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

M. Bildhauer ; M. Fuchs

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 32, Tome 288 (2002), pp. 79-99

Résumé

We consider strictly convex energy dencities $f\colon\mathbb R^n\to\mathbb R$ under nonstandart growth conditions. More precisely, we assume that for some constants $\lambda$, $\Lambda$ and for all $Z,Y\in\mathbb R^n$ the inequality $$ \lambda(1+|Z|^2)^{\frac{-\mu}2}|Y|^2\le D^2f(Z)(Y,Y)\le\Lambda(1+|Z|^2)^{\frac{q-2}2}|Y|^2 $$ holds with exponents $\mu\in\mathbb R$ and $q>1$. If $u$ denotes a bounded local minimizer of the energy $\int f(\nabla\omega)dx$ subject to a constraint of the form $\omega\ge\psi$ a.e. with a given obstacle $\psi\in C^{1,\alpha}(\Omega)$, then we prove local $C^{1,\alpha}$-regularity of $u$ provided that $q4-\mu$. This result substantially improves what is known up to now (see, for instance, [8, 7, 13]), even for the case of unconstrained local minimizers.