Regularity of minimizers for double phase functionals of borderline case with variable exponents

Abstract

The aim of this article is to study regularity properties of a local minimizer of a double phase functional of type ℱ(u)≔∫Ω(∣Du∣p(x)+a(x)∣Du∣p(x)log(e+∣Du∣))dx,{\mathcal{ {\mathcal F} }}\left(u):= \mathop{\int }\limits_{\Omega }({| Du| }^{p\left(x)}+a\left(x){| Du| }^{p\left(x)}\log \left(e+| Du| )){\rm{d}}x, being p(x),a(x)p\left(x),a\left(x) log-continuous functions with p(x)>1p\left(x)\gt 1, a≥0a\ge 0. Double phase functionals ∫(∣Du∣p+a(x)∣Du∣q)dx\int ({| Du| }^{p}+a\left(x){| Du| }^{q}){\rm{d}}x, with constant exponents pp and qq (q≥p≥1)\left(q\ge p\ge 1), appeared in the papers by Zhikov, and C1,α{C}^{1,\alpha }-regularity of their minimizers was given by Colombo and Mingione. Later, by Baroni, Colombo, and Mingione, the above type functionals with logarithm but with constant exponent, regularity properties were given. They obtained sharp regularity results for minimizers of such functionals. In this article, we treat the case that the exponents are functions of p(x)p\left(x) and partly generalize their regularity results.

Keywords

• variational problem
• hölder continuity
• 35j20
• 35j47
• 35j60
• 49n60