RAYLEIGH-FABER-KRAHN INEQUALITIES AND NONLINEAR BOUNDARY VALUE PROBLEMS

CATHERINE BANDLE

bandle@math.unibas.ch

Mathematisches Institut

Universität Basel

Rheinsprung 21, CH-4051 Basel, Switzerland

Universität Basel

Rheinsprung 21, CH-4051 Basel, Switzerland

The classical Rayleigh-Faber-Krahn inequality states that the among all domains of given volume the first eigenvalue of the membrane is smallest for the ball. This is expressed in terms of Sobolev constants as follows:

infvol

where is the unit ball in . This estimate is obtained by means of Schwarz
symmetrization. In this talk we study more general Sobolev constants of the type
inf

where and are positive continuous functions. Without further assumptions on the
weights no general Rayleigh-Faber-Krahn inequalities, that is estimates from below depending
essentially on
, are
to be expected. We describe classes of functions
and for which such inequalities hold. They are then used to construct upper
bounds for the solutions of nonlinear boundary value problems involving the p-Laplacian.
EPSRC Gregynog Workshop, 21-26 July 2002