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Sharp regularity for functionals with (\(p\),\(q\)) growth. (English) Zbl 1072.49024

In this very interesting paper the authors discuss the regularity of solutions to non-autonomous anisotropic variational problems. They impose a condition on the growth exponents which ensures higher integrability results for various classes of energy densities. The main point here is to prove the absence of a Lavrentiev phenomenon. By presenting a counterexample they also show the sharpness of this condition which surprizingly differs from the one to be expected from the known results in the autonomous case. In the final sections they focus on the relaxed functional and give an answer to a question of Marcellini concerning isolated singularities.

MSC:

49N60 Regularity of solutions in optimal control
49J25 Optimal control problems with equations with ret. arguments (exist.) (MSC2000)
49J45 Methods involving semicontinuity and convergence; relaxation
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[1] Acerbi, E.; Bouchitté, G.; Fonseca, I., Relaxation of convex functionalsthe gap problem, Ann. IHP (anal. non lineare), 20, 359-380, (2003) · Zbl 1025.49012
[2] Acerbi, E.; Fusco, N., Regularity of minimizers of non-quadratic functionalsthe case 1<p<2, J. math. anal. appl, 140, 115-135, (1989) · Zbl 0686.49004
[3] Acerbi, E.; Fusco, N., Partial regularity under anisotropic (p,q) growth conditions, J. differential equations, 107, 46-67, (1994) · Zbl 0807.49010
[4] Adams, R.A., Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101
[5] G. Buttazzo, M. Belloni, 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. · Zbl 0852.49001
[6] Buttazzo, G.; Mizel, V.J., Interpretation of the Lavrentiev phenomenon by relaxation, J. functional anal, 110, 2, 434-460, (1992) · Zbl 0784.49006
[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, 409-424, (1994) · Zbl 0799.49012
[9] Choe, H.J., Regularity for the minimizers of certain degenerate functionals with non-standard growth conditions, Comm. partial differential equations, 16, 363-372, (1991) · Zbl 0736.35018
[10] Cianchi, A., Boundedness of solutions to variational problems under general growth conditions, Comm. partial differential equations, 22, 1629-1646, (1997) · Zbl 0892.35048
[11] Cianchi, A.; Fusco, N., Gradient regularity for minimizers under general growth conditions, J. reine angew. math, 509, 15-36, (1999) · Zbl 0913.49024
[12] Coifman, R.R.; Fefferman, C., Weighted norm inequalities for maximal functions and singular integrals, Studia math, 51, 241-250, (1974) · Zbl 0291.44007
[13] Coscia, A.; Mingione, G., Hölder continuity of the gradient of p(x)-harmonic mappings, C.R. acad. sci. Paris, 328, 363-368, (1999) · Zbl 0920.49020
[14] L. Diening, Theoretical and numerical results for electrorheological fluids, Ph.D. Thesis, University of Freiburg, 2002. · Zbl 1022.76001
[15] Edmunds, D.R.; Lang, J.; Nekvinda, A., On Lp(x) norms, R. soc. lond. proc. ser. A math. phys. eng. sci, 455, 219-225, (1999) · Zbl 0953.46018
[16] Edmunds, D.E.; Rakosnik, J., Sobolev embeddings with variable exponent, Studia math, 143, 267-293, (2000) · Zbl 0974.46040
[17] Esposito, L.; Leonetti, F.; Mingione, G., Higher integrability for minimizers of integral functionals with (p,q) growth, J. differential equations, 157, 414-438, (1999) · Zbl 0939.49021
[18] Esposito, L.; Leonetti, F.; Mingione, G., Regularity results for minimizers of irregular integrals with (p,q) growth, Forum math, 14, 245-272, (2002) · Zbl 0999.49022
[19] Esposito, L.; Mingione, G., Partial regularity for minimizers of convex integrals with \(L logL\)-growth, Nonlinear differential equations appl, 7, 107-125, (2000) · Zbl 0954.49026
[20] Fonseca, I.; Fusco, N.; Marcellini, P., On the total variation of the Jacobian, J. funct. anal, 207, 1, 1-32, (2004) · Zbl 1041.49016
[21] I. Fonseca, N. Fusco, P. Marcellini, Topological degree, Jacobian determinants and relaxation, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), to appear. · Zbl 1177.49066
[22] Fonseca, I.; Malý, J., Relaxation of multiple integrals below the growth exponent for the energy density, Ann. IHP (anal. nonlineare), 14, 309-338, (1997) · Zbl 0868.49011
[23] M. Foss, 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, 445-464, (2003) · Zbl 1084.49002
