B_{R_{1}}(x_{0}),y\in V}
      {\left|
        {D}_{y}
        {{\ifthenelse{\equal{*}{*}}{({u}(x,\tau)-h)_{+}}{\left({u}(x,\tau)-h\right)_{+}}}}^{\frac{p+\theta-1}{p}}
      \right|}^{p}
      {\omega}(x,y)
      {\,\textup{\textmd{d}}}\tau
      \\
      &\qquad\le
      \frac{\gamma}{\tau_{1}-\tau_{2}}
      \int_{\tau_{2}}^{t}
      \sum_{x\in B_{R_{2}}(x_{0})}
      {{\ifthenelse{\equal{*}{*}}{({u}(x,\tau)-h)_{+}}{\left({u}(x,\tau)-h\right)_{+}}}}^{\theta+1}
      {d_{{\omega}}}(x)
      {\,\textup{\textmd{d}}}\tau
      +
      \gamma
      A^{\frac{1}{p}}
      B^{\frac{p-1}{p}}
      +
      \gamma
      A
      \,,
    \end{split}$$ where $$\begin{aligned}
    A&=
       \frac{1}{(R_{2}-R_{1})^{p}}
       \int_{\tau_{2}}^{t}
       \sum_{x\in B_{R_{2}}(x_{0})}
       {{\ifthenelse{\equal{*}{*}}{({u}(x,\tau)-k)_{+}}{\left({u}(x,\tau)-k\right)_{+}}}}^{p+\theta-1}
       {d_{{\omega}}}(x)
       {\,\textup{\textmd{d}}}\tau
       \,,
    \\
    B&=
       h^{p}
       (h-k)^{\theta-1}
       \int_{\tau_{2}}^{t}
       {\mu_{{\omega}}}(B_{R_{2}}(x_{0})\cap\{2h\ge {u}(x,\tau)>h\})
       {\,\textup{\textmd{d}}}\tau
       \,.
  \end{aligned}$$

\[r:caccio\] The term $A^{1/p}B^{(p-1)/p}$ in can be reduced to one containing only $A$ by means of Young’s and Chebychev’s inequalities.

We multiply against $\zeta(x)^{p}\eta(t)^{p}{{\ifthenelse{\equal{*}{*}}{({u}(x,t)-h)_{+}}{\left({u}(x,t)-h\right)_{+}}}}^{\theta}$ and apply the well known formula of integration by parts $$\begin{gathered}
    \sum_{x,y\in V}
    {\lvert{D}_{y}{u}(x)\rvert}^{p-2}
    {D}_{y}{u}(x)
    f(x)
    {\omega}(x,y)
    \\=
    -
    \frac{1}{2}
    \sum_{x,y\in V}
    {\lvert{D}_{y}{u}(x)\rvert}^{p-2}
    {D}_{y}{u}(x)
    {D}_{y}f(x)
    {\omega}(x,y)
    \,,
  \end{gathered}$$ where $f:V\to {{\boldsymbol}{R}}$ has finite support. Below we denote $B_{R}(x_{0})=B_{R}$ for simplicity of notation.

We obtain $$\begin{gathered}
    \label{eq:cacc_i}
    J_{1}+J_{2}
    :=
    \frac{1}{\theta+1}
    \sum_{x\in B_{R_{2}}}
    {{\ifthenelse{\equal{*}{*}}{({u}(x,t)-h)_{+}}{\left({u}(x,t)-h\right)_{+}}}}^{\theta+1}
    \zeta(x)^{p}
    \eta(t)^{p}
    {d_{{\omega}}}(x)
    \\
    +
    \frac{1}{2}
    \int_{0}^{t}
    \sum_{x,y\in V}
    {\lvert{D}_{y}{u}(x,\tau)\rvert}^{p-2}
    {D}_{y}{u}(x,\tau)
    {D}_{y}[
    {{\ifthenelse{\equal{*}{*}}{({u}(x,\tau)-h)_{+}}{\left({u}(x,\tau)-h\right)_{+}}}}^{\theta}
    \zeta(x)^{p}
    ]
    {\omega}(x,y)
    \eta(\tau)^{p}
    {\,\textup{\textmd{d}}}\tau
    \\
    =
    \frac{p}{\theta+1}
    \int_{0}^{t}
    \sum_{x\in B_{R_{2}}}
    {{\ifthenelse{\equal{*}{*}}{({u}(x,\tau)-h)_{+}}{\left({u}(x,\tau)-h\right)_{+}}}}^{\theta+1}
    \zeta(x)^{p}
    \eta(\tau)^{p-1}
    \eta'(\tau)
    {d_{{\omega}}}(x)
    {\,\textup{\textmd{d}}}\tau
    =:J_{3}
    \,.
  \end{gathered}$$ We split $J_{2}$ according to the equality $${D}_{y}[
    {{\ifthenelse{\equal{*}{*}}{({u}(x,\tau)-h)_{+}}{\left({u}(x,\tau)-h\right)_{+}}}}^{\theta}
    \zeta(x)^{p}
    ]
    =
    \zeta(y)^{p}
    {D}_{y}
    {{\ifthenelse{\equal{*}{*}}{({u}(x,\tau)-h)_{+}}{\left({u}(x,\tau)-h\right)_{+}}}}^{\theta}
    +
    {{\ifthenelse{\equal{*}{*}}{({u}(x,\tau)-h)_{+}}{\left({u}(x,\tau)-h\right)_{+}}}}^{\theta}
    {D}_{y} \zeta(x)^{p}
    \,.$$ Next we appeal to Lemma \[l:monot\] with ${v}=0$ to get $$\label{eq:cacc_iii}
    {\lvert{D}_{y}{u}(x,\tau)\rvert}^{p-2}
    {D}_{y}{u}(x,\tau)
    {D}_{y}[
    {{\ifthenelse{\equal{*}{*}}{({u}(x,\tau)-h)_{+}}{\left({u}(x,\tau)-h\right)_{+}}}}^{\theta}
    ]
    \ge
    \gamma_{0}
    {\left|
      {D}_{y}
      {{\ifthenelse{\equal{*}{*}}{({u}(x,\tau)-h)_{+}}{\left({u}(x,\tau)-h\right)_{+}}}}^{\frac{p+\theta-1}{p}}
    \right|}^{p}
    \,.$$ Thus 