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

In this paper, we prove global gradient estimates for solutions to linear elliptic and parabolic equations. For a sufficiently smooth bounded convex domain $Ω\subset \mathbb{R}^N$, we show that a solution $ϕ\in W_0^{1,\infty}(Ω)$ to an appropriate elliptic equation $\mathcal{L} ϕ= F$, with $F \in L^{\infty}(Ω;\mathbb{R})$, satisfies $|\nabla ϕ|_{\infty} \leq C |F|_{\infty}$, with a positive constant $C = \exp(C(\mathcal{L})\text{diam}(Ω))$. 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.

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