# A regularity result for a convex functional and bounds for the singular set

• Volume: 16, Issue: 4, page 1002-1017
• ISSN: 1292-8119

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## Abstract

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In this paper we prove a regularity result for local minimizers of functionals of the Calculus of Variations of the type${\int }_{\Omega }f\left(x,Du\right)\phantom{\rule{4pt}{0ex}}\mathrm{d}x$where Ω is a bounded open set in ${ℝ}^{n}$, u∈${W}_{\mathrm{loc}}^{1,p}$(Ω; ${ℝ}^{N}$), p> 1, n≥ 2 and N≥ 1. We use the technique of difference quotient without the usual assumption on the growth of the second derivatives of the function f. We apply this result to give a bound on the Hausdorff dimension of the singular set of minimizers.

## How to cite

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De Maria, Bruno. "A regularity result for a convex functional and bounds for the singular set." ESAIM: Control, Optimisation and Calculus of Variations 16.4 (2010): 1002-1017. <http://eudml.org/doc/250746>.

@article{DeMaria2010,
abstract = { In this paper we prove a regularity result for local minimizers of functionals of the Calculus of Variations of the type$$\int\_\{\Omega\}f(x, Du)\ \{\rm d\}x$$where Ω is a bounded open set in $\mathbb\{R\}^\{n\}$, u∈$W^\{1,p\}_\{\rm loc\}$(Ω; $\mathbb\{R\}^\{N\}$), p> 1, n≥ 2 and N≥ 1. We use the technique of difference quotient without the usual assumption on the growth of the second derivatives of the function f. We apply this result to give a bound on the Hausdorff dimension of the singular set of minimizers. },
author = {De Maria, Bruno},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Partial regularity; singular sets; fractional differentiability; variational integrals; partial regularity},
language = {eng},
month = {10},
number = {4},
pages = {1002-1017},
publisher = {EDP Sciences},
title = {A regularity result for a convex functional and bounds for the singular set},
url = {http://eudml.org/doc/250746},
volume = {16},
year = {2010},
}

TY - JOUR
AU - De Maria, Bruno
TI - A regularity result for a convex functional and bounds for the singular set
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/10//
PB - EDP Sciences
VL - 16
IS - 4
SP - 1002
EP - 1017
AB - In this paper we prove a regularity result for local minimizers of functionals of the Calculus of Variations of the type$$\int_{\Omega}f(x, Du)\ {\rm d}x$$where Ω is a bounded open set in $\mathbb{R}^{n}$, u∈$W^{1,p}_{\rm loc}$(Ω; $\mathbb{R}^{N}$), p> 1, n≥ 2 and N≥ 1. We use the technique of difference quotient without the usual assumption on the growth of the second derivatives of the function f. We apply this result to give a bound on the Hausdorff dimension of the singular set of minimizers.
LA - eng
KW - Partial regularity; singular sets; fractional differentiability; variational integrals; partial regularity
UR - http://eudml.org/doc/250746
ER -

## References

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