Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth

Abstract

We consider the functional ℱ(u)≔∫Ωf(x,Du(x))dx,{\mathcal{ {\mathcal F} }}\left(u):= \mathop{\int }\limits_{\Omega }f\left(x,Du\left(x)){\rm{d}}x, where f(x,z)f\left(x,z) satisfies a (p,q)\left(p,q)-growth condition with respect to zz and can be approximated by means of a suitable sequence of functions. We consider BR⋐Ω{B}_{R}\hspace{0.33em}\Subset \hspace{0.33em}\Omega and the spaces X=W1,p(BR,RN)andY=W1,p(BR,RN)∩Wloc1,q(BR,RN).X={W}^{1,p}\left({B}_{R},{{\mathbb{R}}}^{N})\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}Y={W}^{1,p}\left({B}_{R},{{\mathbb{R}}}^{N})\cap {W}_{\hspace{0.1em}\text{loc}\hspace{0.1em}}^{1,q}\left({B}_{R},{{\mathbb{R}}}^{N}). We prove that the lower semicontinuous envelope of ℱ∣Y{\mathcal{ {\mathcal F} }}{| }_{Y} coincides with ℱ{\mathcal{ {\mathcal F} }} or, in other words, that the Lavrentiev term is equal to zero for any admissible function u∈W1,p(BR,RN)u\in {W}^{1,p}\left({B}_{R},{{\mathbb{R}}}^{N}). We perform the approximations by means of functions preserving the values on the boundary of BR{B}_{R}.

Keywords

• regularity
• minimizer
• variational
• integral
• lavrentiev
• phenomenon
• 35b65
• 35j60
• 35j47