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

Abstract

AbstractLet$$\Omega \subset {\mathbb {R}}^d$$Ω⊂Rdbe a$$C^1$$C1domain or, more generally, a Lipschitz domain with small Lipschitz constant andA(x) be a$$d \times d$$d×duniformly elliptic, symmetric matrix with Lipschitz coefficients. Assumeuis harmonic in$$\Omega $$Ω, or with greater generalityusolves$${\text {div}}(A(x)\nabla u)=0$$div(A(x)∇u)=0in$$\Omega $$Ω, anduvanishes on$$\Sigma = \partial \Omega \cap B$$Σ=∂Ω∩Bfor some ballB. We study thedimension of the singular setofuin$$\Sigma $$Σ, in particular we show that there is a countable family of open balls$$(B_i)_i$$(Bi)isuch that$$u|_{B_i \cap \Omega }$$u|Bi∩Ωdoes not change sign and$$K \backslash \bigcup _i B_i$$K\⋃iBihas Minkowski dimension smaller than$$d-1-\epsilon $$d-1-ϵfor any compact$$K \subset \Sigma $$K⊂Σ. We also find upper bounds for the$$(d-1)$$(d-1)-dimensional Hausdorff measure of the zero set ofuin balls intersecting$$\Sigma $$Σin terms of the frequency. As a consequence, we prove a newunique continuation principle at the boundaryfor this class of functions and show that theorder of vanishingat all points of$$\Sigma $$Σis bounded except for a set of Hausdorff dimension at most$$d-1-\epsilon $$d-1-ϵ.