Higher differentiability of minimizers of convex variational integrals

Carozza, Menita ; Kristensen, Jan ; Passarelli di Napoli, Antonia

Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011), p. 395-411 / Harvested from Numdam

Résumé

In this paper we consider integral functionals of the form 𝔉(v,Ω)=∫ ΩF(Dv(x))dx with convex integrand satisfying $ (p,q)$ growth conditions. We prove local higher differentiability results for bounded minimizers of the functional $ \mathfrak{F}$ under dimension-free conditions on the gap between the growth and the coercivity exponents.As a novel feature, the main results are achieved through uniform higher differentiability estimates for solutions to a class of auxiliary problems, constructed adding singular higher order perturbations to the integrand.

Détail

Publié le : 2011-01-01

MR 2795713 | Zbl 1245.49052

DOI : https://doi.org/10.1016/j.anihpc.2011.02.005
Classification:  49N15,  49N60,  49N99

Bibliographie

[1] E. Acerbi, G. Bouchitté, I. Fonseca, Relaxation of convex functionals: The gap problem, Ann. I. H. Poincaré – AN 20 no. 3 (2003), 359-390 | Numdam | MR 1972867 | Zbl 1025.49012

[2] E. Acerbi, N. Fusco, Partial regularity under anisotropic $ (p,q)$ growth conditions, J. Differential Equations 107 (1994), 46-67 | MR 1260848 | Zbl 0807.49010

[3] E. Acerbi, G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal. 156 (2001), 121-140 | MR 1814973 | Zbl 0984.49020

[4] M. Bildhauer, Convex Variational Problems. Linear, Nearly Linear and Anisotropic Growth Conditions, Lecture Notes in Math. vol. 1818, Springer-Verlag, Berlin (2003) | MR 1998189 | Zbl 1033.49001

[5] M. Bildhauer, M. Fuchs, Partial regularity for variational integrals with $ (s,\mu ,q)$-growth, Calc. Var. Partial Differ. Eq. 13 (2001), 537-560 | MR 1867941 | Zbl 1018.49026

[6] M. Bildhauer, M. Fuchs, Partial regularity for a class of anisotropic variational integrals with convex hull property, Asymptot. Anal. 32 no. 3–4 (2002), 293-315 | MR 1993652 | Zbl 1076.49018

[7] M. Bildhauer, M. Fuchs, Higher integrability of the gradient for vectorial minimizers of decomposable variational integrals, Manuscripta Math. 123 (2007), 269-283 | MR 2314085 | Zbl 1120.49031

[8] M. Bildhauer, M. Fuchs, Variational integrals of splitting-type: higher integrability under general growth conditions, Ann. Mat. Pura Appl. (4) 188 no. 3 (2009), 467-496 | MR 2512159 | Zbl 1181.49035

[9] S. Campanato, Elliptic systems with nonlinearity q greater or equal to two. Regularity of the solution of the Dirichlet problem, Ann. Mat. Pura Appl. (4) 147 (1987), 117-150 | MR 916705 | Zbl 0635.35038

[10] H.J. Choe, Interior behaviour of minimizers for certain functionals with nonstandard growth, Nonlinear Anal. Theory Methods Appl. 19 no. 10 (1992), 933-945 | MR 1192273 | Zbl 0786.35040

[11] F. Duzaar, J. Kristensen, G. Mingione, The existence of regular boundary points for non-linear elliptic systems, J. Reine Angew. Math. 602 (2007), 17-58 | MR 2300451 | Zbl 1214.35021

[12] L. Esposito, F. Leonetti, G. Mingione, Regularity for minimizers of functionals with $ p-q$ growth, NoDEA Nonlinear Differ. Eq. Appl. 6 (1999), 133-148 | MR 1694803 | Zbl 0928.35044

[13] L. Esposito, F. Leonetti, G. Mingione, Higher integrability for minimizers of integral functionals with $ (p,q)$ growth, J. Differential Equations 157 (1999), 414-438 | MR 1713266 | Zbl 0939.49021

[14] L. Esposito, F. Leonetti, G. Mingione, Regularity results for minimizers of irregular integrals with $ (p,q)$ growth, Forum Mathematicum 14 (2002), 245-272 | MR 1880913 | Zbl 0999.49022

[15] L. Esposito, F. Leonetti, G. Mingione, Sharp regularity for functionals with $ (p,q)$ growth, J. Differential Equations 204 no. 1 (2004), 5-55 | MR 2076158 | Zbl 1072.49024

[16] I. Fonseca, N. Fusco, Regularity results for anisotropic image segmentation models, Ann. Sc. Norm. Super. Pisa 24 (1997), 463-499 | Numdam | MR 1612389 | Zbl 0899.49018

