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

Abstract

Let Ω⊂Rd be a C1 domain or, more generally, a Lipschitz domain with small Lipschitz constant and A(x) be a d×d uniformly elliptic, symmetric matrix with Lipschitz coefficients. Assume u is harmonic in Ω, or with greater generality u solves div(A(x)∇u)=0 in Ω, and u vanishes on Σ=∂Ω∩B for some ball B. We study the dimension of the singular set of u in Σ, in particular we show that there is a countable family of open balls (Bi)i such that u|Bi∩Ω does not change sign and K∖⋃iBi has Minkowski dimension smaller than d−1−ϵ for any compact K⊂Σ. We also find upper bounds for the (d−1)-dimensional Hausdorff measure of the zero set of u in balls intersecting Σ 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 Σ is bounded except for a set of Hausdorff dimension at most d−1−ϵ.