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

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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

∫ Ω f(x,Du)dx
where Ω is a bounded open set in ℝ n , u ∈ W 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 : 10.1051/cocv/2009030
Classification : 35J50, 35J60, 35B65
Keywords: partial regularity, singular sets, fractional differentiability, variational integrals 
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     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},
     pages = {1002--1017},
     publisher = {EDP-Sciences},
     volume = {16},
     number = {4},
     year = {2010},
     doi = {10.1051/cocv/2009030},
     mrnumber = {2744159},
     zbl = {1203.35088},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2009030/}
}
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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, Tome 16 (2010) no. 4, pp. 1002-1017. doi: 10.1051/cocv/2009030

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