[17] N. Fusco, C. Sbordone, Local boundedness of minimizers in a limit case, Manuscripta Math. 69 (1990), 19-25 | MR 1070292 | Zbl 0722.49012

[18] N. Fusco, C. Sbordone, Higher integrability of the gradient of minimizers of functionals with nonstandard growth conditions, Comm. Pure Appl. Math. 43 (1990), 673-683 | MR 1057235 | Zbl 0727.49021

[19] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Math. Studies vol. 105, Princeton Univ. Press (1983) | MR 717034 | Zbl 0516.49003

[20] M. Giaquinta, Growth conditions and regularity, a counterexample, Manuscripta Math. 59 (1987), 245-248 | MR 905200 | Zbl 0638.49005

[21] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific (2003) | MR 1962933 | Zbl 1028.49001

[22] M.C. Hong, Some remarks on the minimizers of variational integrals with nonstandard growth conditions, Boll. Un. Mat. Ital. A (7) 6 (1992), 91-101 | MR 1164739 | Zbl 0768.49022

[23] J. Kristensen, C. Melcher, Regularity in oscillatory nonlinear elliptic systems, Math. Z. 260 (2008), 813-847 | MR 2443332 | Zbl 1158.35037

[24] J. Kristensen, G. Mingione, The singular set of ω-minima, Arch. Ration. Mech. Anal. 177 (2005), 93-114 | MR 2187315 | Zbl 1082.49036

[25] J. Kristensen, G. Mingione, The singular set of minima of integral functionals, Arch. Ration. Mech. Anal. 180 (2006), 331-398 | MR 2214961 | Zbl 1116.49010

[26] J. Kristensen, G. Mingione, Boundary regularity in variational problems, Arch. Ration. Mech. Anal. 198 (2010), 369-455 | MR 2721587 | Zbl 1228.49043

[27] F. Leonetti, Maximum principles for vector-valued minimizers of some integral functionals, Boll. Un. Mat. Ital. A (6) 5 (1991), 51-56 | MR 1101010 | Zbl 0729.49015

[28] P. Marcellini, On the definition and the lower semicontinuity of certain quasiconvex integrals, Ann. I. H. Poincaré – AN 3 (1986), 391-409 | Numdam | MR 868523 | Zbl 0609.49009

[29] P. Marcellini, Un example de solution discontinue dʼun probéme variationel dans le cas scalaire, preprint, Ist. U. Dini, Firenze, 1987–1988.

[30] P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with non-standard growth conditions, Arch. Ration. Mech. Anal. 105 (1989), 267-284 | MR 969900 | Zbl 0667.49032

[31] P. Marcellini, Regularity and existence of solutions of elliptic equations with $ (p,q)$ growth conditions, J. Differential Equations 90 (1991), 1-30 | MR 1094446 | Zbl 0724.35043

[32] P. Marcellini, Everywhere regularity for a class of elliptic systems without growth conditions, Ann. Sc. Norm. Super. Pisa Cl. Sci. 23 (1996), 1-25 | Numdam | MR 1401415 | Zbl 0922.35031

[33] P. Marcellini, G. Papi, Nonlinear elliptic systems with general growth, J. Differential Equations 221 (2006), 412-443 | MR 2196484 | Zbl 1330.35131

[34] A. Passarelli Di Napoli, F. Siepe, A regularity result for a class of anisotropic systems, Rend. Ist. Mat di Trieste (1997), 13-31 | MR 1463906 | Zbl 0909.49026

[35] T. Riviére, P. Strzelecki, A sharp nonlinear Gagliardo–Nirenberg-type estimate and applications to the regularity of elliptic systems, Comm. Part. Diff. Eq. 30 (2005), 589-604 | MR 2153509 | Zbl 1157.35376

[36] T. Schmidt, Regularity of minimizers of $ {W}^{1,p}$-quasiconvex variational integrals with $ (p,q)$-growth, Calc. Var. 32 (2008), 1-24 | MR 2377404 | Zbl 1138.49034

[37] V. Šverák, X. Yan, Non-Lipschitz minimizers of smooth uniformly convex variational integrals, Proc. Natl. Acad. Sci. USA 99 (2002), 15269-15276 | MR 1946762 | Zbl 1106.49046

[38] V.V. Zhikov, On Lavrentievʼs phenomenon, Russian J. Math. Phys. 3 (1995), 249-269 | MR 1350506

[39] V.V. Zhikov, On some variational problems, Russian J. Math. Phys. 5 (1997), 105-116 | MR 1486765 | Zbl 0917.49006

[40] W.P. Ziemer, Weakly Differentiable Functions, Grad. Texts in Math. vol. 120, Springer-Verlag (1989) | MR 1014685 | Zbl 0177.08006