Lavrentiev phenomenon, relaxation and some regularity results for anisotropic functionals

Abstract

We study local minimizers of anisotropic variational integrals of the form J[u]=\int_{\Omega}f(\cdot,\nabla u)dx with integrand f satisfying a (p,\bar{q})-growth condition w.r.t. \nabla u and with D_{P}f(x,P) satisfying a Lipschitz condition w.r.t. x\in\Omega. If the Lavrentiev gap functional \mathcal{L} relative to J vanishes for all balls B_{R}\Subset\Omega and if \bar{q}<p(1+1/), then (partial) C^{1,\alpha}-regularity holds. Moreover, the bound on the exponents can be replaced by \bar{q}<p+1 provided we study locally bounded minimizers. We also investigate the relaxation of global minimization problems and discuss the regularity of the corresponding solutions. The importance of the condition \mathcal{L}\equiv0 was recently discovered by Esposito, Leonetti and Mingione in [ELM], where besides other results the higher integrability of the gradient is proved even under weaker assumptions than used here.