Abstract
We study local regularity properties of local minimizers of scalar integral functionals of the form
where the convex integrand F satisfies controlled
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: BE 5922/1-1
Funding statement: P. Bella was partially supported by the German Science Foundation DFG in context of the Emmy Noether Junior Research Group BE 5922/1-1.
References
[1] K. Adimurthi and V. Tewary, On Lipschitz regularity for bounded minimizers of functionals with (p, q) growth, preprint (2021), https://arxiv.org/abs/2108.06153. 10.1515/forum-2022-0108Search in Google Scholar
[2] A. K. Balci, L. Diening and M. Surnachev, New examples on Lavrentiev gap using fractals, Calc. Var. Partial Differential Equations 59 (2020), no. 5, Paper No. 180. 10.1007/s00526-020-01818-1Search in Google Scholar
[3] P. Baroni, Riesz potential estimates for a general class of quasilinear equations, Calc. Var. Partial Differential Equations 53 (2015), no. 3–4, 803–846. 10.1007/s00526-014-0768-zSearch in Google Scholar
[4] P. Baroni, M. Colombo and G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations 57 (2018), no. 2, Paper No. 62. 10.1007/s00526-018-1332-zSearch in Google Scholar
[5] L. Beck and G. Mingione, Lipschitz bounds and nonuniform ellipticity, Comm. Pure Appl. Math. 73 (2020), no. 5, 944–1034. 10.1002/cpa.21880Search in Google Scholar
[6]
P. Bella and M. Schäffner,
On the regularity of minimizers for scalar integral functionals with
[7] P. Bella and M. Schäffner, Local boundedness and Harnack inequality for solutions of linear nonuniformly elliptic equations, Comm. Pure Appl. Math. 74 (2021), no. 3, 453–477. 10.1002/cpa.21876Search in Google Scholar
[8] G. Bertazzoni and S. Riccò, Lipschitz regularity results for a class of obstacle problems with nearly linear growth, J. Elliptic Parabol. Equ. 6 (2020), no. 2, 883–918. 10.1007/s41808-020-00088-4Search in Google Scholar
[9] M. Bildhauer and M. Fuchs, Interior regularity for free and constrained local minimizers of variational integrals under general growth and ellipticity conditions, J. Math. Sci. (N.Y.) 123 (2004), no. 6, 4565–4576. 10.1023/B:JOTH.0000041474.73595.d3Search in Google Scholar
[10] P. Bousquet and L. Brasco, Global Lipschitz continuity for minima of degenerate problems, Math. Ann. 366 (2016), no. 3–4, 1403–1450. 10.1007/s00208-016-1362-9Search in Google Scholar
[11] M. Bulíček, P. Gwiazda and J. Skrzeczkowski, On a range of exponents for absence of Lavrentiev phenomenon for double phase functionals, preprint (2021), https://arxiv.org/abs/2110.13945. Search in Google Scholar
[12] S.-S. Byun and J. Oh, Regularity results for generalized double phase functionals, Anal. PDE 13 (2020), no. 5, 1269–1300. 10.2140/apde.2020.13.1269Search in Google Scholar
[13] M. Carozza, J. Kristensen and A. Passarelli di Napoli, Higher differentiability of minimizers of convex variational integrals, Ann. Inst. H. Poincaré C Anal. Non Linéaire 28 (2011), no. 3, 395–411. 10.1016/j.anihpc.2011.02.005Search in Google Scholar
[14]
I. Chlebicka, C. De Filippis and L. Koch,
Boundary regularity for manifold constrained
[15]
A. Cianchi,
Maximizing the
[16] A. Cianchi and V. G. Maz’ya, Global Lipschitz regularity for a class of quasilinear elliptic equations, Comm. Partial Differential Equations 36 (2011), no. 1, 100–133. 10.1080/03605301003657843Search in Google Scholar
[17] M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal. 215 (2015), no. 2, 443–496. 10.1007/s00205-014-0785-2Search in Google Scholar
[18]
G. Cupini, P. Marcellini, E. Mascolo and A. Passarelli di Napoli,
Lipschitz regularity for degenerate elliptic integrals with
[19] C. De Filippis, Quasiconvexity and partial regularity via nonlinear potentials, preprint (2021), https://arxiv.org/abs/2105.00503. Search in Google Scholar
[20] C. De Filippis and G. Mingione, On the regularity of minima of non-autonomous functionals, J. Geom. Anal. 30 (2020), no. 2, 1584–1626. 10.1007/s12220-019-00225-zSearch in Google Scholar
[21] C. De Filippis and G. Mingione, Lipschitz bounds and nonautonomous integrals, Arch. Ration. Mech. Anal. 242 (2021), no. 2, 973–1057. 10.1007/s00205-021-01698-5Search in Google Scholar
