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On equivalence of weak and viscosity solutions to nonlocal double phase problems with nonhomogeneous data

Sekhar Ghosh, R. Lakshmi, Chao Zhang

Abstract

This work focuses on the nonhomogeneous nonlocal double phase problem \begin{align*} L_au(x)=f(x,u,D_s^p u, D_{a,t}^q u) \text{ in } Ω, \end{align*} where $Ω\subset\mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $0<s,t<1<p\leq q<\infty$ with $tq\leq sp$ and the operator $L_a$ is defined as \begin{align*} L_a u(x)&=2\operatorname{P.V.}\int_{\mathbb{R}^N}|u(x)-u(y)|^{p-2}(u(x)-u(y))K_{s,p}(x,y) &\ \ \ +2\operatorname{P.V.}\int_{\mathbb{R}^N}a(x,y)|u(x)-u(y)|^{q-2}(u(x)-u(y))K_{t,q}(x,y)dy. \end{align*} We establish the equivalence between weak and viscosity solutions under boundedness and continuity assumptions. In addition, the local boundedness of weak solutions in some special cases on $f$ is also obtained using the notion of De Giorgi classes.

On equivalence of weak and viscosity solutions to nonlocal double phase problems with nonhomogeneous data

Abstract

This work focuses on the nonhomogeneous nonlocal double phase problem \begin{align*} L_au(x)=f(x,u,D_s^p u, D_{a,t}^q u) \text{ in } Ω, \end{align*} where is a bounded domain with Lipschitz boundary, with and the operator is defined as \begin{align*} L_a u(x)&=2\operatorname{P.V.}\int_{\mathbb{R}^N}|u(x)-u(y)|^{p-2}(u(x)-u(y))K_{s,p}(x,y) &\ \ \ +2\operatorname{P.V.}\int_{\mathbb{R}^N}a(x,y)|u(x)-u(y)|^{q-2}(u(x)-u(y))K_{t,q}(x,y)dy. \end{align*} We establish the equivalence between weak and viscosity solutions under boundedness and continuity assumptions. In addition, the local boundedness of weak solutions in some special cases on is also obtained using the notion of De Giorgi classes.

Paper Structure

This paper contains 6 sections, 17 theorems, 310 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and space setup
  3. Comparison principle
  4. Weak solutions are viscosity solutions
  5. Viscosity solutions are weak solutions
  6. Local Boundedness of weak solutions

Key Result

Theorem 1.1

Let $u$ be a continuous weak supersolution to problem D and let the comparison principle (Definition D-CP) hold. Assume that the map $f=f(x,t,\zeta,\eta)$ is continuous in $x$ and $t$ and is Lipschitz continuous in $\zeta$ and $\eta$. Then, $u$ is a viscosity supersolution to problem D.

Theorems & Definitions (35)

  • Definition 1.1: Comparison Principle
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 25 more