A remark on quantitative unique continuation from subsets of the boundary of positive measure

TL;DR

This paper establishes quantitative unique continuation results for harmonic functions with boundary data vanishing on subsets of positive measure, extending previous qualitative results to a more explicit, measurable context in certain domains.

Contribution

It provides the first quantitative estimates for unique continuation from boundary subsets of positive measure in W 2,∞ domains, using recent advances by Logunov and Malinnikova.

Findings

Quantitative unique continuation results for harmonic functions with boundary conditions.

Explicit estimates for Laplace eigenfunctions on W 2,∞ domains.

Extension of qualitative boundary unique continuation to a quantitative setting.

Abstract

The question of unique continuation of harmonic functions in a domain $\Omega$ $\subset$ R d with boundary $\partial$$\Omega$, satisfying Dirichlet boundary conditions and with normal derivatives vanishing on a subset $\omega$ of the boundary is a classical problem. When $\omega$ contains an open subset of the boundary it is a consequence of Carleman estimates (uniqueness for second order elliptic operators across an hypersurface). The case where $\omega$ is a set of positive (d -- 1) dimensional measure has attracted a lot of attention, see e.g. [10, 3, 15], where qualitative results have been obtained in various situations. The main open questions (about uniqueness) concern now Lipschitz domains and variable coefficients. Here, using results by Logunov and Malinnikova [13, 14], we consider the simpler case of W 2,$\infty$ domains but prove quantitative uniqueness both for Dirichlet and Neumann boundary conditions. As an application, we deduce quantitative estimates for the Dirichlet and Neumann Laplace eigenfunctions on a W 2,$\infty$ domain with boundary.