Nodal Sets for "Broken" Quasilinear PDEs

TL;DR

This paper investigates the local structure of nodal sets of solutions to a class of elliptic quasilinear PDEs with nonlinear conductivity, revealing smoothness and structural properties of these sets even at degenerate points.

Contribution

It provides new geometric insights into the behavior of nodal sets for broken quasilinear PDEs, extending classical results to more complex nonlinear cases.

Findings

Nodal sets where solutions are nondegenerate are locally smooth graphs.

Degenerate points exhibit structures similar to harmonic functions.

Almost complete characterization of nodal set behavior for these PDEs.

Abstract

We study the local behavior of the nodal sets of the solutions to elliptic quasilinear equations with nonlinear conductivity part, \begin{equation*} \operatorname{div}(A_s(x,u)\nabla u)=\operatorname{div}{\vec f}(x), \end{equation*} where $A_s(x,u)$ has "broken" derivatives of order $s\geq 0$, such as \begin{equation*} A_s(x,u) = a(x) + b(x)(u^+)^s, \end{equation*} with $(u^+)^0$ being understood as the characteristic function on $\{u>0\}$. The vector ${\vec f}(x)$ is assumed to be $C^\alpha$ in case $s=0$, and $C^{1,\alpha}$ (or higher) in case $s>0$. Using geometric methods, we prove almost complete results (in analogy with standard PDEs) concerning the behavior of the nodal sets. More exactly, we show that the nodal sets, where solutions have (linear) nondegeneracy, are locally smooth graphs. Degenerate points are shown to have structures that follow the lines of arguments as that of the nodal sets for harmonic functions, and general PDEs.