Lipschitz regularity of minimizers of variational integrals with variable exponents. (English) Zbl 1518.35307
Summary: In this paper we prove the Lipschitz regularity for local minimizers of convex variational integrals of the form
\[
\mathfrak{F} ( v , \Omega ) = \int_\Omega F ( x , D v ( x ) ) d x ,
\]
where, for \(n \geq 2\) and \(N \geq 1\), \(\Omega\) is a bounded open set in \(\mathbb{R}^n\), \(u \in \operatorname{W}^{1 , 1} ( \Omega , \mathbb{R}^N )\) and the energy density \(F : \Omega \times \mathbb{R}^{N \times n} \to \mathbb{R}\) satisfies the so called variable growth conditions. The main novelty here is that we assume an almost critical regularity in the Orlicz Sobolev setting for the energy density as a function of the \(x\) variable.
MSC:
| 35J50 | Variational methods for elliptic systems |
| 35A15 | Variational methods applied to PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
References:
| [1] | Acerbi, E.; Mingione, G., Regularity results for a class of functionals with nonstandard growth, Arch. Ration. Mech. Anal., 156, 121-140 (2001) · Zbl 0984.49020 |
| [2] | Adams, R. A., Sobolev Spaces (1975), Academic Press, New York · Zbl 0314.46030 |
| [3] | Caselli, M.; Eleuteri, M.; Passarelli di Napoli, A., Regularity results for a class of obstacle problems with \(p , q -\) growth conditions, ESAIM Control Optim. Calc. Var., 27, 27 (2021) · Zbl 1468.35068 |
| [4] | Cianchi, A., Continuity properties of functions from Orlicz-Sobolev spaces and embedding theorems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 23, 3, 575-608 (1996) · Zbl 0877.46023 |
| [5] | Cianchi, A., A sharp embedding theorem for Orlicz-Sobolev spaces, Indiana Univ. Math. J., 45, 1, 39-66 (1996) · Zbl 0860.46022 |
| [6] | Clop, A.; Giova, R.; Hatami, F.; Passarelli di Napoli, A., Very degenerate elliptic equations under almost critical Sobolev regularity, Forum Math., 32, 6, 1515-1537 (2020) · Zbl 1460.35120 |
| [7] | Coscia, A.; Mingione, G., Hölder continuity of the gradient of \(p ( x ) -\) harmonic mappings, C. R. Math. Acad. Sci. Paris, 328, 363-368 (1999) · Zbl 0920.49020 |
| [8] | Cruz-Uribe, D.; Fiorenza, A., (Variable Lebesgue Spaces. Variable Lebesgue Spaces, Applied and Numerical Harmonic Analysis (2013), Springer: Springer Basel) · Zbl 1268.46002 |
| [9] | Cupini, G.; Guidorzi, M.; Mascolo, E., Regularity of minimizers of vectorial integrals with p-q growth, Nonlinear Anal., 54, 591-616 (2003) · Zbl 1027.49032 |
| [10] | Cupini, G.; Marcellini, P.; Mascolo, E.; Passarelli di Napoli, A., Lipschitz regularity for degenerate elliptic integrals with p,q-growth, Adv. Calc. Var. (2021) |
| [11] | De Filippis, C.; Mingione, G., Lipschitz bounds and nonautonomous integrals, Arc. Ration. Mech. Anal., 242, 973-1057 (2021) · Zbl 1483.49050 |
| [12] | Diening, L.; Hǎrjulehto, P.; Hǎsto, P.; Růžička, M., Lebesgue and Sobolev spaces with variable exponents, Lect. Not. Math., 2017 (2011) · Zbl 1222.46002 |
| [13] | Eleuteri, M., Hölder continuity results for a class of functionals with non standard growth, Boll. Unione Mat. Ital., 7- B, 8, 129-157 (2004) · Zbl 1178.49045 |
| [14] | Eleuteri, M.; Marcellini, P.; Mascolo, E., Lipschitz continuity for functionals with variable exponents, Rend. Lincei Mat. Appl., 27, 61-87 (2016) · Zbl 1338.35169 |
| [15] | Eleuteri, M.; Marcellini, P.; Mascolo, E., Lipschitz estimates for systems with ellipticity conditions at infinity, Ann. Mat. Pura Appl., 195, 1575-1603 (2016) · Zbl 1354.35035 |
