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 nonstandard growth conditions and nonuniform 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 nonautonomous functionals. Finally, we disclose some new interpolation type effects that, as we conjecture, should draw a general phenomenon in the setting of nonuniformly elliptic problems. Such effects naturally connect with the Lavrentiev phenomenon.
 Publication:

arXiv eprints
 Pub Date:
 August 2017
 arXiv:
 arXiv:1708.09147
 Bibcode:
 2017arXiv170809147B
 Keywords:

 Mathematics  Analysis of PDEs