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

Let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ 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 $\mathrm{\Omega }$, or with greater generality $u$ solves $\mathrm{div}(A(x)\mathrm{\nabla }u)=0$ in $\mathrm{\Omega }$, and $u$ vanishes on $\Sigma = \partial\Omega \cap B$forsomeball$B$. We study the dimension of the singular set of $u$ in $\mathrm{\Sigma }$, in particular we show that there is a countable family of open balls $(B_i{)}_i$ such that $u{\mathrm{\mid }}_{B_i\cap \mathrm{\Omega }}$ does not change sign and $K\mathrm{\backslash }{\bigcup }_i{B}_i$ has Minkowski dimension smaller than $d-1-ϵ$ for any compact $K\subset \mathrm{\Sigma }$. We also find upper bounds for the $(d-1)$-dimensional Hausdorff measure of the zero set of $u$ in balls intersecting $\mathrm{\Sigma }$ 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 $\mathrm{\Sigma }$ is bounded except for a set of Hausdorff dimension at most $d-1-ϵ$.