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Variational integrals of splitting-type: higher integrability under general growth conditions

Authors:

M Bildhauer M Fuchs

Institution:

1.Universit?t des Saarlandes,Saarbrücken,Germany

Abstract:

Besides other things we prove that if ${u\in L^\infty_{loc}(\Omega;{\mathbb{R}}^M)}$ , ${\Omega \subset {\mathbb{R}}^n}$ , locally minimizes the energy

${\int \limits_{\Omega}} \lefta(|{\tilde{\nabla}} u|) + b(|\partial_n u|)\right] dx,$

${{\tilde{\nabla}} := (\partial_1,\dots, \partial_{n-1})}$ , with N-functions a ≤ b having the Δ₂-property, then ${|\partial_n u|^2 b(|\partial_n u|) \in L^1_{loc}(\Omega)}$ . Moreover, the condition

$b(t) \leq const t^2 a(t^2) \quad \quad \quad (\ast)$

for all large values of t implies ${|{\tilde{\nabla}} u|^2 a(|{\tilde{\nabla}} u|)\in L^1_{loc}(\Omega)}$ . If n = 2, then these results can be improved up to ${|\nabla u|\in L^s_{loc}(\Omega)}$ for all s < ∞ without the hypothesis ${(\ast)}$ . If n ≥ 3 together with M = 1, then higher integrability for any exponent holds under more restrictive assumptions than ${(\ast)}$ .

Keywords:

Decomposable variational integrals Local minimizers Higher integrability Anisotropic problems Nonstandard growth conditions

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