Mixed local and nonlocal equations with measure data

Abstract

We study nonlinear measure data problems involving elliptic operators modeled after the mixed local and nonlocal p-Laplacian. We establish existence, regularity and Wolff potential estimates for solutions. As a consequence, we are able to obtain Calderón-Zygmund type estimates and continuity criteria for solutions.

Appendix A: Existence of weak solutions

Here we sketch the proof of the existence and uniqueness of weak solutions to (1.10) for the sake of completeness.

Proposition A.1

Let $\mu \in W^{-1,p'}(\Omega )$ and $g\in W^{1,p}({\mathbb {R}}^{n})$ be fixed. Under assumptions (1.2)–(1.4) with $p>1$ and $s\in (0,1)$, there exists a unique weak solution $u \in W^{1,p}({\mathbb {R}}^{n})$ to the Dirichlet problem (1.10).

Proof

Let us denote

$$\begin{aligned} {\mathcal {X}}^{1,p}_{g}(\Omega ) {:}{=}\left\{ u \in W^{1,p}({\mathbb {R}}^{n}): u-g \in {\mathcal {X}}^{1,p}_{0}(\Omega ) \right\} . \end{aligned}$$

We use standard monotonicity methods. We define the operator ${\mathcal {A}}:{\mathcal {X}}^{1,p}_{g}(\Omega ) \rightarrow ({\mathcal {X}}^{1,p}_{0}(\Omega ))^{*}$ by

$$\begin{aligned} \langle {\mathcal {A}}v,\varphi \rangle = \int _{\Omega }A(x,Dv)\cdot D\varphi \,dx + \int _{{\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\Phi _{p}(v(x)-v(y))(\varphi (x)-\varphi (y))K(x,y)\,dxdy \end{aligned}$$

for any $\varphi \in {\mathcal {X}}^{1,p}_{0}(\Omega )$, where $\langle \cdot ,\cdot \rangle$ denotes the standard bilinear paring. We then check the following:

• (1) For any $u,v\in {\mathcal {X}}^{1,p}_{g}(\Omega )$, it holds that $\langle {\mathcal {A}}u - {\mathcal {A}}v,u-v \rangle \ge 0$.

• (2) If $u_{j},u \in {\mathcal {X}}^{1,p}_{g}(\Omega )$ and $u_{j} \rightarrow u$ in ${\mathcal {X}}^{1,p}_{0}(\Omega )$, then

  $$\begin{aligned} \lim _{j\rightarrow \infty }\langle {\mathcal {A}}u_{j}-{\mathcal {A}}u,v\rangle = 0 \qquad \forall \;v\in {\mathcal {X}}^{1,p}_{0}(\Omega ). \end{aligned}$$

• (3) We have

  $$\begin{aligned} \lim _{\Vert Du \Vert _{L^{p}(\Omega )}\rightarrow \infty }\frac{\langle {\mathcal {A}}u-{\mathcal {A}}g,u-g\rangle }{\Vert Du \Vert _{L^{p}(\Omega )}} = \infty . \end{aligned}$$

Recalling (1.4), (2.1) and (2.2), it is straightforward to check that (1) holds:

$$\begin{aligned}&\langle {\mathcal {A}}u-{\mathcal {A}}v,u-v \rangle \\&= \int _{\Omega }(A(x,Du)-A(x,Dv))\cdot (Du-Dv)\,dx \\&\quad + \int _{{\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\left( \Phi _{p}(u(x)-u(y))-\Phi _{p}(v(x)-v(y))\right) (u(x)-v(x)-u(y)+v(y))K(x,y)\,dxdy \\&\approx \int _{\Omega }|V(Du)-V(Dv)|^{2}\,dx \\&\quad + \int _{{\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}(|u(x)-u(y)|+|v(x)-v(y)|)^{p-2}|u(x)-v(x)-u(y)+v(y)|^{2}\frac{dxdy}{|x-y|^{n+sp}} \\&\ge 0. \end{aligned}$$

In fact, we also have the strong monotonicity: $\langle {\mathcal {A}}u-{\mathcal {A}}v,u-v\rangle = 0$ if and only if $u=v$.

To show (2), we write

$$\begin{aligned}&|\langle {\mathcal {A}}u_{j}-{\mathcal {A}}u,v \rangle | \le \int _{\Omega }|A(x,Du_{j})-A(x,Du)|| Dv|\,dx \\&\quad + \int _{{\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}|\Phi _{p}(u_{j}(x)-u_{j}(y))-\Phi _{p}(u(x)-u(y))||v(x)-v(y)|K(x,y)\,dxdy. \end{aligned}$$

