Lipschitz continuity and convexity preserving for solutions of semilinear evolution equations in the Heisenberg group

In this paper we study viscosity solutions of semilinear parabolic equations in the Heisenberg group. We show uniqueness of viscosity solutions with exponential growth at space infinity. We also study Lipschitz and horizontal convexity preserving properties under appropriate assumptions. Counterexam...

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Bibliographic Details
Published inCalculus of variations and partial differential equations Vol. 55; no. 4; pp. 1 - 25
Main Authors Liu, Qing, Manfredi, Juan J., Zhou, Xiaodan
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2016
Springer Nature B.V
Subjects
Analysis
Calculus of Variations and Optimal Control; Optimization
Control
Convexity
Euclidean space
Mathematical analysis
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Nonlinear equations
Systems Theory
Theoretical
Viscosity
35D40
35R03
35K58
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Summary:In this paper we study viscosity solutions of semilinear parabolic equations in the Heisenberg group. We show uniqueness of viscosity solutions with exponential growth at space infinity. We also study Lipschitz and horizontal convexity preserving properties under appropriate assumptions. Counterexamples show that in general such properties that are well known for semilinear and fully nonlinear parabolic equations in the Euclidean spaces do not hold in the Heisenberg group.
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-016-1024-5