[22] C. De Filippis and G. Mingione, Nonuniformly elliptic Schauder theory, preprint (2022), https://arxiv.org/abs/2201.07369. Search in Google Scholar
[23] M. Eleuteri, P. Marcellini and E. Mascolo, Regularity for scalar integrals without structure conditions, Adv. Calc. Var. 13 (2020), no. 3, 279–300. 10.1515/acv-2017-0037Search in Google Scholar
[24]
L. Esposito, F. Leonetti and G. Mingione,
Higher integrability for minimizers of integral functionals with
[25]
L. Esposito, F. Leonetti and G. Mingione,
Sharp regularity for functionals with
[26] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, River Edge, 2003. 10.1142/5002Search in Google Scholar
[27] P. Harjulehto, P. Hästö and O. Toivanen, Hölder regularity of quasiminimizers under generalized growth conditions, Calc. Var. Partial Differential Equations 56 (2017), no. 2, Paper No. 22. 10.1007/s00526-017-1114-zSearch in Google Scholar
[28] P. Hästö and J. Ok, Maximal regularity for local minimizers of non-autonomous functionals, J. Eur. Math. Soc. (JEMS) 24 (2022), no. 4, 1285–1334. 10.4171/jems/1118Search in Google Scholar
[29] J. Hirsch and M. Schäffner, Growth conditions and regularity, an optimal local boundedness result, Commun. Contemp. Math. 23 (2021), no. 3, Paper No. 2050029. 10.1142/S0219199720500297Search in Google Scholar
[30]
L. Koch,
Global higher integrability for minimisers of convex functionals with
[31] T. Kuusi and G. Mingione, Linear potentials in nonlinear potential theory, Arch. Ration. Mech. Anal. 207 (2013), no. 1, 215–246. 10.1007/s00205-012-0562-zSearch in Google Scholar
[32] T. Kuusi and G. Mingione, A nonlinear Stein theorem, Calc. Var. Partial Differential Equations 51 (2014), no. 1–2, 45–86. 10.1007/s00526-013-0666-9Search in Google Scholar
[33] P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch. Ration. Mech. Anal. 105 (1989), no. 3, 267–284. 10.1007/BF00251503Search in Google Scholar
[34]
P. Marcellini,
Regularity and existence of solutions of elliptic equations with
[35] P. Marcellini, Growth conditions and regularity for weak solutions to nonlinear elliptic pdes, J. Math. Anal. Appl. 501 (2021), no. 1, Paper No. 124408. 10.1016/j.jmaa.2020.124408Search in Google Scholar
[36] E. Mascolo, E. A. Passarelli di Napoli, Higher differentiability for a class of problems under p, q subquadratic growth, preprint (2021), https://arxiv.org/abs/2110.15874. Search in Google Scholar
[37] G. Mingione and V. Rǎdulescu, Recent developments in problems with nonstandard growth and nonuniform ellipticity, J. Math. Anal. Appl. 501 (2021), no. 1, Paper No. 125197. 10.1016/j.jmaa.2021.125197Search in Google Scholar
[38] C. Mooney, A proof of the Krylov–Safonov theorem without localization, Comm. Partial Differential Equations 44 (2019), no. 8, 681–690. 10.1080/03605302.2019.1581807Search in Google Scholar
[39] M. Schäffner, Higher integrability for variational integrals with non-standard growth, Calc. Var. Partial Differential Equations 60 (2021), no. 2, Paper No. 77. 10.1007/s00526-020-01907-1Search in Google Scholar
[40]
E. M. Stein,
Editor’s note: The differentiability of functions in
[41] V. Sverák and X. Yan, Non-Lipschitz minimizers of smooth uniformly convex functionals, Proc. Natl. Acad. Sci. USA 99 (2002), no. 24, 15269–15276. 10.1073/pnas.222494699Search in Google Scholar PubMed PubMed Central
[42] L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Lect. Notes Unione Mat. Ital. 3, Springer, Berlin, 2007. Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- An inequality for the normal derivative of the Lane–Emden ground state
- Homogenization of high-contrast composites under differential constraints
- Bounds for eigenfunctions of the Neumann p-Laplacian on noncompact Riemannian manifolds
- The first Grushin eigenvalue on cartesian product domains
- Lipschitz bounds for integral functionals with (p,q)-growth conditions
- Polygons as maximizers of Dirichlet energy or first eigenvalue of Dirichlet-Laplacian among convex planar domains
- Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets
- Functional inequalities and applications to doubly nonlinear diffusion equations
- Quasistatic crack growth in elasto-plastic materials with hardening: The antiplane case
- Stability of the ball under volume preserving fractional mean curvature flow