Minimax Theory and its Applications 08 (2023), No. 1, 037--060
Copyright Heldermann Verlag 2023
Weak Comparison Principles for Fully Nonlinear Degenerate Parabolic Equations with Discontinuous Source Terms
Nao Hamamuki
Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo, Japan
hnao@math.sci.hokudai.ac.jp
Kuniyasu Misu
Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo, Japan
kuniyasu.misu@math.sci.hokudai.ac.jp
We study the initial value problem for a fully nonlinear degenerate parabolic equation with discontinuous source terms, to which a usual type of comparison principles does not apply. Examples include singular equations appearing in surface evolution problems such as the level-set mean curvature flow equation with a driving force term and a discontinuous source term. By a suitable scaling, we establish weak comparison principles for a viscosity sub- and supersolution to the equation. We also present uniqueness and existence results of possibly discontinuous viscosity solutions.
Keywords: Weak comparison principle, viscosity solution, fully nonlinear equation, discontinuous source term, level-set mean curvature flow equation.
MSC: 35B51, 35D40, 35K15.
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