ANTI-MAXIMUM PRINCIPLES

GUIDO SWEERS

G.H.Sweers@its.tudelft.nl

Dept. of Applied Mathematical Analysis
ITS Faculty
Delft University of Technology
PO Box 5031, 2600 GA Delft, The Netherlands

For second order elliptic boundary value problems such as in and on it is well known that for any reasonable bounded domain in a first eigenvalue exists and moreover, for any a positive source term implies that the solution is positive. Clément and Peletier showed that for in a right neighbourhood of opposite behaviour occurs, an phenomenon which they named the anti-maximum principle: certain imply Such behaviour is roughly explained by the pole of the resolvent at and the positive sign of the corresponding eigenfunction

with the remaining regular part. A more precise statement reads as: if with then there is such that for the solution satisfies For higher order elliptic boundary value problems with appropriate boundary conditions sometimes a stronger uniform anti-maximum principle holds, that is, for in with on there exists such that for all and one obtains

The anti-maximum principle for higher order elliptic boundary value problems is joint work with Ph.Clément. Sharp conditions for the uniform anti-maximum principle will appear in a joint work with H.-Ch.Grunau.

EPSRC Gregynog Workshop, 21-26 July 2002