| [16] | Foralli, N.; Giliberti, G., Higher differentiability of solutions for a class of obstacle problems with variable exponents, J. Differential Equations, 313, 244-268 (2021) · Zbl 1482.49035 |
| [17] | Giannetti, F.; Passarelli di Napoli, A., Higher differentiability of minimizers of variational integrals with variable exponents, Math. Z., 280, 3-4, 873-892 (2015) · Zbl 1320.49025 |
| [18] | Giannetti, F.; Passarelli di Napoli, A.; Ragusa, M. A.; Tachikawa, A., Partial regularity for minimizers of a class of non autonomous functionals with nonstandard growth, Calc. Var. Partial Differential Equations, 56, 153 (2017) · Zbl 1383.49047 |
| [19] | Giannetti, F.; Passarelli di Napoli, A.; Tachikawa, A., Partial regularity results for non autonomous functionals with \(\Phi \)-growth conditions, Ann. Mat. Pura Appl., 196, 2147-2165 (2017) · Zbl 1381.49040 |
| [20] | Giaquinta, M., (Multiple Integrals in the Calculus of Variations an Nonlinear Elliptic Systems. Multiple Integrals in the Calculus of Variations an Nonlinear Elliptic Systems, Ann. Math. Studies, vol. 105 (1983), Princeton University Press: Princeton University Press Princeton) · Zbl 0516.49003 |
| [21] | Giusti, E., Direct Methods in the Calculus of Variations (2003), World scientific publishing Co. · Zbl 1028.49001 |
| [22] | Hǎrjulehto, P.; Hǎsto, P., (Orlicz Spaces and Generalized Orlicz Spaces. Orlicz Spaces and Generalized Orlicz Spaces, Lecture Notes in Mathematics, vol. 2236 (2019), Springer: Springer Cham) · Zbl 1436.46002 |
| [23] | Hǎrjulehto, P.; Hǎsto, P.; Le, U. V.; Nuortio, M., Overview of differential equations with non-standard growth, Nonlinear Anal., 72, 4551-4574 (2010) · Zbl 1188.35072 |
| [24] | Iwaniec, T.; Migliaccio, L.; Moscariello, G.; Passarelli di Napoli, A., A priori estimates for nonlinear elliptic complexes, Adv. Differ. Equ., 8,5, 513-546 (2003) · Zbl 1290.35074 |
| [25] | Ladyzhenskaya, O. A.; Ural’tseva, N. N., (Linear and Quasilinear Elliptic Equations. Linear and Quasilinear Elliptic Equations, Mathematics in Science and Engineering, vol. 46 (1968), Academic Press: Academic Press New York-London) · Zbl 0164.13002 |
| [26] | 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 |
| [27] | Marcellini, P., Regularity for elliptic equations with general growth conditions, J. Differential Equations, 105, 296-333 (1993) · Zbl 0812.35042 |
| [28] | Marcellini, P., Regularity for some scalar variational problems under general growth conditions, J. Optim. Theory Appl., 90, 161-181 (1996) · Zbl 0901.49030 |
| [29] | Marcellini, P., Anisotropy versus inhomogeneity in the calculus of variations, J. Convex Anal., 28, 2, 613-618 (2021) · Zbl 1467.35126 |
| [30] | Marcellini, P., Growth conditions and regularity for weak solutions to nonlinear elliptic pdes, J. Math. Anal. Appl., 501, 1, Article 124408 pp. (2021) · Zbl 1512.35301 |
| [31] | Marcellini, P., Local Lipschitz continuity for \(p , q -\) PDEs with explicit \(u -\) dependence, Nonlinear Anal., 226 (2023) · Zbl 1501.35188 |
| [32] | M.M. Rao, Z.D. Ren, Theory of Orlicz Spaces, in: Monogr. Textb. Pure Appl. Math., vol. 146, New York, 1991. · Zbl 0724.46032 |
| [33] | Zhikov, V. V., On Lavrentiev’s phenomenon, Russ. J. Math. Phys., 3, 249-269 (1995) · Zbl 0910.49020 |
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