# Seminario di Equazioni Differenziali e Applicazioni: Regularity of stable solutions to semilinear elliptic equations on Riemannian models

## Venerdì 14 Marzo 2014, ore 12:15 - Aula 1BC45 - Daniele Castorina

ARGOMENTI: Seminars

SEMINARIO DI EQUAZIONI DIFFERENZIALI E APPLICAZIONI

Venerdì 14 Marzo 2014 alle ore 12:15 in aula 1BC45, Daniele Castorina (Università di Padova) terrà un seminario dal titolo "Regularity of stable solutions to semilinear elliptic equations on Riemannian models".

Abstract
We consider the reaction-diffusion problem Delta_g u + f(u) = 0 in B with zero Dirichlet boundary condition, posed in a geodesic ball B of a Riemannian model (M, g). This class of Riemannian manifolds includes the classical space forms, i.e., the Euclidean, elliptic, and hyperbolic spaces. For the class of semistable solutions we prove radial symmetry and monotonicity. Furthermore, we establish Linfty, Lp, and W1,p estimates which are optimal and do not depend on the nonlinearity f. As an application, under standard assumptions on the nonlinearity f(u), we prove that the corresponding extremal solution is bounded whenever the dimension is less than ten. To establish the optimality of our regularity results we find the extremal solution for some exponential and power nonlinearities using an improved weighted Hardy inequality. This is a joint work with Manel Sanchon (UAB, Barcelona).

Ref. Int. M. Bardi, E. Feleqi, L. Rossi
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