Quantitative unique continuation for Robin boundary value problems on $C^{1,1}$ domains

TL;DR

This paper establishes new quantitative unique continuation results for second order elliptic equations with Robin boundary conditions on $C^{1,1}$ domains, extending previous results to more general potentials and boundary behaviors.

Contribution

It provides the first sharp vanishing order estimate for Robin problems with variable coefficients and potentials, and generalizes boundary unique continuation results to Robin boundary conditions.

Findings

Sharp vanishing order estimate for Robin problems with Lipschitz coefficients.

Unique continuation from the boundary for solutions with Robin boundary conditions.

Extension of previous results to more general potentials and boundary behaviors.

Abstract

In this paper, we prove two unique continuation results for second order elliptic equations with Robin boundary conditions on $C^{1,1}$ domains. The first one is a sharp vanishing order estimate of Robin problems with Lipschitz coefficients and differentiable, sign-changing potentials. This generalizes the result for the "Robin eigenfunctions" in [26], which deals with the case with constant potentials. The second result is a unique continuation result from the boundary -- any non-trivial solution cannot vanish at infinite order from the boundary or vanish on an open subset on the boundary. Such result generalizes the one in [1] for the Laplace equation on $C^{1,1}$ domains with zero Neumann boundary conditions.