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 $\Omega \subset \mathbb{R}^N$, 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.