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arXiv · 1808.01851

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

Yannick Sire·Susanna Terracini·Giorgio Tortone
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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}.

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Yannick Sire, Susanna Terracini, Giorgio Tortone. 2018-08-06. On the nodal set of solutions to degenerate or singular elliptic equations with an application to $s-$harmonic functions. https://arxiv.org/abs/1808.01851

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