If $p > 2$, we have

$$\begin{aligned}&|\langle {\mathcal {A}}u_{j}-{\mathcal {A}}u,v \rangle |\\&\le c\int _{\Omega }(|Du_{j}|+|Du|)^{p-2}|Du_{j}-Du||Dv|\,dx \\&\quad + c\int _{{\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}(|u_{j}(x)-u_{j}(y)|+|u(x)-u(y)|)^{p-2}\\&\quad |u_{j}(x)-u_{j}(y)-u(x)+u(y)| |v(x)-v(y)| \frac{dxdy}{|x-y|^{n+sp}} \\&\le c\left( \int _{\Omega }(|Du_{j}|^{p}+|Du|^{p})\,dx\right) ^{\frac{p-2}{p}}\left( \int _{\Omega }|Du_{j}-Du|^{p}\,dx\right) ^{\frac{1}{p}}\left( \int _{\Omega }|Dv|^{p}\,dx\right) ^{\frac{1}{p}} \\&\quad + c\left( \int _{{\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\frac{|u_{j}(x)-u_{j}(y)|^{p}}{|x-y|^{n+sp}} + \frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\,dxdy\right) ^{\frac{p-2}{p}} \\&\qquad \quad \cdot \left( \int _{{\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\frac{|u_{j}(x)-u(x)-u_{j}(y)+u(y)|^{p}}{|x-y|^{n+sp}}\,dxdy\right) ^{\frac{1}{p}}\left( \int _{{\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}}\,dxdy\right) ^{\frac{1}{p}}. \end{aligned}$$

If $1<p\le 2$, we have

$$\begin{aligned}&|\langle {\mathcal {A}}u_{j}-{\mathcal {A}}u,v \rangle | \\&\le c\int _{\Omega }|Du_{j}-Du|^{p-1}|Dv|\,dx \\&\quad + c\int _{{\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}|u_{j}(x)-u_{j}(y)-u(x)+u(y)|^{p-1}|v(x)-v(y)|\frac{dxdy}{|x-y|^{n+sp}} \\&\le c\left( \int _{\Omega }|Du_{j}-Du|^{p}\,dx\right) ^{\frac{p-1}{p}}\left( \int _{\Omega }|Dv|^{p}\,dx\right) ^{\frac{1}{p}} \\&\quad + c\left( \int _{{\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\frac{|u_{j}(x)-u(x)-u_{j}(y)+u(y)|^{p}}{|x-y|^{n+sp}}\,dxdy\right) ^{\frac{p-1}{p}}\left( \int _{{\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}}\,dxdy\right) ^{\frac{1}{p}}. \end{aligned}$$

We now apply Lemma 2.2 to $u_{j}-u \in {\mathcal {X}}^{1,p}_{0}(\Omega )$ to observe that

$$\begin{aligned}{}[u_{j} - u]_{s,p;{\mathbb {R}}^{n}} \rightarrow 0 \qquad \text {as} \qquad \Vert u_{j}-u \Vert _{W^{1,p}(\Omega )} \rightarrow 0. \end{aligned}$$

Hence, in any case, (2) follows.

By using (1.2), (2.2) and Young’s inequality, we observe that

$$\begin{aligned} \langle {\mathcal {A}}u-{\mathcal {A}}g,u-g \rangle&= \int _{\Omega }A(x,Du)\cdot (Du-Dg)\,dx - \int _{\Omega }A(x,Dg)\cdot (Du-Dg)\,dx \\&\quad + \int _{{\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\Phi _{p}(u(x)-u(y))(u(x)-g(x)-u(y)+g(y))K(x,y)\,dxdy \\&\quad - \int _{{\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\Phi _{p}(g(x)-g(y))(u(x)-g(x)-u(y)+g(y))K(x,y)\,dxdy \\&\ge \frac{1}{c}\int _{\Omega }|Du|^{p}\,dx - c\int _{\Omega }|Dg|^{p}\,dx \\&\quad + \frac{1}{c}\int _{{\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\,dxdy - c\int _{{\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\frac{|g(x)-g(y)|^{p}}{|x-y|^{n+sp}}\,dxdy, \end{aligned}$$

from which (3) follows.

We now define the modified operator ${\mathcal {A}}_{0}:{\mathcal {X}}^{1,p}_{0}(\Omega )\rightarrow ({\mathcal {X}}^{1,p}_{0}(\Omega ))^{*}$ by

$$\begin{aligned} {\mathcal {A}}_{0}v {:}{=}{\mathcal {A}}(v+g), \qquad v \in {\mathcal {X}}^{1,p}_{0}(\Omega ). \end{aligned}$$

Then the properties (1)–(3) imply that ${\mathcal {A}}_{0}$ is monotone, hemicontinuous and coercive (see [47, Chapter II, Section 2] for the relevant definitions). Moreover, we have $\mu \in W^{-1,p'}(\Omega ) \subset ({\mathcal {X}}^{1,p}_{0}(\Omega ))^{*}$. Then, by [47, Corollary 2.2], there exists $v \in {\mathcal {X}}^{1,p}_{0}(\Omega )$ such that

$$\begin{aligned} \langle {\mathcal {A}}_{0}v,\varphi \rangle = \langle \mu ,\varphi \rangle \qquad \forall \;\varphi \in {\mathcal {X}}^{1,p}_{0}(\Omega ), \end{aligned}$$

which is equivalent to

$$\begin{aligned} \langle {\mathcal {A}}(v+g),\varphi \rangle = \langle \mu ,\varphi \rangle \qquad \forall \;\varphi \in {\mathcal {X}}^{1,p}_{0}(\Omega ). \end{aligned}$$

Therefore, $u {:}{=}v+g$ is a weak solution to (1.10) as desired. Also, the uniqueness easily follows from the strict monotonicity of ${\mathcal {A}}_{0}$. $\square$