On the nodal set of solutions to degenerate or singular elliptic equations with an application to $s-$harmonic functions

TL;DR

This paper studies the geometric structure of the zero set of solutions to certain degenerate or singular elliptic equations, including fractional Laplacian extensions, providing a comprehensive stratification theory.

Contribution

It develops a complete stratification theory for the nodal sets of solutions to degenerate or singular elliptic equations involving the operator $L_a$, extending classical results to this class.

Findings

Established stratification properties of nodal sets

Extended geometric analysis to fractional Laplacian related equations

Provided new insights into the structure of solutions' zero sets

Abstract

This work is devoted to the geometric-theoretic analysis of the nodal set of solutions to degenerate or singular equations involving a class of operators including $$ L_a = \mbox{div}(\abs{y}^a \nabla), $$ with $a\in(-1,1)$ and their perturbations. As they belong to the Muckenhoupt class $A_2$, these operators appear in the seminal works of Fabes, Kenig, Jerison and Serapioni \cite{fkj,fjk2,fks} and have recently attracted a lot of attention in the last decade due to their link to the localization of the fractional Laplacian via the extension in one more dimension \cite{CS2007}. Our goal in the present paper is to develop a complete theory of the stratification properties for the nodal set of solutions of such equations in the spirit of the seminal works of Hardt, Simon, Han and Lin \cite{MR1010169,MR1305956,MR1090434}.