An a priori Campanato type regularity condition for local minimisers in the calculus of variations

Dodd, Thomas J.

doi:10.1051/cocv:2008066

Dodd, Thomas J.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 1, pp. 111-131.

Résumé

An a priori Campanato type regularity condition is established for a class of W¹X local minimisers $\bar{u}$ of the general variational integral $\int_{Ω} F (\nabla u (x)) d x$ where $Ω \subset ℝ^{n}$ is an open bounded domain, F is of class C², F is strongly quasi-convex and satisfies the growth condition $F (ξ) \leq c (1 + | ξ |^{p})$ for a p > 1 and where the corresponding Banach spaces X are the Morrey-Campanato space $ℒ^{p, μ} (Ω, ℝ^{N \times n})$ , µ < n, Campanato space $ℒ^{p, n} (Ω, ℝ^{N \times n})$ and the space of bounded mean oscillation $BMO Ω, ℝ^{N \times n})$ . The admissible maps $u : Ω \to ℝ^{N}$ are of Sobolev class W^1,p, satisfying a Dirichlet boundary condition, and to help clarify the significance of the above result the sufficiency condition for W¹BMO local minimisers is extended from Lipschitz maps to this admissible class.

MR Zbl

DOI : 10.1051/cocv:2008066

Classification : 49N60, 49K10
Mots clés : calculus of variations, local minimiser, partial regularity, strong quasiconvexity, Campanato space, Morrey space, Morrey-Campanato space, space of bounded mean oscillation, extremals, positive second variation

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     title = {An a priori {Campanato} type regularity condition for local minimisers in the calculus of variations},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {111--131},
     publisher = {EDP-Sciences},
     volume = {16},
     number = {1},
     year = {2010},
     doi = {10.1051/cocv:2008066},
     mrnumber = {2598091},
     zbl = {1183.49037},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2008066/}
}

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Dodd, Thomas J. An a priori Campanato type regularity condition for local minimisers in the calculus of variations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 1, pp. 111-131. doi : 10.1051/cocv:2008066. http://www.numdam.org/articles/10.1051/cocv:2008066/

Bibliographie
Cité par

[1] E. Acerbi and N. Fusco, A regularity theorem for quasiconvex integrals. Arch. Ration. Mech. Anal. 99 (1987) 261-281. | Zbl

[2] E. Acerbi and N. Fusco, Local regularity for minimizers of non convex integrals. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 16 (1989) 603-636. | Numdam | Zbl

[3] E. Acerbi and N. Fusco, Regularity for minimizers of non-quadratic functionals: the case 1 < p < 2. J. Math. Anal. Appl. 140 (1989) 115-135. | Zbl

[4] S. Campanato, Proprietà di hölderianità di alcune classi di funzioni. Ann. Scuola Norm. Sup. Pisa (3) 17 (1963) 175-188. | Numdam | Zbl

[5] M. Carozza and A. Passarelli Di Napoli, Partial regularity of local minimisers of quasiconvex integrals with sub-quadratic growth. Proc. Roy. Soc. Edinburgh 133 (2003) 1249-1262. | Zbl

[6] M. Carozza, N. Fusco and G. Mingione, Partial regularity of minimisers of quasiconvex integrals with subquadratic growth. Ann. Mat. Pura Appl. 175 (1998) 141-164. | Zbl

[7] F. Duzaar, J.F. Grotowski and M. Kronz, Regularity of almost minimizers of quasi-convex variational integrals with subquadratic growth. Ann. Mat. Pura Appl. 184 (2005) 421-448.

[8] L.C. Evans, Quasiconvexity and partial regularity in the calculus of variations. Arch. Ration. Mech. Anal. 95 (1986) 227-252. | Zbl

[9] C. Fefferman and E.M. Stein, Hp spaces of several variables. Acta Math. 129 (1972) 137-193. | Zbl

[10] N.B. Firoozye, Positive second variation and local minimisers in BMO-Sobolev spaces. SFB 256: Preprint No. 252, University of Bonn, Germany (1992).

[11] E. Giusti, Direct methods in the calculus of variations. World Scientific Publishing, Singapore (2003). | Zbl

[12] E. Giusti and M. Miranda, Sulla regolaritá delle soluzioni di una classe di sistemi ellittici quasi-lineari. Arch. Ration. Mech. Anal. 31 (1968) 173-184. | Zbl

[13] Y. Grabovsky and T. Mengesha, Direct approach to the problem of strong local minima in calculus of variations. Calc. Var. Partial Differential Equations 29 (2007) 59-83. | Zbl

[14] F. John and L. Nirenberg, On functions of bounded mean oscillation. Comm. Pure Appl. Math. 14 (1961) 415-426. | Zbl

[15] J. Kristensen and A. Taheri, Partial regularity of strong local minimizers in the multi-dimensional calculus of variations. Arch. Ration. Mech. Anal. 170 (2003) 63-89. | Zbl

[16] R. Moser, Vanishing mean oscillation and regularity in the calculus of variations. Preprint No. 96, MPI for Mathematics in the Sciences, Leipzig, Germany (2001).

[17] S. Müller and V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity. Ann. Math. 157 (2003) 715-742. | Zbl

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