Regularity for general functionals with double phase

TL;DR

This paper establishes sharp regularity results for a broad class of functionals with non-standard growth and ellipticity, especially focusing on the double phase integral, and introduces new methods applicable to non-autonomous functionals.

Contribution

It provides new regularity theorems for non-uniformly elliptic functionals, including the double phase case, and introduces innovative methods and interpolation effects related to the Lavrentiev phenomenon.

Findings

Sharp regularity results for double phase functionals.

New methods for non-autonomous functional regularity.

Identification of interpolation effects linked to Lavrentiev phenomenon.

Abstract

We prove sharp regularity results for a general class of functionals of the type $$ w \mapsto \int F(x, w, Dw) \, dx\;, $$ featuring non-standard growth conditions and non-uniform ellipticity properties. The model case is given by the double phase integral $$ w \mapsto \int b(x,w)(|Dw|^p+a(x)|Dw|^q) \, dx\;,\quad 1 <p < q\,, \quad a(x)\geq 0\;, $$ with $0<\nu \leq b(\cdot)\leq L $. This changes its ellipticity rate according to the geometry of the level set $\{a(x)=0\}$ of the modulating coefficient $a(\cdot)$. We also present new methods and proofs, that are suitable to build regularity theorems for larger classes of non-autonomous functionals. Finally, we disclose some new interpolation type effects that, as we conjecture, should draw a general phenomenon in the setting of non-uniformly elliptic problems. Such effects naturally connect with the Lavrentiev phenomenon.