Regularity theory for elliptic equations under unbalanced growths

One of the most basic and important questions in PDE is that of regularity. It is also a unifying problem in the field, since it affects all kinds of PDEs. A classical example is Hilbert’s XIXth problem (1900), which roughly speaking asked to determine whether all solutions to uniformly elliptic variational PDEs are smooth. Starting from De Giorgi’s groundbreaking approach to this problem (1957), the first part of this talk will review the core ideas of elliptic regularity theory, emphasizing the main differences between the linear and nonlinear settings. We will then turn to the more recent theory of elliptic PDEs with p, q-growth — that is, elliptic equations whose ellipticity and growth are governed by different powers of the gradient. In this setting, a central feature is that regularity does not always hold: as shown by counterexamples due to Marcellini (1987) and Giaquinta (1987), certain variational integrals admit unbounded minimizers as soon as p and q are to far apart. Ensuring regularity for all solutions therefore requires an appropriate balance between p and q. Finally, we will discuss some current developments in which the growth of the stress field is prescribed by distinct Young functions, leading to an Orlicz-type framework that captures a broad range of nonstandard behaviors and provides a natural setting for genuinely non-homogeneous problems. - Seminario all'interno del ciclo di seminari ASK -