A regularity result for a convex functional and bounds for the singular set

Bruno De Maria

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 16, Issue: 4, page 1002-1017
  • ISSN: 1292-8119

Abstract

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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 , D u ) d x where Ω is a bounded open set in n , u∈ W loc 1 , p (Ω; 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.

How to cite

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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 16.4 (2010): 1002-1017. <http://eudml.org/doc/250746>.

@article{DeMaria2010,
abstract = { In this paper we prove a regularity result for local minimizers of functionals of the Calculus of Variations of the type$$ \int\_\{\Omega\}f(x, Du)\ \{\rm d\}x $$where Ω is a bounded open set in $\mathbb\{R\}^\{n\}$, u∈$W^\{1,p\}_\{\rm loc\}$(Ω; $\mathbb\{R\}^\{N\}$), 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. },
author = {De Maria, Bruno},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Partial regularity; singular sets; fractional differentiability; variational integrals; partial regularity},
language = {eng},
month = {10},
number = {4},
pages = {1002-1017},
publisher = {EDP Sciences},
title = {A regularity result for a convex functional and bounds for the singular set},
url = {http://eudml.org/doc/250746},
volume = {16},
year = {2010},
}

TY - JOUR
AU - De Maria, Bruno
TI - A regularity result for a convex functional and bounds for the singular set
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/10//
PB - EDP Sciences
VL - 16
IS - 4
SP - 1002
EP - 1017
AB - In this paper we prove a regularity result for local minimizers of functionals of the Calculus of Variations of the type$$ \int_{\Omega}f(x, Du)\ {\rm d}x $$where Ω is a bounded open set in $\mathbb{R}^{n}$, u∈$W^{1,p}_{\rm loc}$(Ω; $\mathbb{R}^{N}$), 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.
LA - eng
KW - Partial regularity; singular sets; fractional differentiability; variational integrals; partial regularity
UR - http://eudml.org/doc/250746
ER -

References

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  1. E. Acerbi and N. Fusco, A regularity theorem for minimizers of quasiconvex integrals. Arch. Ration. Mech. Anal.99 (1987) 261–281.  Zbl0627.49007
  2. E. Acerbi and N. Fusco, Regularity of minimizers of non-quadratic functionals: the case 1 < p < 2 . J. Math. Anal. Appl.140 (1989) 115–135.  Zbl0686.49004
  3. R.A. Adams, Sobolev Spaces. Academic Press, New York (1975).  
  4. M. Carozza and A. Passarelli di Napoli, A regularity theorem for minimizers of quasiconvex integrals: the case 1 < p < 2 . Proc. R. Math. Soc. Edinb. A126 (1996) 1181–1199.  Zbl0955.49021
  5. M. Carozza and A. Passarelli di Napoli, Model problems from nonlinear elasticity: partial regularity results. ESAIM: COCV13 (2007) 120–134.  Zbl1221.35132
  6. M. Carozza, N. Fusco and R. Mingione, Partial regularity of minimizers of quasiconvex integrals with subquadratic growth. Annali di matematica pura e applicata (IV)CLXXV (1998) 141–164.  Zbl0960.49025
  7. G. Cupini, N. Fusco and R. Petti, Hölder continuity of local minimizers. J. Math. Anal. Appl.235 (1999) 578–597.  Zbl0949.49022
  8. E. De Giorgi, Un esempio di estremali discontinue per un problema variazionale di tipo ellittico. Boll. Un. Mat. It.1 (1968) 135–137.  Zbl0155.17603
  9. L. Esposito, F. Leonetti and G. Mingione, Higher integrability for minimizers of integral functionals with ( p , q ) growth. J. Differ. Equ.157 (1999) 414–438.  Zbl0939.49021
  10. L. Esposito, F. Leonetti and G. Mingione, Regularity results for minimizers of irregular integrals with ( p , q ) growth. Forum Math.14 (2002) 245–272.  Zbl0999.49022
  11. L. Esposito, F. Leonetti and G. Mingione, Sharp regularity for functionals with ( p , q ) growth. J. Differ. Equ.204 (2004) 5–55.  Zbl1072.49024
  12. L.C. Evans, Quasiconvexity and partial regularity in the Calculus of Variations. Arch. Ration. Mech. Anal.95 (1984) 227–252.  Zbl0627.49006
  13. L.C. Evans and R.F. Gariepy, Blow-up, compactness and partial regularity in the Calculus of Variations. Indiana Univ. Math. J.36 (1987) 361–371.  Zbl0626.49007
  14. I. Fonseca and N. Fusco, Regularity results for anisotropic image segmentation models. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)3 (1997) 463–499.  Zbl0899.49018
  15. I. Fonseca, N. Fusco and P. Marcellini, An existence result for a nonconvex variational problem via regularity. ESAIM: COCV7 (2002) 69–95.  Zbl1044.49011
  16. M. Giaquinta and G. Modica, Remarks on the regularity of minimizers of certain degenerate functionals. Manuscripta Math.47 (1986) 55–99.  Zbl0607.49003
  17. E. Giusti, Direct methods in the calculus of variations. World Scientific, River Edge, USA (2003).  Zbl1028.49001
  18. O. John, J. Malý and J. Stará, Nowhere continuous solutions to elliptic systems. Comm. Math. Univ. Carolin.30 (1989) 33–43.  Zbl0691.35024
  19. J. Kristensen and G. Mingione, Non-differentiable functionals and singular sets of minima. C. R. Acad. Sci. Paris Ser. I Math.340 (2005) 93–98.  Zbl1058.49012
  20. J. Kristensen and G. Mingione, The singular set of minima of integral functionals. Arch. Ration. Mech. Anal.180 (2006) 331–398.  Zbl1116.49010
  21. G. Mingione, The singular set of solutions to non differentiable elliptic systems. Arch. Ration. Mech. Anal.166 (2003) 287–301.  Zbl1142.35391
  22. G. Mingione, Bounds for the singular set of solutions to non linear elliptic system. Calc. Var.18 (2003) 373–400.  Zbl1045.35024
  23. G. Mingione, Regularity of minima: an invitation to the dark side of calculus of variations. Appl. Math.51 (2006) 355–426.  Zbl1164.49324
  24. J. Nečas, Example of an irregular solution to a nonlinear elliptic system with analytic coefficients and conditions for regularity, in Theory of nonlinear operators, Proc. Fourth Internat. Summer School, Acad. Sci., Berlin (1975) 197–206.  
  25. A. Passarelli di Napoli, A regularity result for a class of polyconvex functionals. Ric. di Matem.XLVIII (1994) 379–393.  Zbl0947.35052
  26. V. Šverák and X. Yan, A singular minimizer of a smooth strongly convex functional in three dimensions. Calc. Var.10 (2000) 213–221.  Zbl1013.49027
  27. V. Šverák and X. Yan, Non Lipschitz minimizers of smooth strongly convex variational integrals. Proc. Nat. Acad. Sc. USA99 (2002) 15269–15276.  Zbl1106.49046

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