Seminari M.A.P.

Abstract

The question of regularity — whether solutions inherit the smoothness of their underlying equations — is a central, unifying theme in partial differential equations. This talk begins by reviewing the foundational ideas of elliptic regularity, from De Giorgi’s groundbreaking resolution of Hilbert’s XIXth problem to the key distinctions between linear and nonlinear regimes. We then turn to nonuniformly elliptic problems, with a particular focus on equations exhibiting (p,q)-growth, where ellipticity and growth are controlled by distinct powers of the gradient. In this context, regularity is no longer guaranteed: classical counterexamples by Marcellini and Giaquinta demonstrate that minimizers can be unbounded if the gap between p and q is too large. Finally, we discuss recent developments in quite general frameworks, providing a natural setting for analyzing genuinely non-homogeneous problems.