In this paper we establish a comparison result through symmetrization for solutions to some boundary value problems involving the fractional Laplacian. This allows to get sharp estimates for the solutions, obtained by comparing them with solutions of suitable radial problems. Furthermore, we use such result to prove a priori estimates for solutions in terms of the data, providing several regularity results which extend the well-known ones for the classical Laplacian. (C) 2012 Elsevier Inc. All rights reserved

Abstract. In this paper we study a Dirichlet problem relative to the equationwhere L is a linear elliptic operator with lower-order terms whose ellipticity condition is given in terms of the function ϕ(x) = (2π) − n 2 exp − |x| 2 /2 , the density of the Gaussian measure.

We provide isoperimetric Szego-Weinberger type inequalities for the first nontrivial Neumann eigenvaltie mu(1) (Omega) in Gauss space. where Omega is a possibly unbounded domain of R-N. Our main result consists in showing that among all sets Omega of R-N symmetric about the origin, having prescribed Gaussian measure, mu(1) (Omega) is maximum if and only if Omega is the Euclidean ball centered at the origin. (C) 2011 Elsevier Masson SAS. All rights reserved

We obtain a comparison result for solutions to nonlinear fully anisotropic elliptic problems by means of anisotropic symmetrization. As consequence we deduce a priori estimates for norms of the relevant solutions.

Integral estimates for weak solutions to a class of Dirichlet problems for nonlinear, fully anisotropic, elliptic equations with a zero order term are obtained using symmetrization techniques. * Istituto per le Applicazioni del Calcolo "M. Picone"(I.A.C.), Sede di Napoli, Consiglio Nazionale delle Ricerche (C.N.R.