Calculus of Variations and Geometric Measure Theory

Y. Sire - S. Terracini - G. Tortone

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

created by tortone on 01 Nov 2023


Published Paper

Inserted: 1 nov 2023
Last Updated: 1 nov 2023

Journal: Journal de Mathématiques Pures et Appliquées
Year: 2020
Doi: 10.1016/j.matpur.2020.01.010

ArXiv: 1808.01851 PDF


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 $$ La = \mbox{div}(\abs{y}a \nabla), $$ 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}.