Global Lipschitz regularizing effects for linear and nonlinear parabolic equations

TL;DR

This paper establishes global gradient bounds and regularity estimates for viscosity solutions of linear and nonlinear parabolic equations with unbounded and minimally regular coefficients, extending classical results and providing new insights into regularizing effects.

Contribution

It introduces new global gradient and Hölder estimates for solutions with unbounded data and coefficients, extending prior linear results to nonlinear and more general cases.

Findings

Global bounds on spatial gradients of solutions.

Extension of estimates to unbounded data and nonlinear operators.

Comparison of analytic and probabilistic methods for regularity analysis.

Abstract

In this paper we prove global bounds on the spatial gradient of viscosity solutions to second order linear and nonlinear parabolic equations in $(0,T) \times \R^N$. Our assumptions include the case that the coefficients be both unbounded and with very mild local regularity (possibly weaker than the Dini continuity), the estimates only depending on the parabolicity constant and on the modulus of continuity of coefficients (but not on their $L^{\infty}$-norm). Our proof provides the analytic counterpart to the probabilistic proof used in Priola and Wang (J. Funct. Anal. 2006) to get this type of gradient estimates in the linear case. We actually extend such estimates to the case of possibly unbounded data and solutions as well as to the case of nonlinear operators including Bellman-Isaacs equations. We investigate both the classical regularizing effect (at time $t>0$) and the possible conservation of Lipschitz regularity from $t=0$, and similarly we prove global H\"older estimates under weaker assumptions on the coefficients. The estimates we prove for unbounded data and solutions seem to be new even in the classical case of linear equations with bounded and H\"older continuous coefficients. Applications to Liouville type theorems are also given in the paper. Finally, we compare in an appendix the analytic and the probabilistic approach discussing the analogy between the doubling variables method of viscosity solutions and the probabilistic coupling method.