Self-Improving inequalities for bounded weak solutions to nonlocal double phase equations

James M. Scott; Tadele Mengesha

doi:10.3934/cpaa.2021174

Article Contents

2022, Volume 21, Issue 1: 183-212. Doi: 10.3934/cpaa.2021174

This issue Previous Article Marcinkiewicz integral operators along twisted surfaces Next Article Large deviation principle for stochastic Burgers type equation with reflection

Self-Improving inequalities for bounded weak solutions to nonlocal double phase equations

James M. Scott^1, , and
Tadele Mengesha^2,

1.
Department of Mathematics, 301 Thackeray Hall, Pittsburgh, PA 15260
2.
Department of Mathematics, 1403 Circle Drive, Knoxville, TN 37996

^* Corresponding author
^* Corresponding author

Received: January 2021

Early access: October 2021

Published: January 2022

This work is partially supported by NSF grant number 1910180

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Abstract

We prove higher Sobolev regularity for bounded weak solutions to a class of nonlinear nonlocal integro-differential equations. The leading operator exhibits nonuniform growth, switching between two different fractional elliptic "phases" that are determined by the zero set of a modulating coefficient. Solutions are shown to improve both in integrability and differentiability. These results apply to operators with rough kernels and modulating coefficients. To obtain these results we adapt a particular fractional version of the Gehring lemma developed by Kuusi, Mingione, and Sire in their work "Nonlocal self-improving properties" Analysis & PDE, 8(1):57–114 for the specific nonlinear setting under investigation in this manuscript.

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Mathematics Subject Classification: Primary: 35B65, 45K05; Secondary: 35J70, 42B37.

