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Interior Regularity for Free and Constrained Local Minimizers of Variational Integrals Under General Growth and Ellipticity Conditions

Authors:

M Bildhauer M Fuchs

Institution:

(1) Universität des Saarlandes Fachrichtung 6.1, Mathematik Postfach 15 11 50 D-66041 Saarbrüucken, Germany

Abstract:

We consider strictly convex energy densities f: mgr

(x) under nonstandard growth conditions. More precisely, we assume that for some constants lambda

and for all Z, Y isin

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 mgr

and q< 1. If u denotes a bounded local minimizer of the energy int

w)dx subject to a constraint of the form w

a.e. with a given obstacle psgr

C^1, (

), then we prove the local C ^1,-regularity of u provided that q < 4 — mgr

. This result substantially improves what is known up to now even for the case of unconstrained local minimizers. Bibliography: 27 titles.

Keywords:

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