A global second-order Sobolev regularity for p-Laplacian type equations with variable coefficients in bounded domains

Abstract

Let $\Omega \subset {{\mathbb {R}}}^n$ be a bounded convex domain with $n\ge 2$. Suppose that A is uniformly elliptic and belongs to $W^{1,n}$ when $n\ge 3$ or $W^{1,q}$ for some $q>2$ when $n=2$. For $1<p<\infty$, we establish a global second-order regularity estimate

$$\begin{aligned}\Vert D[|Du|^{p-2} Du]\Vert _{L^2(\Omega )}+\Vert D[ \langle ADu,Du\rangle ^{\frac{p-2}{2} }A Du]\Vert _{L^2(\Omega )} \le C \Vert f\Vert _{L^2(\Omega )} \end{aligned}$$

for the inhomogeneous p-Laplace type equation

$$\begin{aligned} -\textrm{div}\big (\langle A Du,Du\rangle ^{\frac{p-2}{2}} A Du\big )=f \end{aligned}$$

in $\Omega$ with Dirichlet or Neumann homogeneous boundary condition. Similar result was also established for certain bounded Lipschitz domains whose boundary is weakly second-order differentiable and satisfies some smallness assumptions.

Appendix A: Proof of Lemma 7.1

To prove Lemma 7.1, given any bounded uniform domain $\Omega$, below we briefly recall the construction of the extension operator $\Lambda : \dot{W}^{1,q} (\Omega )\rightarrow \dot{W}^{1,q} ({{{{\mathbb {R}}}}^n})$ by Jones [12] (see also [13]). For $1\le q<\infty$, denote by $\dot{W}^{1,q} (\Omega )$ the homogeneous Sobolev space in any domain $\Omega \subset {{{{\mathbb {R}}}}^n}$, that is, the collection of all function $v\in L^q_{\mathop \mathrm {\,loc\,}}(\Omega )$ with its distributional derivative $Dv\in L^q(\Omega )$.

Recall that $\Omega$ is an $\epsilon _0$-uniform domain for some $\epsilon _0>0$ if for any $x,y\in \Omega$ one can find a rectifiable curve ${\gamma }:[0,T] \rightarrow \Omega$ joining x, y so that

$$\begin{aligned} T=\ell ({\gamma }) \le \frac{1}{\epsilon _0} |x-y| \,\hbox {and}\, {\mathop \mathrm {\,dist\,}}({\gamma }(t),\partial \Omega )\ge \epsilon _0\min \{t,T-t\}\quad \forall t\in [0,T], \end{aligned}$$

where C is a constant. Note that $|\partial \Omega |=0$. It is well-known that Lipschitz domains are always $\epsilon _0$-uniform domains, where $\epsilon _0$ depends on Lipschitz constant of $\Omega$. In the case $\Omega$ is convex, $\epsilon _0$ depends on ${\mathop \mathrm {\,diam\,}}\Omega$ and $|\Omega |$.

Denote by $W_1=\{S_j\}$ the Whitney decomposition of $\Omega$ and $W_2=\{Q_j\}$ as the Whitney decomposition of $({\overline{\Omega }})^\complement$ as [13, Section 2]. Set also $W_3=\{Q\in W_2, \ell (Q)\le \frac{\epsilon _0}{16n}{\mathop \mathrm {\,diam\,}}\Omega \}$ as [13, Section 2]. By Jones and also [13], any cube $Q\in W_3$ has a reflection cube $Q^*\in W_1$ such that $\ell (Q)\le \ell (Q ^*)\le 4\ell (Q)$ and hence ${\mathop \mathrm {\,dist\,}}(Q^*,Q)\le C\ell (Q)$ for some constant $C\ge 1$ depending only on $\epsilon _0$ and n. For any $Q\in W_2\setminus W_3$ we just write $Q^*=\Omega$.

Let $\{\varphi _Q\}_{Q\in W_2}$ be a partition of unit associated to $W_2$ so that ${\text { supp}}\varphi _Q\subset \frac{17}{16}Q$. The extension operator is then defined by

Such extension operator is a slight modification of that in [13] and also [12].

