Exponential bounds for gradient of solutions to linear 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 $\varphi \in W_0^{1,\infty }(\Omega )$ to an appropriate elliptic equation $\mathcal{L}\varphi =F$, with $F\in L^{\infty }(\Omega ;\mathbb{R})$, satisfies $|\nabla \varphi {|}_{\infty }\le C|F{|}_{\infty }$, 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.