Sharp regularity for functionals with (p,q) growth

Journal of Differential Equations

Volumn 204, Issue 1, 2004, Pages 5-55

  Esposito, Luca ^a   Leonetti, Francesco ^b   Mingione, Giuseppe ^c  

^a UNIVERSITY OF SALERNO   (Italy)

^b UNIVERSITY OF L'AQUILA   (Italy)

^c UNIVERSITY OF PARMA   (Italy)

Author keywords

Lavrentiev phenomenon; Non standard growth conditions; Regularity; Variational integrals

Indexed keywords

EID: 3543102067     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jde.2003.11.007     Document Type: Article

References (57)

1. Acerbi E. Bouchitté G. Fonseca I. Relaxation of convex functionals: the gap problem Ann. IHP (Anal. non Lineare) 20 2003 359-380
2. Acerbi E. Fusco N. Regularity of minimizers of non-quadratic functionals: the case 1
3. Acerbi E. Fusco N. Partial regularity under anisotropic (p,q) growth conditions J. Differential Equations 107 1994 46-67
4. Adams R.A. Sobolev Spaces 1975 Academic Press New York
5. Buttazzo G. Belloni M. A survey of old and recent results about the gap phenomenon in the Calculus of variations, in: R. Lucchetti, J. Revalski (Eds.), Recent Developments in Well-posed Variational Problems, Mathematical Applictions, Vol. 331, Kluwer Academic Publishers, Dordrecht, 1995, pp. 1-27
6. Buttazzo G. Mizel V.J. Interpretation of the Lavrentiev Phenomenon by relaxation J. Functional Anal. 110 2 1992 434-460
7. Cesari L. Optimization: theory and applications, Applied Mathematics 1983 Springer Berlin
8. Chiadò P.V. Serra C.F. Relaxation of degenerate variational integrals Nonlinear Anal. TMA 22 4 1994 409-424
9. Choe H.J. Regularity for the minimizers of certain degenerate functionals with non-standard growth conditions Comm. Partial Differential Equations 16 1991 363-372
10. Cianchi A. Boundedness of solutions to variational problems under general growth conditions Comm. Partial Differential Equations 22 1997 1629-1646
11. Cianchi A. Fusco N. Gradient regularity for minimizers under general growth conditions J. Reine Angew. Math. 509 1999 15-36
12. Coifman R.R. Fefferman C. Weighted norm inequalities for maximal functions and singular integrals Studia Math. 51 1974 241-250
13. Coscia A. Mingione G. Hölder continuity of the gradient of p(x)-harmonic mappings C.R. Acad. Sci. Paris 328 1999 363-368
14. Diening L. Theoretical and numerical results for electrorheological fluids, Ph.D. Thesis, University of Freiburg, 2002
15. p(x) norms R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 1999 219-225
16. Edmunds D.E. Rakosnik J. Sobolev embeddings with variable exponent Studia Math. 143 2000 267-293
17. Esposito L. Leonetti F. Mingione G. Higher integrability for minimizers of integral functionals with (p,q) growth J. Differential Equations 157 1999 414-438
18. Esposito L. Leonetti F. Mingione G. Regularity results for minimizers of irregular integrals with (p,q) growth Forum Math. 14 2002 245-272
19. Esposito L. Mingione G. Partial regularity for minimizers of convex integrals with L log L-growth Nonlinear Differential Equations Appl. 7 2000 107-125
20. Fonseca I. Fusco N. Marcellini P. On the total variation of the Jacobian J. Funct. Anal. 207 1 2004 1-32
21. Fonseca I. Fusco N. Marcellini P. Topological degree, Jacobian determinants and relaxation, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), to appear.
22. Fonseca I. Malý J. Relaxation of multiple integrals below the growth exponent for the energy density Ann. IHP (Anal. Nonlineare) 14 1997 309-338
23. Foss M. On Lavrentiev's phenomenon, Ph.D. Thesis, Carnegie Mellon University, 2001.
24. Foss M. Examples of the Lavrentiev phenomenon with continuous Sobolev exponent dependence J. Convex Anal. 10 2 2003 445-464
25. Foss M. Hrusa W. Mizel V. The Lavrentiev gap phenomenon in nonlinear elasticity Arch. Ration. Mech. Anal. 167 2003 337-365
26. Franchi B. Serapioni R. Serra C.F. Irregular solutions of linear degenerate elliptic equations Potential Anal. 9 1998 201-216
27. Fuchs M. Seregin G. Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids Springer Lecture Notes in Mathematics Vol. 1749 2000 Springer Berlin
