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

TL;DR

This paper proves radial symmetry for positive solutions of mixed local-nonlocal equations with singular nonlinearities and introduces quantitative symmetry results, including new inequalities, advancing the understanding of these complex equations.

Contribution

It establishes the first quantitative symmetry results and inequalities for solutions of mixed local-nonlocal equations with singular nonlinearities.

Findings

Proved radial symmetry of solutions using the moving plane method.

Developed a weak Harnack inequality for mixed equations.

Established an analogue of the Alexandroff-Bakelman-Pucci inequality in this setting.

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.