A regularity result for a convex functional and bounds for the singular set
ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 4, p. 1002-1017

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.

DOI : https://doi.org/10.1051/cocv/2009030
Classification:  35J50,  35J60,  35B65
Keywords: partial regularity, singular sets, fractional differentiability, variational integrals
@article{COCV_2010__16_4_1002_0,
author = {De Maria, Bruno},
title = {A regularity result for a convex functional and bounds for the singular set},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {16},
number = {4},
year = {2010},
pages = {1002-1017},
doi = {10.1051/cocv/2009030},
zbl = {1203.35088},
mrnumber = {2744159},
language = {en},
url = {http://www.numdam.org/item/COCV_2010__16_4_1002_0}
}

De Maria, Bruno. A regularity result for a convex functional and bounds for the singular set. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 4, pp. 1002-1017. doi : 10.1051/cocv/2009030. http://www.numdam.org/item/COCV_2010__16_4_1002_0/

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