28. Garcia C.J. Rubio de Francia J.L. Weighted Norm Inequalities and Related Topics 1985 North-Holland Amsterdam
29. Giaquinta M. Giusti E. On the regularity of the minima of variational integrals Acta Math. 148 1982 31-46
30. Giaquinta M. Giusti E. Differentiability of minima of nondifferentiable functionals Invent. Math. 72 1983 285-298
31. Giusti E. Direct Methods in the Calculus of Variations 2003 World Scientific Singapore
32. Iwaniec T. The Gehring lemma Duren P.L. et. al. Quasiconformal Mappings and Analysis: Papers Honoring F.W. Gehring 1995 181-204 Springer Berlin, Heidelberg, New York
33. Lieberman G. The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations Comm. Partial Differential Equations 16 1991 311-361
34. Lieberman G. Gradient estimates for a class of elliptic systems Ann. Mat. Pura Appl. (IV) 164 1993 103-120
35. Lieberman G. Gradient estimates for a new class of degenerate elliptic and parabolic equations Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21 4 1994 497-522
36. Manfredi J. Regularity for minima of functionals with p-growth J. Differential Equations 76 1988 203-212
37. Marcellini P. On the definition and the lower semicontinuity of certain quasiconvex integrals Ann. IHP (Anal. Nonlineare) 3 1986 391-409
38. Marcellini P. The stored-energy for some discontinuous deformations in nonlinear elasticity, in: Partial Differential Equations and the Calculus of Variations, Vol. II, Programmes in Nonlinear Differential Equations Applications, Vol. 2, Birkhauser, Boston, MA, 1989, pp. 767-786.
39. Marcellini P. Regularity of minimizers of integrals of the calculus of variations with non-standard growth conditions Arch. Rat. Mech. Anal. 105 1989 267-284
40. Marcellini P. Regularity and existence of solutions of elliptic equations with p,q-growth conditions J. Differential Equations 90 1991 1-30
41. Marcellini P. Regularity for elliptic equations with general growth conditions J. Differential Equations 105 1993 296-333
42. Marcellini P. Everywhere regularity for a class of elliptic systems without growth conditions Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23 1996 1-25
43. Marcellini P. Regularity for some scalar variational problems under general growth conditions J. Optim. Theory Appl. 90 1996 161-181
44. Mascolo E. Migliorini A. Everywhere regularity for vectorial functionals with general growth ESAIM: Control Optim. Calc. Var. 9 2003 399-418
45. Mingione G. Mucci D. Integral functionals and the gap problem: sharp bounds for relaxation and energy concentration, SIAM J. Math. Anal., 2003, to appear.
46. Muckenhoupt B. Weighted norm inequalities for the Hardy maximal function Trans. Amer. Math. Soc. 165 1972 207-226
47. Musielak J. Orlicz Spaces and Modular Spaces 1983 Springer Berlin
48. Pagano A. Higher integrability for minimizers of variational integrals with nonstandard growth Ann. Univ. Ferrara Sez. VII (N.S.) 39 1993 1-17
49. Rao M.M. Ren Z.D. Theory of Orlicz Spaces Monographs of Pure Applied Mathematics Vol. 146 1991 Marcel Dekker New York
50. Růžička M. Flow of shear dependent electrorheological fluids C. R. Acad. Sci. Paris 329 1999 393-398
51. Růžička M. Electrorheological Fluids: Modeling and Mathematical Theory Springer Lecture Notes in Math. Vol. 1748 2000 Springer Berlin
52. Stein E.M. Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals 1993 Princeton University Press Princeton, NJ
53. Talenti G. Boundedness of minimizers Hokkaido Math. J. 19 1990 259-279
54. Tang Q. Regularity of minimizers of nonisotropic integrals in the calculus of variations Ann. Mat. Pura Appl. (IV) 164 1993 77-87
55. Ural'tseva N. Urdaletova A.B. The boundedness of the gradients of generalized solutions of degenerate quasilinear nonuniformly elliptic equations Vestnik Leningrad Univ. Math. 19 1983
56. Zhikov V.V. On Lavrentiev's phenomenon Russian J. Math. Phys. 3 1995 249-269
57. Zhikov V.V. Meyer-type estimates for solving the nonlinear Stokes system Differential Equations 33 1997 108-115