Regularity for general functionals with double phase

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 $$w\mapsto \int b(x,w)(|Dw{|}^p+a(x)|Dw{|}^q)dx,1<p<q,a(x)\ge 0,$$ with $0<\hat{I}½\le b(\cdot )\le L$. 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.