On the microscopic spacetime convexity principle for fully nonlinear parabolic equations II: Spacetime quasiconcave solutions

Chuanqiang  Chen

doi:10.3934/dcds.2016007

Article Contents

2016, Volume 36, Issue 9: 4761-4811. Doi: 10.3934/dcds.2016007

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On the microscopic spacetime convexity principle for fully nonlinear parabolic equations II: Spacetime quasiconcave solutions

Chuanqiang Chen^1,

1.
Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou, 310023, Zhejiang Province

Received: October 31, 2014

Revised: February 29, 2016

Published: May 2017

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Abstract

In [13], Chen-Ma-Salani established the strict convexity of spacetime level sets of solutions to heat equation in convex rings, using the constant rank theorem and a deformation method. In this paper, we generalize the constant rank theorem in [13] to fully nonlinear parabolic equations, that is, establish the corresponding microscopic spacetime convexity principles for spacetime level sets. In fact, the results hold for fully nonlinear parabolic equations under a general structural condition, including the $p$-Laplacian parabolic equations ($p >1$) and some mean curvature type parabolic equations.

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Mathematics Subject Classification: Primary: 35K10; Secondary: 35B99.

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