Rigidity for a semilinear Neumann problem with exponential nonlinearity in the large diffusion limit

TL;DR

This paper proves that for a semilinear Neumann problem with exponential nonlinearity in a 2D domain, solutions become constant when the diffusion parameter exceeds a certain threshold, confirming a recent conjecture.

Contribution

It establishes a threshold for the diffusion parameter beyond which all classical solutions are constant, addressing a conjecture in the field.

Findings

Solutions are constant for large diffusion parameter $ ext{varepsilon} > ar{ ext{varepsilon}}$

The proof combines $L^1$-estimates, Green's function analysis, and elliptic regularity

Confirms a recent conjecture about solution rigidity in the large diffusion limit

Abstract

We consider a semilinear Neumann problem with exponential nonlinearity in a smooth bounded domain $\Omega \subset \mathbb{R}^2$. We prove that there exists a threshold $\bar{\varepsilon}>0$ such that for all $\varepsilon>\bar{\varepsilon}$, any classical solution must be constant. This result provides a positive answer to a conjecture recently posed by Calanchi, Ciraolo, and Messina (2026). Our proof relies on a combination of $L^1$-estimates, a Jensen-type argument via the Neumann Green's function to obtain uniform exponential integrability, and elliptic regularity.