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January  2022, 21(1): 183-212. doi: 10.3934/cpaa.2021174

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

Received  January 2021 Published  January 2022 Early access  October 2021

Fund Project: This work is partially supported by NSF grant number 1910180

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.

Keywords: self-improving inequalities, integro-differential equations, double phase equations, nonlocal Gehring lemma, Sobolev regularity.
Mathematics Subject Classification: Primary: 35B65, 45K05; Secondary: 35J70, 42B37.
Citation: James M. Scott, Tadele Mengesha. Self-Improving inequalities for bounded weak solutions to nonlocal double phase equations. Communications on Pure and Applied Analysis, 2022, 21 (1) : 183-212. doi: 10.3934/cpaa.2021174
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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