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) no. 3, pp. 395-411

In this paper we consider integral functionals of the form

$$𝔉(v,\Omega )=\underset{\Omega }{\int }F\left(Dv\left(x\right)\right)\mathrm{d}x$$

with convex integrand satisfying $(p,q)$ growth conditions. We prove local higher differentiability results for bounded minimizers of the functional $𝔉$ 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.

Classification : 49N15, 49N60, 49N99

[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 | MR | EuDML | Zbl | Numdam

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

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

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

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

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

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

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

[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 | Zbl

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

[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 | MR | EuDML | Zbl | Numdam

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

[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 | Zbl

[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 | Zbl

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

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

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

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

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