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Log-concavity in some parabolic problems. (English) Zbl 0921.35030
Summary: We improve a concavity maximum principle for parabolic equations of the second order, which was initially established by Korevaar, and then we use this result to investigate some boundary value problems. In particular, we find structural conditions on the equation, and suitable conditions on the domain of the problem and on the boundary data, that suffice to yield spatial log-concavity of the (positive) solution. Examples and applications are provided, and some unsolved problems are pointed out. We also survey some classical as well as recent contributions to the subject.

MSC:
35B50 Maximum principles in context of PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35K55 Nonlinear parabolic equations
35K65 Degenerate parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
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