$H^2$-regularity on convex domains for Robin eigenfunctions with parameter of arbitrary sign

TL;DR

This paper proves that Robin eigenfunctions on convex domains are always $H^2$ regular, regardless of the sign of the boundary parameter, using adapted classical methods and a Rellich-Pohozaev identity.

Contribution

It establishes $H^2$ regularity for Robin eigenfunctions on convex domains for any sign of the boundary parameter, extending previous results limited to positive parameters.

Findings

Robin eigenfunctions are $H^2$ regular on convex domains for any parameter sign.

The proof combines classical arguments with a Rellich-Pohozaev identity.

Regularity holds regardless of the boundary parameter's sign.

Abstract

We prove that the Robin eigenfunctions on convex domains of $\mathbb R^n$ are $H^2$ regular regardless of the sign of the parameter involved in the boundary conditions. The proof is an adaptation of a classical argument used in the case of positive parameters combined with a Rellich-Pohozaev identity.