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[1]	K. Adimurthi, T. Mengesha and N. C. Phuc, Gradient weighted norm inequalities for linear elliptic equations with discontinuous coefficients, Appl. Math. Optim., 2018, 1–45. doi: 10.1007/s00245-018-9542-5.
[2]	P. Baroni, M. Colombo and G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Analysis: Theory, Methods & Applications, 121 (2015), 206-222. doi: 10.1016/j.na.2014.11.001.
[3]	P. Baroni, M. Colombo and G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differ. Equ., 57 (2018), 48 pp. doi: 10.1007/s00526-018-1332-z.
[4]	R. F. Bass and M. Kassmann, Hölder continuity of harmonic functions with respect to operators of variable order, Commun. Partial Differ. Equ., 30 (2005), 1249-1259. doi: 10.1080/03605300500257677.
[5]	L. Brasco and E. Lindgren, Higher Sobolev regularity for the fractional p-Laplace equation in the superquadratic case, Adv. Math., 304 (2017), 300-354. doi: 10.1016/j.aim.2016.03.039.
[6]	L. Brasco, E. Lindgren and A. Schikorra, Higher Hölder regularity for the fractional p-Laplacian in the superquadratic case, Adv. Math., 338 (2018), 782-846. doi: 10.1016/j.aim.2018.09.009.
[7]	S-S. Byun and H-S. Lee, Calderón-Zygmund estimates for elliptic double phase problems with variable exponents, J. Math. Anal. Appl., 25 (2020), 3843-3855. doi: 10.1016/j.jmaa.2020.124015.
[8]	S-S. Byun and J. Oh, Global gradient estimates for the borderline case of double phase problems with BMO coefficients in nonsmooth domains, J. Differ. Equ., 263 (2017), 1643-1693. doi: 10.1016/j.jde.2017.03.025.
[9]	S-S. Byun, J. Ok and K. Song, Holder Regularity for weak solutions to nonlocal double phase problems, preprint, arXiv: 2108.09623.
[10]	S-S. Byun and Y. Youn, Riesz potential estimates for a class of double phase problems, J. Differ. Equ., 264 (2018), 1263-1316. doi: 10.1016/j.jde.2017.09.038.
[11]	M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal., 218 (2015), 219-273. doi: 10.1007/s00205-015-0859-9.
[12]	M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal., 215 (2015), 443-496. doi: 10.1007/s00205-014-0785-2.
[13]	M. Colombo and G. Mingione, Calderón–Zygmund estimates and non-uniformly elliptic operators, J. Funct. Anal., 270 (2016), 1416-1478. doi: 10.1016/j.jfa.2015.06.022.
[14]	C. De Filippis, On the regularity of the $\omega$-minima of $\varphi$-functionals, preprint, arXiv: 1810.06050. doi: 10.1016/j.na.2019.02.017.
[15]	C. De Filippis and G. Mingione, A borderline case of Calderón-Zygmund estimates for nonuniformly elliptic problems, St. Petersburg Math. J., 31 (2020), 455-477. doi: 10.1090/spmj/1608.
[16]	C. De Filippis and J. Oh, Regularity for multi-phase variational problems, J. Differ. Equ., 267 (2019), 1631-1670. doi: 10.1016/j.jde.2019.02.015.
[17]	C. De Filippis and G. Palatucci, Hölder regularity for nonlocal double phase equations, J. Differ. Equ., 267 (2019), 547-586. doi: 10.1016/j.jde.2019.01.017.
[18]	M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems (AM-105), Princeton University Press, 1983.
[19]	E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, Singapore, 2003. doi: 10.1142/9789812795557.
[20]	Y. Fang and C. Zhang, On weak and viscosity solutions of nonlocal double phase equations, arXiv: 2106.04412.
[21]	T. Kuusi, G. Mingione and Y. Sire, A fractional Gehring lemma, with applications to nonlocal equations, Rendiconti Lincei-Matematica e Applicazioni, 25 (2014), 345-358. doi: 10.4171/RLM/683.
[22]	T. Kuusi, G. Mingione and Y. Sire, Nonlocal self-improving properties, Anal. PDE, 8 (2015), 57-114. doi: 10.2140/apde.2015.8.57.
[23]	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. doi: 10.1007/BF00251503.
[24]	P. Marcellini, Regularity and existence of solutions of elliptic equations with p, q-growth conditions, J. Differ. Equ., 90 (1991), 1-30. doi: 10.1016/0022-0396(91)90158-6.
[25]	P. Marcellini, Regularity for elliptic equations with general growth conditions, J. Differ. Equ., 105 (1993), 296-333. doi: 10.1006/jdeq.1993.1091.
[26]	P. Marcellini, Everywhere regularity for a class of elliptic systems without growth conditions, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 23 (1996), 1-25.
[27]	T. Mengesha and J. M. Scott, A fractional Korn-type inequality for smooth domains and a regularity estimate for nonlinear nonlocal systems of equations, preprint, arXiv: 2011.12407. doi: 10.3934/dcds.2019137.
[28]	T. Mengesha and J. M. Scott, A note on estimates of level sets and their role in demonstrating regularity of solutions to nonlocal double phase equations, preprint, arXiv: 2011.12407
[29]	G. Mingione, The singular set of solutions to non-differentiable elliptic systems, Arch. Ration. Mech. Anal., 166 (2003), 287-301. doi: 10.1007/s00205-002-0231-8.
[30]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[31]	J. Ok, Partial regularity for general systems of double phase type with continuous coefficients, Nonlinear Anal., 177 (2018), 673-698. doi: 10.1016/j.na.2018.03.021.
[32]	G. Palatucci, The Dirichlet problem for the p-fractional Laplace equation, Nonlinear Anal., 177 (2018), 699-732. doi: 10.1016/j.na.2018.05.004.
[33]	P. Pucci and V. Radulescu, The maximum principle with lack of monotonicity, Electron. J. Qual. Theory Differ. Equ., 2018 (2018), 1-11. doi: 10.14232/ejqtde.2018.1.58.
[34]	A. Schikorra, Nonlinear commutators for the fractional p-Laplacian and applications, Math. Ann., 366 (2016), 695-720. doi: 10.1007/s00208-015-1347-0.
[35]	V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR-Izvestiya, 29 (1987), 34 pp.
[36]	V. V. Zhikov, On Lavrentiev's phenomenon., Russian J. Math. Phys., 3 (1995), 249-269.