Strict convexity and the regularity of solutions to variational problems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 3, pp. 862-871.

We consider the problem of minimizing

Ω [L(v(x))+g(x,v(x))]dxonu 0 +W 0 1,2 (Ω)
where Ω is a bounded open subset of N and L is a convex function that grows quadratically outside the unit ball, while, when |v|<1, it behaves like |v| p with 1<p<2. We show that, for each ωΩ, there exists a constant H, depending on ω but not on p, such that both
u W 1,2 (ω) Handu |u| 2-p W 1,2 (ω) H (p-1) 2 ;
in particular, for every i=1,...N, we have max{|u x i | |u| 2-p ,|u x i |}W loc 1,2 (Ω).

Reçu le :
DOI : 10.1051/cocv/2015034
Classification : 49K10
Mots clés : Regularity of solutions, higher differentiability, strict convexity
Cellina, Arrigo 1

1 Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, Via R. Cozzi 53, 20125 Milano, Italy
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Cellina, Arrigo. Strict convexity and the regularity of solutions to variational problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 3, pp. 862-871. doi : 10.1051/cocv/2015034. http://www.numdam.org/articles/10.1051/cocv/2015034/

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