Calculus of Variations and Geometric Measure Theory
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L. Esposito - F. Leonetti - G. Mingione

Regularity results for minimizers of irregular integrals with (p,q) growth

created on 05 Dec 2001
modified on 09 Aug 2002


Published Paper

Inserted: 5 dec 2001
Last Updated: 9 aug 2002

Journal: Forum Mathematicum
Volume: 14
Number: 2
Pages: 245-272
Year: 2002


We consider variational integrals $$ \int f(Du)\ dx $$ where $u:\Omega \to R^{N}$ and the convex function $f$ has $(p,q)$ growth $ \mid z \mid^{p} \leq f(z) \leq L(\mid z\mid ^{q} +1)$, $ p<q$. We prove local Lipschitz continuity of minimizers in the scalar case and in some vectorial cases. We further prove higher integrability and differentiability for local minimizers. The results cover the case in which $f$ is degenerate convex. A main feature of the paper is that we do not $f$ to be differentiable everywhere. The essential smallness condition on the numbers $(p,q)$ is: $$ qp < (n+1)n$$

Keywords: regularity, Non standard growth, minimizer

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