By essentially the argument of Jones [12] (see also [13]), for $1\le q<\infty$ one has that $\Lambda :\dot{W}^{1,q} (\Omega )\rightarrow \dot{W}^{1,q} (\Omega )$ is a linear bounded extension operator, that is, for any $v\in W^{1,q}(\Omega )$ we have $\Lambda v\in \dot{W}^{1,q}({{{{\mathbb {R}}}}^n})$ so that $\Lambda v|_\Omega =v$ and $\Vert D \Lambda v\Vert _{L^q({{{{\mathbb {R}}}}^n})}\le C\Vert Dv\Vert _{L^q(\Omega )}$ for some C depending on $n,\epsilon _0$ and q.

Moreover, by the arguments in [13], for any $x\in {\overline{\Omega }}$ and $r\le \frac{\epsilon _0}{16n}{\mathop \mathrm {\,diam\,}}\Omega$, one has $\Vert D \Lambda v\Vert _{L^n(B(x,r))}\le C\Vert Dv\Vert _{L^n(\Omega \cap B({\bar{x}},Cr))}.$ Indeed, the choice of r implies that $B(x,r)\cap Q=\emptyset$ for any $Q\in W_2\setminus W_3$ and hence one only need to bound $\textbf{H}_{1,1}$ in [13, P.1422] and $\textbf{H}_{1,2}=0$ and $\textbf{H}_2=0$ in [13, P.1422]. Thus $\Vert D \Lambda v\Vert _{L^n(B(x,r){\setminus } \Omega )}\le C\Vert Dv\Vert _{L^n(\Omega \cap B({\bar{x}},Cr))}$. Moreover, for any $x\notin {\overline{\Omega }}$, denote by ${\bar{x}}\in \partial \Omega$ is the nearest point of x. If ${\mathop \mathrm {\,dist\,}}(x,\partial \Omega )<r<{\mathop \mathrm {\,diam\,}}\Omega$, one has

$$\begin{aligned} \Vert D \Lambda v\Vert _{L^q(B(x,r))}\le \Vert D \Lambda v\Vert _{L^q(B({\bar{x}},2r))} \le C\Vert Dv\Vert _{L^q(\Omega \cap B({\bar{x}},Cr))}. \end{aligned}$$

Proof of Lemma 7.1

Let $A=(a_{ij})\in {\mathcal {E}}_L(\Omega )$ with $DA\in L^q(\Omega )$ with $q\ge n$. Write $\widetilde{A}=(\Lambda a_{ij})$. By the boundedness of $\Lambda$, we have $\Vert D\widetilde{A}\Vert _{L^q({{{{\mathbb {R}}}}^n})}<C\Vert DA\Vert _{L^q(\Omega )}$. Noting

we know that $\widetilde{A}\in {\mathcal {E}}_L ({{{{\mathbb {R}}}}^n})$. Moreover in the case $\Vert DA\Vert _{L^n(\Omega )}<\infty$, we have $\Phi _{\widetilde{A}}(\Omega _\eta ,r)\le C\Phi _{ A}(\Omega , Cr)$ whenever $x\in \Omega _\eta$ and $0< \eta<r<{\mathop \mathrm {\,diam\,}}\Omega$.

For $\epsilon >0$, $A^\epsilon = \widetilde{A}*\eta _\epsilon$, where $\eta _\epsilon$ is the standard smooth mollifier. Since $\langle \widetilde{A}*\eta (x)\xi , \xi \rangle =\langle \widetilde{A} \xi , \xi \rangle *\eta (x)$, we know that $A^\epsilon \in {\mathcal {E}}_L({{{{\mathbb {R}}}}^n})$. Moreover, in the case $\Vert DA\Vert _{L^n(\Omega )}<\infty$, we have $\Vert \widetilde{A}*\eta _\epsilon \Vert _{L^n(B(x,r))}\le \Vert \widetilde{A} \Vert _{L^n(B(x,r+\epsilon ))}$. For any $x\in \Omega _\eta$ and $0< \epsilon \le \eta<r<{\mathop \mathrm {\,diam\,}}\Omega$, we know that $\Phi _{ A^\epsilon }(\Omega _\eta ,r)\le \Phi _{ \widetilde{A} }(\Omega _{\epsilon +\eta },r+\epsilon )$. and hence $\Phi _{ A^\epsilon }(\Omega _\eta ,r)\le C\Phi _{ A}(\Omega , Cr)$ as desired. $\square$