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 Ω⊂RN, we show that a solution ϕ∈W01,∞(Ω;R) to an appropriate elliptic equation Lϕ=F, with F∈L∞(Ω;R), satisfies |∇ϕ|∞≤C|F|∞, with a positive constant C=exp⁡(C(L)diam(Ω)). We also obtain similar 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.