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: μ(x) under nonstandard growth conditions. More precisely, we assume that for some constants λ, Λ and for all Z, Y∈ ℝ n the inequality $$\lambda \left( {1 + \left| Z \right|^2 } \right)^{ - \frac{\mu }{2}} \left| Y \right|^2 \leqslant D^2 f\left( Z \right)\left( {Y,Y} \right) \leqslant \Lambda \left( {1 + \left| Z \right|^2 } \right)^{\frac{{q - 2}}{2}} \left| Y \right|^2 $$ holds with exponents μ ∈ ℝ and q< 1. If u denotes a bounded local minimizer of the energy ∫ f(▽w)dx subject to a constraint of the form w ≥ ψ a.e. with a given obstacle ψ ∈ C1,α (Ω), then we prove the local C1,α-regularity of u provided that q < 4 — μ. This result substantially improves what is known up to now even for the case of unconstrained local minimizers. Bibliography: 27 titles.