Calculus of Variations and Geometric Measure Theory
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P. Baroni - M. Colombo - G. Mingione

Regularity for general functionals with double phase

created by baroni on 13 Feb 2018

[BibTeX]

Preprint

Inserted: 13 feb 2018
Last Updated: 13 feb 2018

Year: 2018

ArXiv: 1708.09147 PDF

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.


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