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Self-Improving inequalities for bounded weak solutions to nonlocal double phase equations
1. | Department of Mathematics, 301 Thackeray Hall, Pittsburgh, PA 15260 |
2. | Department of Mathematics, 1403 Circle Drive, Knoxville, TN 37996 |
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.
References:
[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.
|
show all references
References:
[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.
|
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