Gradient bounds for viscosity solutions to certain elliptic equations
Thalia Jeffres, Xiaolong Li
TL;DR
This work develops an elliptic analogue of a recent approach tying the modulus of continuity of viscosity solutions to a one-dimensional subsolution via a structure condition on the operator $F$. For degenerate elliptic equations of the form $F(x,u,∇u,D^{2}u)=0$, the modulus of continuity $ω$ of a periodic or uniformly vanishing solution is shown to satisfy a one-dimensional equation $f(s,φ,φ',φ'')=0$ in the viscosity sense, provided a suitable structure condition and comparison principle hold. When a one-dimensional comparison principle and an appropriate supersolution exist, one obtains gradient bounds for the original solution; the paper demonstrates this in both abstract form and through two concrete examples, extending Li21’s parabolic results to the elliptic setting. The method offers a constructive route to quantitative gradient and Hölder bounds for nonsmooth solutions of fully nonlinear elliptic equations by connecting two-point maximum principles, modulus of continuity, and viscosity solutions, with potential broad applicability to nonlinear PDE theory.
Abstract
Our principal object of study is the modulus of continuity of a periodic or uniformly vanishing function \( u: \mathbb{R} ^{n} \rightarrow \mathbb{R} \) which satisfies a degenerate elliptic equation \( F(x, u, \nabla u, D^{2} u) = 0 \) in the viscosity sense. The equations under consideration here have second-order terms of the form \( -{\rm Trace} \, (\mathcal{A} (\|\nabla u \|) \cdot D^{2} u) , \) where \( \mathcal{A} \) is an \( n\times n\) matrix which is symmetric and positive semi-definite. Following earlier work, \cite{Li21}, of the second author, which addressed the parabolic case, we identify a one-dimensional equation for which the modulus of continuity is a subsolution. In favorable cases, this one-dimensional operator can be used to derive a gradient bound on $u$ or to draw other conclusions about the nature of the solution.