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

DOI10.1007/S00526-022-02426-XarXiv2201.12307MaRDI QIDQ6389511

Josep M. Gallegos

Publication date: 28 January 2022

Abstract: Let $Omegasubsetmathbb{R}^d$ be a $C^1$ domain or, more generally, a Lipschitz domain with small Lipschitz constant and $A(x)$ be a $dimesd$ uniformly elliptic, symmetric matrix with Lipschitz coefficients. Assume $u$ is harmonic in $Omega$, or with greater generality $u$ solves $operatornamediv(A(x)ablau)=0$ in $Omega$, and $u$ vanishes on $Sigma=partialOmegacapB$ 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 $u{|}_{B_{i}capOmega}$ does not change sign and $$ has Minkowski dimension smaller than $d-1-epsilon$ for any compact $KsubsetSigma$. 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 $d-1-epsilon$.

Mathematics Subject Classification ID

Boundary value problems for second-order elliptic equations (35J25) Harmonic, subharmonic, superharmonic functions in higher dimensions (31B05) Boundary value and inverse problems for harmonic functions in higher dimensions (31B20) Quasilinear elliptic equations (35J62)

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