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Symmetry and Approximate Symmetry for Solutions of Mixed Local-Nonlocal Singular Equations

Sanjit Biswas

Abstract

In this article, we establish radial symmetry for positive weak solutions of a class of mixed local-nonlocal equations with possibly singular nonlinearity via the moving plane method. Furthermore, we provide a quantitative version of Gidas-Ni-Nirenberg type theorem for mixed local-nonlocal equations. To this regard, we establish a weak Harnack-type inequality and an analogue of the Alexandroff-Bakelman-Pucci inequality in the mixed nonhomogeneous setting with a lower order term, which appear to be new. To the best of our knowledge, this paper initiates the study of the quantitative properties of solutions to mixed problems.

Symmetry and Approximate Symmetry for Solutions of Mixed Local-Nonlocal Singular Equations

Abstract

In this article, we establish radial symmetry for positive weak solutions of a class of mixed local-nonlocal equations with possibly singular nonlinearity via the moving plane method. Furthermore, we provide a quantitative version of Gidas-Ni-Nirenberg type theorem for mixed local-nonlocal equations. To this regard, we establish a weak Harnack-type inequality and an analogue of the Alexandroff-Bakelman-Pucci inequality in the mixed nonhomogeneous setting with a lower order term, which appear to be new. To the best of our knowledge, this paper initiates the study of the quantitative properties of solutions to mixed problems.
Paper Structure (8 sections, 15 theorems, 154 equations)

This paper contains 8 sections, 15 theorems, 154 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Functional analytical setting
  4. Weak Harnack estimate
  5. Auxiliary result
  6. Preliminaries for Theorem \ref{['T1']}
  7. Proof of Theorem \ref{['T1']}
  8. Proof of Theorem \ref{['T2']}

Key Result

Theorem 1.1

(Symmetry) Let $\Omega\subset\mathbb{R}^n$ be a smooth domain that is convex with respect to the $x_1$-direction and symmetric with respect to the hyperplane $\{x_1=0\}$. Assume that $\Gamma\subset \{x\in\Omega: x_1=0\}$ is a closed set such that $\Gamma$ is a point when $n=2$ and $\mathrm{cap}_2(\G and Suppose that $u\in H^1_{\mathrm{loc}}(\Omega\setminus\Gamma)\cap C(\overline{\Omega}\setminus\

Theorems & Definitions (30)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Remark 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 20 more