A general low-order partial regularity theory for asymptotically convex functionals with asymptotic dependence on the minimizer

Abstract

We consider a minimizer $$u\in W^{1,p}(\Omega )$$ , where $$\Omega \subseteq \mathbb {R}^{n}$$ is a bounded open set, of the integral functional $$\begin{aligned} u\mapsto \int _{\Omega }f(x,u,Du)\ dx \end{aligned}$$ in the case when $$f : \Omega \subseteq \mathbb {R}^{n}\times \mathbb {R}^{N}\times \mathbb {R}^{N\times n}\rightarrow \mathbb {R}$$ is asymptotically related to a more regular function; since we assume that $$N\ge 2$$ , we study here the vectorial case. The asymptotical relatedness condition is such that dependence on u is allowed even as $$|\xi |\rightarrow +\infty $$ . Unlike in previous work f is allowed to satisfy a more general growth condition, which permits a coupling between the x and u variables. Due to the generality of the asymptotical relatedness condition this coupling induces a subtle restriction on the regularity required of the various functions in the growth condition, and we study these restrictions in this paper. In addition, here we do not obtain the full spectrum of Holder continuity for the minimizer, but rather a restricted range of Holder exponents that depend on the initial data.