),
ETHICS Hendrycks et al. (2021) and Moral Stories Emelin et al. (2021).
Specifically, we pretrain T5-small on these corpora and then finetune it on
the downstream Social-IQ and DeSIQ datasets. We call this variant
T5-smallDelphi.

###  4.1 Results on DeSIQ

We analyze the effectiveness of our proposed architecture, and the effect of
the distillation of commonsense knowledge. The results of our new model
architecture are shown in the bottom portions of Tables 5 and 5, where the
inputs are concatenated (“✓” for the column “Concat”). We can make the
following observations:

  * •

Both T5-small and T5-smallDelphi outperform the LSTM baseline on both DeSIQd
and DeSIQs while not achieving near perfect accuracy, showing the
effectiveness of our proposed architecture as well as the unbiased nature of
DeSIQ.

  * •

When the question is given as part of the model input, T5-small and
T5-smallDelphi (✓) significantly outperform the vanilla versions (✗), showing
the effectiveness of our model architecture.

  * •

Injecting commonsense knowledge can indeed improve model performance on social
intelligence. T5-smallDelphi with “q+a+t” inputs shows the best A2 score as
76.77% and A4 and 74.51% on DeSIQd, and 67.70% in A2 on DeSIQs. On DeSIQd, it
outperforms T5-small in all but one settings (q+a+v for A2). On DeSIQs,
however, T5-small shows competitive performance, and significantly outperforms
T5-smallDelphi on A4 for both q+a+t. We leave the investigation of this result
to future work.

  * •

In many cases, adding the transcript can help improve model performance, and
usually more effective than adding the video modality. Since T5-smallDelphi is
pretrained on a textual corpus, it is reasonable that adding the video
modality may decrease model performance.

  * •

Compared to DeSIQd, DeSIQs is a more challenging dataset, as except for “a”,
performance of T5-small and T5-smallDelphi drops for all others.

  * •

Comparing the performance of q+a+t/q+a+v and q+a, we can observe that both
T5-small and T5-smallDelphi can learn some shortcuts, as they achieve
comparable performance when only given the question and answers as input.

Some examples are shown in Appendix A Figure 5, illustrating the influence of
different modalities. The first two examples show how the transcript and video
features may provide clues for answering the question. For instance, the first
example cannot be correctly answered based on “q+a”, since the transcript
contains the required information. T5-smallDelphi is the only model that
predicts correct options for the last example in Figure 5, which we attribute
to Delphi’s commonsense knowledge.

###  4.2 Results on DeSIQ-2.0

For DeSIQ-2.0, we can apply multi-modal model using the raw videos and audios.
The experimental results are in Table 6. Apart from some similar observations
on DeSIQ-1.0 above, some new conclusions can be made as follows:

  * •

Adding audios or videos can help improve model performance. Moreover, audios
are more effective as the model achieves overall best A4 score of 74.13% under
the “q+a+s” setting.

  * •

Employing raw transcripts can reduce model performance ($57.23\%\rightarrow
52.02\%$ ) as they are usually 5 times longer than other input features in
length, which can largely influence the representation learning procedure of
other inputs.

  * •

Compared with ChatGPT in Table 7, our best result outperforms 24.52% on A4,
which shows DeSIQ-2.0 to be a challenging dataset.

We also conduct experiments with settings “a+t” and “a+v”, but don’t include
them in the paper. After debiasing, both settings for the proposed model on
DeSIQ2.0 are near the random guess performance: “a+t” 22.66% and “a+v” 26.46%.
Thus, questions are necessary when compared with the performance of “q+a+t”
and “q+a+v” inputs in Table 6.

###  4.3 Further Research Questions

The above results show the lack of biases and challenging nature of our DeSIQ
datasets as well as promising performance by modestly-sized language models.
These results lead to the following important research directions for further
investigation:

  * •

Are there still noticeable biases in DeSIQ, and if so, how to further debias
it?

  * •

What is the performance of stronger language models on DeSIQ?

  * •

How to effectively incorporate socio-cultural and commonsense knowledge into
large language models for this task?

  * •

How to utilize multi-modal language models to better exploit video and audio
input?

##  5 Related Work

#### Debiasing.

Shah et al. (2020) proposed a number of _expectations_ to examine a model’s
performance on a number of multiple-choice QA datasets and observed that the
model (RoBERTa) falls short of the expectations. Different from this work, we
establish a systematic methodology, consisting of six novel methods, to
examine a dataset. And we design some experimental settings on both Social-IQ
and Social-IQ-2.0.

Language Dependence/Prior is actually a MODEL side bias resulting in the model
largely depending on one major modality (usually text). Reducing it can be
regarded as an optimization problem. Gat et al. (2020) try to balance the
influence of text and image from the MODEL side. Though the paper includes
Social-IQ dataset and gets positive results, it doesn’t realise the bias’s
existence in the original Social-IQ dataset.

