The nodal set of solutions to some elliptic problems: sublinear equations, and unstable two-phase membrane problem

TL;DR

This paper investigates the structure and regularity of the nodal set of solutions to certain elliptic equations with sublinear and two-phase membrane problems, revealing finiteness of vanishing order, regularity properties, and stratification of the nodal set.

Contribution

It provides a comprehensive analysis of the nodal set for sublinear and two-phase membrane elliptic equations, including regularity, order spectrum, and stratification results, which were previously not fully understood.

Findings

Finiteness of vanishing order at every point

Nodal set is a union of regular manifolds with a small singular set

Admissible vanishing orders are bounded by a critical value

Abstract

We are concerned with the nodal set of solutions to equations of the form \begin{equation*} -\Delta u = \lambda_+ \left(u^+\right)^{q-1} - \lambda_- \left(u^-\right)^{q-1} \quad \text{in $B_1$} \end{equation*} where $\lambda_+,\lambda_- > 0$, $q \in [1,2)$, $B_1=B_1(0)$ is the unit ball in $\mathbb{R}^N$, $N \ge 2$, and $u^+:= \max\{u,0\}$, $u^-:= \max\{-u,0\}$ are the positive and the negative part of $u$, respectively. This class includes, the \emph{unstable two-phase membrane problem} ($q=1$), as well as \emph{sublinear} equations for $1<q<2$. We prove the following main results: (a) the finiteness of the vanishing order at every point and the complete characterization of the order spectrum; (b) a weak non-degeneracy property; (c) regularity of the nodal set of any solution: the nodal set is a locally finite collection of regular codimension one manifolds up to a residual singular set having Hausdorff dimension at most $N-2$ (locally finite when $N=2$); (d) a partial stratification theorem. Ultimately, the main features of the nodal set are strictly related with those of the solutions to linear (or superlinear) equations, with two remarkable differences. First of all, the admissible vanishing orders can not exceed the critical value $2/(2-q)$. At threshold, we find a multiplicity of homogeneous solutions, yielding the \emph{non-validity} of any estimate of the $(N-1)$-dimensional measure of the nodal set of a solution in terms of the vanishing order. The proofs are based on monotonicity formul\ae \ for a $2$-parameter family of Weiss-type functionals, blow-up arguments, and the classification of homogenous solutions.