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 $\mathrm{\Omega }\subset {\mathbb{R}}^N$, we show that a solution $\phi \in W_0^{1,\infty}(\Omega)$toanappropriateellipticequation$$\mathcal{L}$ \phi = F$,with$F \in L^{\infty}(\Omega;$\mathbb{R}$)$,satisfies$|\nabla \phi|{\infty} \leq C |F|{\infty}$,withapositiveconstant$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.