Exponential bounds for gradient of solutions to linear elliptic and parabolic equations

TL;DR

This paper establishes exponential bounds for the gradients of solutions to linear elliptic and parabolic equations in smooth convex domains, extending previous gradient estimates and motivated by the Landis conjecture.

Contribution

It provides new exponential gradient bounds for solutions to elliptic and parabolic equations, generalizing prior estimates and inspired by recent work on related conjectures.

Findings

Gradient bounds are exponential in the domain diameter.

Solutions with bounded right-hand side have controlled gradient magnitude.

The approach extends to both elliptic and parabolic equations.

Abstract

In this paper, we prove global gradient estimates for solutions to linear elliptic and parabolic equations. For a sufficiently smooth bounded convex domain $\Omega \subset \mathbb{R}^N$, we show that a solution $\phi \in W_0^{1,\infty}(\Omega)$ to an appropriate elliptic equation $\mathcal{L} \phi = F$, with $F \in L^{\infty}(\Omega;\mathbb{R})$, satisfies $|\nabla \phi|_{\infty} \leq C |F|_{\infty}$, with a positive constant $C = \exp(C(\mathcal{L})\text{diam}(\Omega))$. We also obtain similiar estimates in the parabolic setting. The proof of these exponential bounds relies on global gradient estimates inspired by a series of papers by Ben Andrews and Julie Clutterbuck. This work is motivated by a dual version of the Landis conjecture.