Shortcut is a DATA side bias resulting in the model easily learning the
pattern/repeated word in one dataset. For example, some keywords can occur
both in the question and the correct answer, but not in the incorrect answers,
so that the model directly gets clues from this overlap. Ye and Kovashka
(2021) identify the shortcut and show its negative effects. However, they only
modify the validation data and propose a masking approach to perform more
robust training on the MODEL side.

In this paper, we start from the DATA side and also peform debiasing on the
DATA side. Moreover, the bias we identify in the Social-IQ dataset is not the
same kind, which is mainly in the answers and much harder to be debiased in
the DATA side. Thus, though they share some similarities, we consider it a new
task.

#### Multi-modal Question Answering.

With different multiple input modalities, such as image and video, multi-modal
question answering problem is more challenging and has been rising more and
more attention in the past few years. Datasets like MovieQA Tapaswi et al.
(2016), TGIF-QA Jang et al. (2017), TVQA Lei et al. (2018) and TVQA+ Lei et
al. (2020) provide images, GIFs or video clips in addition to text-based
single-turn questions. There are some datasets like AVSD Alamri et al. (2019)
that require dialogue history to predict answers for multi-turn questions. All
these datasets evaluate model capacity of perceive the contextual information
contained in both text and non-text modalities.

#### Social Intelligence Learning.

Understanding and reasoning about social commonsense knowledge and human
interactions is essential for cognitive social intelligence learning. Bosselut
et al. (2019) present a comprehensive study on automatic commonsense knowledge
base construction, which mines the intents and reasons behind human behaviors.
Jiang et al. (2021) propose a commonsense moral model to better understand
social norms and make reliable ethical judgments on real-world human actions.
In this paper, we focus on the Social-IQ dataset Zadeh et al. (2019), a
benchmark provides a diverse annotated set of videos and question-answer
pairs. We run all the experiments on this dataset because it is much more
related to social intelligence learning than other datasets.

##  6 Conclusion

Social intelligence is an essential ingredient for effective human-computer
communications. In this paper, we analyze Social-IQ, a multiple-choice
question answering benchmark dataset for social intelligence. Our empirical
analysis reveal the severe biases present in Social-IQ, which can be easily
exploited by modestly-sized language models such as T5-small to achieve
perfect accuracy on its development set. We construct the DeSIQ benchmark by
applying simple perturbation-based techniques on Social-IQ and show that the
DeSIQ vastly reduce the biases in Social-IQ. Moreover, we propose a new model
architecture on DeSIQ and set strong performance baselines for this
challenging new dataset. Finally, our comprehensive analyses open up a number
of important research questions for further investigation.

## Limitations

For the proposed model architecture designed to address the new DeSIQ
benchmark, we mainly employ text-based language models and pretrain them on
text-based corpora. The exploration of powerful multi-modal language models,
instead of using the projection function as is done in this paper, is thus an
important future research work direction. Due to resource constraints, all the
experiments in this work were under conducted only once with the same random
seed equals 42. Multiple runs with different random seeds would enable us to
performance statistical significance tests of the results, and thus make the
findings more reliable.

## Ethics Statement

Although the benchmark is designed for studying human behaviors and research
purposes only, the resources and findings could be used unexpectedly. For
example, it is possible that harmful content exists in the Social-IQ dataset,
thus also in our DeSIQ datasets, based on which trainable models could turn
from a positive to a negative perspective. Thus, it is prudent for researchers
working on social intelligence to pledge to only make ethical use of our
benchmark datasets.

## Acknowledgement

This material is based on research partially sponsored by the DARPA Assured
Neuro Symbolic Learning and Reasoning (ANSR) program under award number
FA8750-23-2-1016, the DARPA Knowledge Management at Scale and Speed (KMASS)
program under award number HR00112220047, and the DARPA Computational Cultural
Understanding (CCU) program under the agreement number HR001122C0029. The U.S.
Government is authorised to reproduce and distribute reprints for Governmental
purposes notwithstanding any copyright notation thereon. The authors are
grateful to the anonymous reviewers for their helpful comments.

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##  Appendix A Appendix

###  A.1 Experimental Settings

For all experiments, we fix our random seed at 42. We run fully supervised
learning on one A40 GPU with 40GB memory, and set the learning rate as 1e-4 as
well as early stopping by monitoring the loss on the development set.
Typically, it takes 2-3 hours using only features to finish 100-epoch training
with a batch size of 8.

We also employ GPT-3 (175B parameters) as it is the current state-of-the-art
language model in a variety of NLP tasks for in-context learning For the few-
shot evaluation, we select top-3 most similar questions from the training set.
We did not test other choice of k due to both budget and time constraints,
which we leave for future work.

Figure 5: Predictions of different models on DeSIQd benchmark. We show the
answers, question, transcript and video modalities in the first four columns,
and predictions of different models using our settings of input features. Red
ellipses are correct predictions based on certain features. Some explainable
clues are shaded in yellow.