Size of the zero set of solutions of elliptic PDEs near the boundary of Lipschitz domains with small Lipschitz constant

TL;DR

This paper investigates the size and structure of the zero set of solutions to elliptic PDEs near Lipschitz domain boundaries with small Lipschitz constants, establishing bounds on the singular set and unique continuation properties.

Contribution

It provides new bounds on the Minkowski and Hausdorff dimensions of the zero set and singular set of elliptic PDE solutions near Lipschitz boundaries with small constants.

Findings

Bound on the Minkowski dimension of the singular set

Upper bounds for the Hausdorff measure of the zero set

A new boundary unique continuation principle

Abstract

Let $\Omega \subset \mathbb R^d$ be a $C^1$ domain or, more generally, a Lipschitz domain with small Lipschitz constant and $A(x)$ be a $d \times d$ uniformly elliptic, symmetric matrix with Lipschitz coefficients. Assume $u$ is harmonic in $\Omega$, or with greater generality $u$ solves $\operatorname{div}(A(x)\nabla u)=0$ in $\Omega$, and $u$ vanishes on $\Sigma = \partial\Omega \cap B$ for some ball $B$. We study the dimension of the singular set of $u$ in $\Sigma$, in particular we show that there is a countable family of open balls $(B_i)_i$ such that $u|_{B_i \cap \Omega}$ does not change sign and $K \backslash \bigcup_i B_i$ has Minkowski dimension smaller than $d-1-\epsilon$ for any compact $K \subset \Sigma$. We also find upper bounds for the $(d-1)$-dimensional Hausdorff measure of the zero set of $u$ in balls intersecting $\Sigma$ in terms of the frequency. As a consequence, we prove a new unique continuation principle at the boundary for this class of functions and show that the order of vanishing at all points of $\Sigma$ is bounded except for a set of Hausdorff dimension at most $d-1-\epsilon$.