# Regularity for general functionals with double phase

created by baroni on 13 Feb 2018
modified on 30 Mar 2018

[BibTeX]

Published Paper

Inserted: 13 feb 2018
Last Updated: 30 mar 2018

Journal: Calc. Var. Partial Differ. Equ.
Year: 2018
Doi: 10.1007/s00526-018-1332-z

ArXiv: 1708.09147 PDF
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.