$H^2$-regularity of Steklov eigenfunctions on convex domains via Rellich-Pohozaev identity

TL;DR

The paper proves $H^2$-regularity for Steklov eigenfunctions on bounded convex domains in $\mathbb{R}^n$ by adapting Grisvard’s framework to the Steklov setting and pairing the Rellich-Pohozaev identity with Reilly's formula to control boundary contributions. It derives a Hessian bound $\int_{\Omega}|D^2u|^2\le 2\sigma C_{D,\rho,\sigma}$ on smooth convex domains, where $D$ is the diameter, $\rho$ the inradius, and $C_{D,\rho,\sigma}$ depends only on these geometric quantities and the eigenvalue $\sigma$. The authors then approximate a general convex domain by smooth convex domains and use spectral stability to pass the Hessian estimates to the limit, yielding $H^2$-regularity for all Steklov eigenfunctions on the original domain. The approach clarifies how geometry and domain perturbations interact with spectral properties, and contrasts with the Dirichlet/Neumann/Robin cases where regularity follows more directly from Reilly's formula.

Abstract

We prove that the Steklov eigenfunctions on convex domains of $\mathbb R^n$ are $H^2$ regular by adapting a classical argument combined with the Rellich-Pohozaev identity.

Theorems & Definitions (15)

• Theorem 1
• Theorem 2: Rellich-Pohožaev identity
• Theorem 3: Reilly's formula
• Lemma 4
• proof
• Lemma 5
• proof
• Definition 6
• Lemma 7
• Lemma 8