The theorem of Rayleigh-Faber-Krahn for the characteristic values associated with a class of nonlinear boundary value problems

This paper is concerned with functionals which were introduced by Nehari [8,9] and also discussed by Coffman [2,3] in connection with the study of nonlinear boundary value problems. Their behavior under the Schwarz syitimetrization [12] is studied, and an isoperimetric inequality analogous to that of RayleighFaber-Krahn [12] for the fundamental frequency of a vibrating membrane is derived. v This work was supported by NSF Grant GU-2056. THE THEOREM OF RAYLEIGH-FABER-KRAHN FOR THE CHARACTERISTIC VALUES ASSOCIATED WITH A CLASS OF NONLINEAR BOUNDARY VALUE PROBLEMS by Catherine Bandle 1. Let 0 be a bounded region in R for which the Green's function for the Laplace operator exists. We shall write P for an arbitrary point in R and R for the positive real axis. Let F(s,P) be a positive function on "R+ y Cl with the following properties (A) F(,x) is continuous on R for almost all xefh F(s,•) is measurable for all seR . (B) There exists a positive number £ such that for almost all Pen and for all s.. < s2 l* F ( s i ' p ) 1 S 2 * F(s2,P) We define the function G(t,P) by t 6(t,P) = f F(s,P)ds, 0 and consider the functional H(v) = fi(v) J G(v,P)dx n [dx volume element in R, j&(v) = j gradvdx:, (x ^ , . . . ,x) Cartesian coordinates], within the class T of piecewise continuously differentiable functions which vanish on the boundary SO . This note will be concerned with isoperimetric inequalities for the functional