Absence of Lavrentiev gap for non-autonomous functionals with (p,q)-growth

Abstract

We consider non-autonomous functionals of the form ℱ⁢(u,Ω)=∫Ωf⁢(x,D⁢u⁢(x))⁢𝑑x{\mathcal{F}(u,\hskip-0.569055pt\Omega)\hskip-0.853583pt=\hskip-0.853583pt\int% _{\Omega}f(x,\hskip-0.569055ptDu(x))\hskip-0.569055pt\,dx}, where u:Ω→ℝN{u\colon\kern-0.711319pt\Omega\hskip-0.569055pt\to\hskip-0.569055pt\mathbb{R}^% {N}}, Ω⊂ℝn{\Omega\subset\mathbb{R}^{n}}. We assume that f⁢(x,z){f(x,z)} grows at least as |z|p{|z|^{p}} and at most as |z|q{|z|^{q}}. Moreover, f⁢(x,z){f(x,z)} is Hölder continuous with respect to x and convex with respect to z. In this setting, we give a sufficient condition on the density f⁢(x,z){f(x,z)} that ensures the absence of a Lavrentiev gap.

Keywords

• variational integrals
• non-standard growth
• regularity
• lavrentiev gap
• 49n60