Existence and regularity of minimizers of nonconvex integrals with p-q growth

Pietro Celada; Giovanni Cupini; Marcello Guidorzi

ESAIM: Control, Optimisation and Calculus of Variations (2007)

  • Volume: 13, Issue: 2, page 343-358
  • ISSN: 1292-8119

Abstract

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We show that local minimizers of functionals of the form Ω f ( D u ( x ) ) + g ( x , u ( x ) ) d x u u 0 + W 0 1 , p ( Ω ) , are locally Lipschitz continuous provided f is a convex function with p - q growth satisfying a condition of qualified convexity at infinity and g is Lipschitz continuous in u. As a consequence of this, we obtain an existence result for a related nonconvex functional.

How to cite

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Celada, Pietro, Cupini, Giovanni, and Guidorzi, Marcello. "Existence and regularity of minimizers of nonconvex integrals with p-q growth." ESAIM: Control, Optimisation and Calculus of Variations 13.2 (2007): 343-358. <http://eudml.org/doc/250001>.

@article{Celada2007,
abstract = { We show that local minimizers of functionals of the form $\int_\{\Omega\} \left[f(Du(x)) + g(x\,,u(x))\right]\,\{\rm d\}x$,  $u \in u_0 + W_0^\{1,p\}(\Omega)$, are locally Lipschitz continuous provided f is a convex function with $p-q$ growth satisfying a condition of qualified convexity at infinity and g is Lipschitz continuous in u. As a consequence of this, we obtain an existence result for a related nonconvex functional. },
author = {Celada, Pietro, Cupini, Giovanni, Guidorzi, Marcello},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Nonstandard growth; existence; Lipschitz continuity; Non standard growth},
language = {eng},
month = {5},
number = {2},
pages = {343-358},
publisher = {EDP Sciences},
title = {Existence and regularity of minimizers of nonconvex integrals with p-q growth},
url = {http://eudml.org/doc/250001},
volume = {13},
year = {2007},
}

TY - JOUR
AU - Celada, Pietro
AU - Cupini, Giovanni
AU - Guidorzi, Marcello
TI - Existence and regularity of minimizers of nonconvex integrals with p-q growth
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2007/5//
PB - EDP Sciences
VL - 13
IS - 2
SP - 343
EP - 358
AB - We show that local minimizers of functionals of the form $\int_{\Omega} \left[f(Du(x)) + g(x\,,u(x))\right]\,{\rm d}x$,  $u \in u_0 + W_0^{1,p}(\Omega)$, are locally Lipschitz continuous provided f is a convex function with $p-q$ growth satisfying a condition of qualified convexity at infinity and g is Lipschitz continuous in u. As a consequence of this, we obtain an existence result for a related nonconvex functional.
LA - eng
KW - Nonstandard growth; existence; Lipschitz continuity; Non standard growth
UR - http://eudml.org/doc/250001
ER -

