Interior regularity for free and constrained local minimizers of variational integrals under general growth and ellipticity conditions

Abstract

We consider strictly convex energy densities f:&#92;mathbb{R}^{n}&#92;rightarrow&#92;mathbb{R} under nonstandard growth conditions. More precisely, we assume that for some constants &#92;lambda, &#92;Lambda and for all Z,Y&#92;in&#92;mathbb{R}^{n} the inequality &#92;lambda(1+&#92;left|Z&#92;right|^{2})^{-&#92;frac{&#92;mu}{2}}&#92;left|Y&#92;right|^{2}&#92;leq D^{2}f(Z)(Y,Y)&#92;leq&#92;Lambda(1+&#92;left|Z&#92;right|^{2})^{&#92;frac{q-2}{2}}&#92;left|Y&#92;right|^{2} holds with exponents &#92;mu&#92;in&#92;mathbb{R} and q>1. If u denotes a bounded local minimizer of the energy &#92;int f(&#92;nabla w)dx subject to a constraint of the form w&#92;geq&#92;psi a.e. with a given obstacle &#92;psi&#92;in C^{1,&#92;alpha}(&#92;Omega), then we prove local C^{1,&#92;alpha}-regularity of u provided that q<4-&#92;mu. This result substantially improves what is known up to now (see, for instance, [CH], [BFM], [FM]), even for the case of unconstrained local minimizers.