On a dynamic boundary condition for singular degenerate parabolic equations in a half space
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We consider the initial value problem for a fully-nonlinear degenerate parabolic equation with a dynamic boundary condition in a half space. Our setting includes geometric equations with singularity such as the level-set mean curvature flow equation. We establish a comparison principle for a viscosity sub- and supersolution. We also prove existence of solutions and Lipschitz regularity of the unique solution. Moreover, relation to other types of boundary conditions is investigated by studying the asymptotic behavior of the solution with respect to a coefficient of the dynamic boundary condition.
KeywordsDynamic boundary condition Geometric equations Comparison principle Viscosity solutions
Mathematics Subject ClassificationPrimary 35K20 Secondary 35B51 35D40
The authors are grateful to Professor Moto-Hiko Sato who initiated this project with useful suggestions. The authors also thank the anonymous referees for his/her careful reading of the manuscript and valuable comments.
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