Quasiconvexity preserving property for fully nonlinear nonlocal parabolic equations

TL;DR

This paper proves that solutions to a broad class of fully nonlinear nonlocal parabolic equations preserve quasiconvexity over time, extending previous results and providing practical examples.

Contribution

It establishes the quasiconvexity preserving property for solutions of nonlinear nonlocal parabolic equations, generalizing earlier convexity preservation results.

Findings

Viscosity solutions with quasiconvex initial data remain quasiconvex over time.

The proof extends to a limit case of power convexity preservation.

Concrete examples illustrate applications of the main theorem.

Abstract

This paper is concerned with a general class of fully nonlinear parabolic equations with monotone nonlocal terms. We investigate the quasiconvexity preserving property of positive, spatially coercive viscosity solutions. We prove that if the initial value is quasiconvex, the viscosity solution to the Cauchy problem stays quasiconvex in space for all time. Our proof can be regarded as a limit version of that for power convexity preservation as the exponent tends to infinity. We also present several concrete examples to show applications of our result.