References

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  1. M. Bildhauer, Convex variational problems. Linear, nearly linear and anisotropic growth conditions, Springer-Verlag, Berlin and New York. Lect. Notes Math.1818 (2003).  Zbl1033.49001
  2. P. Celada, Existence and regularity of minimizers of non convex functionals depending on u and u . J. Math. Anal. Appl.230 (1999) 30–56.  Zbl0928.49028
  3. P. Celada and S. Perrotta, Minimizing non convex, multiple integrals: a density result. Proc. Roy. Soc. Edinburgh130A (2000) 721–741.  Zbl0963.49001
  4. P. Celada and S. Perrotta, On the minimum problem for nonconvex, multiple integrals of product type. Calc. Var. Partial Differential Equations12 (2001) 371–398.  Zbl0985.49007
  5. P. Celada, G. Cupini and M. Guidorzi, A sharp attainment result for nonconvex variational problems. Calc. Var. Partial Differential Equations20 (2004) 301–328.  Zbl1058.49002
  6. A. Cellina, On minima of a functional of the gradient: necessary conditions. Nonlinear Anal.20 (1993) 337–341.  Zbl0784.49021
  7. A. Cellina, On minima of a functional of the gradient: sufficient conditions. Nonlinear Anal.20 (1993) 343–347.  Zbl0784.49022
  8. G. Cupini and A.P. Migliorini, Hölder continuity for local minimizers of a nonconvex variational problem, J. Convex Anal.10 (2003) 389–408.  Zbl1084.49030
  9. G. Cupini, M. Guidorzi and E. Mascolo, Regularity of minimizers of vectorial integrals with p-q growth. Nonlinear Anal.54 (2003) 591–616.  Zbl1027.49032
  10. G. Dal Maso, An introduction to Γ -convergence, Birkhäuser, Boston. Progr. Nonlinear Differential Equations Appl.8 (1993).  
  11. F.S. De Blasi and G. Pianigiani, On the Dirichlet problem for Hamilton-Jacobi equations. A Baire category approach. Nonlinear Differential Equations Appl.6 (1999) 13–34.  Zbl0922.35039
  12. 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
  13. L. Esposito, F. Leonetti and G. Mingione, Sharp regularity for functionals with ( p , q ) growth. J. Differential Equations204 (2004) 5–55.  Zbl1072.49024
  14. I. Fonseca and N. Fusco, Regularity results for anisotropic image segmentation models. Ann. Scuola Norm. Sup. Pisa Cl. Sci.244 (1997) 463–499.  Zbl0899.49018
  15. I. Fonseca, N. Fusco and P. Marcellini, An existence result for a nonconvex variational problem via regularity. ESAIM: COCV. 7 (2002) 69–95.  Zbl1044.49011
  16. G. Friesecke, A necessary and sufficient condition for nonattainment and formation of microstructure almost everywhere in scalar variational problems. Proc. Roy. Soc. Edinburgh Sect. A124 (1994) 437–471.  Zbl0809.49017
  17. M. Giaquinta and E. Giusti, On the regularity of the minima of variational integrals. Acta Math.148 (1982) 31–46.  Zbl0494.49031
  18. M. Giaquinta and E. Giusti, Differentiability of minima of non-differentiable functionals. Invent. Math.72 (1983) 285–298.  Zbl0513.49003
  19. M. Giaquinta and G. Modica, Remarks on the regularity of the minimizers of certain degenerate functionals. Manuscripta Math.57 (1986) 55–99.  Zbl0607.49003
  20. E. Giusti, Direct methods in the calculus of variations, World Scientific Publishing Co., Inc., River Edge, NJ (2003).  Zbl1028.49001
  21. J.J. Manfredi, Regularity for minima of functionals with p-growth. J. Differential Equations76 (1988) 203–212.  Zbl0674.35008
  22. P. Marcellini, Alcune osservazioni sull'esistenza del minimo di integrali del calcolo delle variazioni senza ipotesi di convessità. Rend. Mat.13 (1980) 271–281.  Zbl0454.49015
  23. P. Marcellini, A relation between existence of minima for non convex integrals and uniqueness for non strictly convex integrals of the Calculus of Variations, in Mathematical theories of optimization (S. Margherita Ligure (1981)), J.P. Cecconi and T. Zolezzi Eds., Springer, Berlin. Lect. Notes Math.979 (1983) 216–231.  
  24. P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions. Arch. Rational Mech. Anal.105 (1989) 267–284.  Zbl0667.49032
  25. P. Marcellini, Regularity for elliptic equations with general growth conditions. J. Differential Equations105 (1993) 296–333.  Zbl0812.35042
  26. M.A. Sychev, Characterization of homogeneous scalar variational problems solvable for all boundary data. Proc. Roy. Soc. Edinburgh Sect. A130 (2000) 611–631.  Zbl0971.49004
  27. S. Zagatti, Minimization of functionals of the gradient by Baire's theorem. SIAM J. Control Optim.38 (2000) 384–399.  Zbl0948.49004

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