no. 4
A Haar-Rado type theorem for minimizers in Sobolev spaces
ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 4, pp. 1133-1143

Let u∈ϕ+W 0 1,1 (Ω) be a minimum for

I(v)=∫ Ω g(x,v(x))+f(∇v(x))dx
where f is convex, v↦g(x,v) is convex for a.e. x. We prove that u shares the same modulus of continuity of ϕ whenever Ω is sufficiently regular, the right derivative of g satisfies a suitable monotonicity assumption and the following inequality holds
∀γ∈∂Ω|u(x)-ϕ(γ)|≤ω(|x-γ|)a.e.x∈Ω.
This result generalizes the classical Haar-Rado theorem for Lipschitz functions.

DOI : 10.1051/cocv/2010038
Classification : 49K20
Keywords: Hölder, regularity, Lipschitz 
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     title = {A {Haar-Rado} type theorem for minimizers in {Sobolev} spaces},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1133--1143},
     year = {2011},
     publisher = {EDP Sciences},
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Mariconda, Carlo; Treu, Giulia. A Haar-Rado type theorem for minimizers in Sobolev spaces. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 4, pp. 1133-1143. doi: 10.1051/cocv/2010038

[1] H. Brezis, Analyse fonctionnelle : théorie et applications. Collection Mathématiques Appliquées pour la Maîtrise [Collection of Applied Mathematics for the Master's Degree], Masson, Paris (1983). | Zbl | MR

[2] H. Brezis and M. Sibony, Équivalence de deux inéquations variationnelles et applications. Arch. Rational Mech. Anal. 41 (1971) 254-265. | Zbl | MR

[3] A. Cellina, On the bounded slope condition and the validity of the Euler Lagrange equation. SIAM J. Control Optim. 40 (2002) 1270-1279. | Zbl | MR

[4] A. Cellina, Comparison results and estimates on the gradient without strict convexity. SIAM J. Control Optim. 46 (2007) 738-749. | MR

[5] F. Clarke, Continuity of solutions to a basic problem in the calculus of variations. Ann. Sc. Norm. Super. Pisa Cl. Sci. 4 (2005) 511-530. | Zbl | MR | Numdam

[6] L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. Studies in Advanced Mathematics, CRC Press, Boca Raton (1992). | Zbl | MR

[7] M. Giaquinta and L. Martinazzi, An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs, Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)] 2. Edizioni della Normale, Pisa (2005). | Zbl | MR

[8] P. Hartman, On the bounded slope condition. Pacific J. Math. 18 (1966) 495-511. | Zbl | MR

[9] P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations. Acta Math. 115 (1966) 271-310. | Zbl | MR

[10] C. Mariconda and G. Treu, Lipschitz regularity for minima without strict convexity of the Lagrangian. J. Differ. Equ. 243 (2007) 388-413. | Zbl | MR

[11] C. Mariconda and G. Treu, Local Lipschitz regularity of minima for a scalar problem of the calculus of variations. Commun. Contemp. Math. 10 (2008) 1129-1149. | Zbl | MR

[12] C. Mariconda and G. Treu, Hölder regularity for a classical problem of the calculus of variations. Adv. Calc. Var. 2 (2009) 311-320. | Zbl | MR

[13] M. Miranda, Un teorema di esistenza e unicità per il problema dell'area minima in n variabili. Ann. Scuola Norm. Sup. Pisa 19 (1965) 233-249. | Zbl | MR | Numdam

[14] G. Treu and M. Vornicescu, On the equivalence of two variational problems. Calc. Var. Partial Differential Equations 11 (2000) 307-319. | Zbl | MR

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