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We give a "generalized' version of the isoperimetric inequality when the perimeter is defined with respect to a convex, positively homogeneous function on W". We use it to prove that, for any function u compactly supported in R", the integral of a convex function of DU decreases when u is rearranged in the corresponding "convex" way. Similar arguments allow us, for example, to prove comparison results for solutions of the Dirichlet problem for elliptic equations when the differential operator satisfies suitable structure assumptions. R~~SUMI? -Nous donnons une version (< generalisee >) de l'inegalite isoperimetrique lorsque la definition du perimetre depend d'une fonction convexe et positivement homogene sur W". Cette inegalite est employee pour demontrer que, pour toutes les fonctions u avec support compact dans R", l'integrale d'une fonction convexe de DU decro'it quand u est rearrangee a une faGon << convexe B. Avec des arguments du m&me type nous demontrons, par exemple, les resultats de comparaison pour les solutions du probleme de Dirichlet pour des equations elliptiques quand l'operateur differentiel satisfait des hypotheses de structure convenables.Work partially supported by MURST (40%)

We investigate the first eigenvalue of a highly nonlinear class of elliptic operators which includes the p -Laplace operator ∆pu =) and others. We derive the positivity of the first eigenfunction, simplicity of the first eigenvalue, Faber-Krahn and Payne-Rayner type inequalities. In another chapter we address the question of symmetry for positive solutions to more general equations. Using a Pohozaev-type inequality and isoperimetric inequalities as well as convex rearrangement methods we generalize a symmetry result of Kesavan and Pacella. Our optimal domains are level sets of a convex function H o . They have the so-called Wulff shape associated with H and only in special cases they are Euclidean balls. Mathematics Subject Classification (2000). 35J20, 35J70, 49R05, 49Q20, 52A38.

We consider the integral functionalwhere Ω ⊂ R n , n ≥ 2, is a nonempty bounded connected open subset of R n with smooth boundary, and R s → f (|s|) is a convex, differentiable function. We prove that if J admits a minimizer in W 1,1 0 (Ω) depending only on the distance from the boundary of Ω, then Ω must be a ball.

Abstract. We investigate elementary properties of a Finsler-Laplacian operator Q that is associated with functionals containing (H(∇u)) 2 . Here H is convex and homogeneous of degree 1, and its polar H o represents a Finsler metric on R n . In particular we study the Dirichlet problem −Qu = 2n on a ball K o = {x ∈ R n : H o (x) < 1} and present a fundamental solution for Q, suitable maximum and comparison principles, and a mean value property for solutions of Qu = 0.

Abstract. Let Ω be an open set in Euclidean space with finite Lebesgue measure |Ω|. We obtain some properties of the set function F : Ω → R + defined bywhere T (Ω) and λ1(Ω) are the torsional rigidity and the first eigenvalue of the Dirichlet Laplacian respectively. We improve the classical Pólya bound F (Ω) ≤ 1, and show that