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Unique continuation for Robin problems with non-smooth potentials

Zongyuan Li

TL;DR

This work establishes strong unique continuation (SUCP) for Robin problems with non-smooth Robin potentials by proving SUCP at boundary points when the Robin coefficient lies in $L_{s}$ with $s> d-1$, extending prior Neumann and differentiable-η results. The authors combine an auxiliary-function reduction that converts Robin data to a homogeneous conormal problem with a robust blowup analysis that yields limiting homogeneous profiles without requiring Almgren-type monotonicity, together with a dimension-reduction argument. The key outcomes are SUCP at the boundary, a bound on the boundary zero set's Hausdorff dimension $\le d-2$ (finite for $d=2$), and a blowup corollary guaranteeing homogeneous leading-order profiles of degree $m$; a flattening map and convergence results underpin these findings. The results advance the understanding of unique continuation under rough coefficients and have potential implications for inverse problems and non-destructive testing where surface Robin data model interactions.

Abstract

In this work, we study the unique continuation properties of Robin boundary value problems with Robin potentials $η\in L_{d-1+\varepsilon}$. Our results generalize earlier ones in which $η$ was assumed to be either zero (Neumann problem) or differentiable.

Unique continuation for Robin problems with non-smooth potentials

TL;DR

This work establishes strong unique continuation (SUCP) for Robin problems with non-smooth Robin potentials by proving SUCP at boundary points when the Robin coefficient lies in with , extending prior Neumann and differentiable-η results. The authors combine an auxiliary-function reduction that converts Robin data to a homogeneous conormal problem with a robust blowup analysis that yields limiting homogeneous profiles without requiring Almgren-type monotonicity, together with a dimension-reduction argument. The key outcomes are SUCP at the boundary, a bound on the boundary zero set's Hausdorff dimension (finite for ), and a blowup corollary guaranteeing homogeneous leading-order profiles of degree ; a flattening map and convergence results underpin these findings. The results advance the understanding of unique continuation under rough coefficients and have potential implications for inverse problems and non-destructive testing where surface Robin data model interactions.

Abstract

In this work, we study the unique continuation properties of Robin boundary value problems with Robin potentials . Our results generalize earlier ones in which was assumed to be either zero (Neumann problem) or differentiable.
Paper Structure (8 sections, 7 theorems, 56 equations)

This paper contains 8 sections, 7 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Strong unique continuation property
  3. Blowup analysis
  4. Proof of Lemma \ref{['lem:convergence']}
  5. Proof of Corollary \ref{['cor-230725-1118']}
  6. Dimension reduction
  7. Monotonicity and rigidity property of Neumann problems
  8. A solvability result of Neumann problems

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{R}^d$ be a $C^{1,1}$ domain, $d\geq 2$. Suppose $u \in H^1(\Omega \cap B_1)$ is non-trivial and solves eqn-230419-0739-1-eqn-230419-0739-2 with coefficients satisfying $a_{ij} \in C^{0,1}(\Omega)$ and Then we have the following.

Theorems & Definitions (10)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.2
  • Lemma 3.1
  • Corollary 3.1
  • Remark 3.2
  • Lemma 4.1: Theorem A.4 in MR756417, Lemma 2.2 in MR1090434
  • Lemma A.1
  • proof
  • Lemma B.1