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

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 $d-1-\epsilon$.