Electron.n. J. Diff. Eqns., Vol. 1999(1999), No. 19, pp. 1-12.

Log-concavity in some parabolic problems

Antonio Greco & Bernd Kawohl

Abstract:
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.

Submitted March 23, 1999. Published May 28, 1999.
Math Subject Classification: 35B50, 35K55, 35K65, 35B05.
Key Words: Concavity maximum principle, Log-concavity, Parabolic equation.

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Antonio Greco
Dipartimento di Matematica, Universita di Cagliari
via Ospedale 72
I 09124 Cagliari, Italy
E-mail: greco@vaxca1.unica.it , http://riemann.unica.it/~antoniog

Bernd Kawohl
Mathematisches Institut, Universitat zu Koln
D 50923 Koln, Germany
E-mail: kawohl@mi.uni-koeln.de , http://www.mi.uni-koeln.de/mi/Forschung/Kawohl

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