We prove regularity theorems for minimizers of integral functionals of the Calculus of Variations \int f(x,Du) with non-standard growth conditions of (p,q) type |z|^p \leq f(x,z) \leq L(|z|^q+1), p < q In particular,we find that a sufficient condition for minimizers to be regular is \frac{q}{p}< \frac{n+\alpha}{n} where the function f(x,z) is \alpha-Holder continuous with respect to the x-variable and x \in \Omega \subset R^n; this condition is also sharp. We include results in the setting of Orlicz spaces; moreover,we treat certain relaxed functionals too. Finally,we address a problem posed by Marcellini in [43], showing a minimizer with an isolated singularity.
Sharp regularity for functionals with (p,q) growth / L., Esposito; F., Leonetti; Mingione, Giuseppe. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - 204:(2004), pp. 5-55. [10.1016/j.jde.2003.11.007]
Sharp regularity for functionals with (p,q) growth
MINGIONE, Giuseppe
2004-01-01
Abstract
We prove regularity theorems for minimizers of integral functionals of the Calculus of Variations \int f(x,Du) with non-standard growth conditions of (p,q) type |z|^p \leq f(x,z) \leq L(|z|^q+1), p < q In particular,we find that a sufficient condition for minimizers to be regular is \frac{q}{p}< \frac{n+\alpha}{n} where the function f(x,z) is \alpha-Holder continuous with respect to the x-variable and x \in \Omega \subset R^n; this condition is also sharp. We include results in the setting of Orlicz spaces; moreover,we treat certain relaxed functionals too. Finally,we address a problem posed by Marcellini in [43], showing a minimizer with an isolated singularity.File | Dimensione | Formato | |
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