[25] Foss, M.; Hrusa, W.; Mizel, V., The Lavrentiev gap phenomenon in nonlinear elasticity, Arch. ration. mech. anal, 167, 337-365, (2003) · Zbl 1090.74010
[26] Franchi, B.; Serapioni, R.; Serra, C.F., Irregular solutions of linear degenerate elliptic equations, Potential anal, 9, 201-216, (1998) · Zbl 0919.35050
[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 · Zbl 0964.76003
[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, 31-46, (1982) · Zbl 0494.49031
[30] Giaquinta, M.; Giusti, E., Differentiability of minima of nondifferentiable functionals, Invent. math, 72, 285-298, (1983) · Zbl 0513.49003
[31] Giusti, E., Direct methods in the calculus of variations, (2003), World Scientific Singapore · Zbl 1028.49001
[32] Iwaniec, T., The Gehring lemma, (), 181-204 · Zbl 0888.30017
[33] Lieberman, G., The natural generalization of the natural conditions of ladyzhenskaya and ural’tseva for elliptic equations, Comm. partial differential equations, 16, 311-361, (1991) · Zbl 0742.35028
[34] Lieberman, G., Gradient estimates for a class of elliptic systems, Ann. mat. pura appl. (IV), 164, 103-120, (1993) · Zbl 0819.35019
[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, 497-522, (1994) · Zbl 0839.35018
[36] Manfredi, J., Regularity for minima of functionals with p-growth, J. differential equations, 76, 203-212, (1988) · Zbl 0674.35008
[37] Marcellini, P., On the definition and the lower semicontinuity of certain quasiconvex integrals, Ann. IHP (anal. nonlineare), 3, 391-409, (1986) · Zbl 0609.49009
[38] P. Marcellini, 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, 267-284, (1989) · Zbl 0667.49032
[40] Marcellini, P., Regularity and existence of solutions of elliptic equations with p,q-growth conditions, J. differential equations, 90, 1-30, (1991) · Zbl 0724.35043
[41] Marcellini, P., Regularity for elliptic equations with general growth conditions, J. differential equations, 105, 296-333, (1993) · Zbl 0812.35042
[42] Marcellini, P., Everywhere regularity for a class of elliptic systems without growth conditions, Ann. scuola norm. sup. Pisa cl. sci, 23, 1-25, (1996) · Zbl 0922.35031
[43] Marcellini, P., Regularity for some scalar variational problems under general growth conditions, J. optim. theory appl, 90, 161-181, (1996) · Zbl 0901.49030
[44] Mascolo, E.; Migliorini, A., Everywhere regularity for vectorial functionals with general growth, ESAIM: control optim. calc. var, 9, 399-418, (2003) · Zbl 1066.49023
[45] G. Mingione, D. Mucci, Integral functionals and the gap problem: sharp bounds for relaxation and energy concentration, SIAM J. Math. Anal., 2003, to appear. · Zbl 1080.49014
[46] Muckenhoupt, B., Weighted norm inequalities for the Hardy maximal function, Trans. amer. math. soc, 165, 207-226, (1972) · Zbl 0236.26016
[47] Musielak, J., Orlicz spaces and modular spaces, (1983), Springer Berlin · Zbl 0557.46020
[48] Pagano, A., Higher integrability for minimizers of variational integrals with nonstandard growth, Ann. univ. ferrara sez. VII (N.S.), 39, 1-17, (1993) · Zbl 0855.49028
[49] Rao, M.M.; Ren, Z.D., Theory of Orlicz spaces, Monographs of pure applied mathematics, Vol. 146, (1991), Marcel Dekker New York · Zbl 0724.46032
[50] Růžička, M., Flow of shear dependent electrorheological fluids, C. R. acad. sci. Paris, 329, 393-398, (1999) · Zbl 0954.76097
[51] Růžička, M., Electrorheological fluids: modeling and mathematical theory, Springer lecture notes in math, Vol. 1748, (2000), Springer Berlin · Zbl 0968.76531
[52] Stein, E.M., Harmonic analysis: real-variable methods, orthogonality and oscillatory integrals, (1993), Princeton University Press Princeton, NJ · Zbl 0821.42001
[53] Talenti, G., Boundedness of minimizers, Hokkaido math. J, 19, 259-279, (1990) · Zbl 0723.58015
[54] Tang, Q., Regularity of minimizers of nonisotropic integrals in the calculus of variations, Ann. mat. pura appl. (IV), 164, 77-87, (1993) · Zbl 0796.49037
[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), (in Russian) (English translation: 16 (1984) 263-270) · Zbl 0569.35029
[56] Zhikov, V.V., On Lavrentiev’s phenomenon, Russian J. math. phys, 3, 249-269, (1995) · Zbl 0910.49020
[57] Zhikov, V.V., Meyer-type estimates for solving the nonlinear Stokes system, Differential equations, 33, 108-115, (1997) · Zbl 0911.35089
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