Generalized structural conditions for a Calderón-Zygmund theory on double phase problems

Pilsoo Shin; Yeonghun Youn; Pilsoo Shin; Yeonghun Youn

doi:10.3934/math.20251126

AIMS Mathematics

2025, Volume 10, Issue 11: 25434-25451. doi: 10.3934/math.20251126

Previous Article Next Article

Research article

Generalized structural conditions for a Calderón-Zygmund theory on double phase problems

Pilsoo Shin ¹,
Yeonghun Youn ^{2
,
,}

1.
Department of Mathematics, Kyonggi University, 154-42 Daehak-ro, Yeongtong-gu, Suwon-si, Gyeonggi-do, 16227, Republic of Korea
2.
Department of Mathematics Education, Incheon National University, 119 Academy-ro, Yeonsu-gu, Incheon, 22012, Republic of Korea

Received: 14 August 2025 Revised: 23 October 2025 Accepted: 29 October 2025 Published: 05 November 2025
MSC : 35A15, 35B65, 35J60

In this paper, we considered a general class of double-phase problems of the form
$ - {{\rm{div}}} (A(x, Du)) = - {{\rm{div}}}( B(x, F)). $
Here, the ellipticity, growth, and continuity assumptions on the main operator $ A $ were described using an auxiliary vector field $ G $. The proposed structural condition allowed us to treat, in a single setting, different structural forms that have appeared in earlier studies. Under this generalized structure, we derived Calderón-Zygmund estimates for the gradients of solutions.
- double phase problems,
- discontinuous coefficients,
- Calderón-Zygmund theory,
- nonlinear elliptic equations,
- calculus of variations
Citation: Pilsoo Shin, Yeonghun Youn. Generalized structural conditions for a Calderón-Zygmund theory on double phase problems[J]. AIMS Mathematics, 2025, 10(11): 25434-25451. doi: 10.3934/math.20251126

Related Papers:

Abstract

In this paper, we considered a general class of double-phase problems of the form

$ - {{\rm{div}}} (A(x, Du)) = - {{\rm{div}}}( B(x, F)). $

Here, the ellipticity, growth, and continuity assumptions on the main operator $ A $ were described using an auxiliary vector field $ G $. The proposed structural condition allowed us to treat, in a single setting, different structural forms that have appeared in earlier studies. Under this generalized structure, we derived Calderón-Zygmund estimates for the gradients of solutions.

References

[1]	E. Acerbi, G. Mingione, Gradient estimates for a class of parabolic systems, Duke Math. J., 136 (2007), 285–320. https://doi.org/10.1215/S0012-7094-07-13623-8 doi: 10.1215/S0012-7094-07-13623-8
[2]	S. Baasandorj, S. S. Byun, J. Oh, Calderón-Zygmund estimates for generalized double phase problems, J. Funct. Anal., 279 (2020), 108670. https://doi.org/10.1016/j.jfa.2020.108670 doi: 10.1016/j.jfa.2020.108670
[3]	P. Baroni, M. Colombo, G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations, 57 (2018), 62. https://doi.org/10.1007/s00526-018-1332-z doi: 10.1007/s00526-018-1332-z
[4]	V. Bögelein, F. Duzaar, G. Mingione, Degenerate problems with irregular obstacles, J. Reine Angew. Math., 2011 (2011), 107–160. https://doi.org/10.1515/crelle.2011.006 doi: 10.1515/crelle.2011.006
[5]	S. S. Byun, Y. Kim, Elliptic equations with measurable nonlinearities in nonsmooth domains, Adv. Math., 288 (2016), 152–200. https://doi.org/10.1016/j.aim.2015.10.015 doi: 10.1016/j.aim.2015.10.015
[6]	S. S. Byun, H. S. Lee, Calderón-Zygmund estimates for elliptic double phase problems with variable exponents, J. Math. Anal. Appl., 501 (2021), 124015. https://doi.org/10.1016/j.jmaa.2020.124015 doi: 10.1016/j.jmaa.2020.124015
[7]	S. S. Byun, H. S. Lee, Gradient estimates for non-uniformly elliptic problems with BMO nonlinearity, J. Math. Anal. Appl., 520 (2023), 126894. https://doi.org/10.1016/j.jmaa.2022.126894 doi: 10.1016/j.jmaa.2022.126894
[8]	S. S. Byun, J. Oh, Global gradient estimates for non-uniformly elliptic equations, Calc. Var. Partial Differential Equations, 56 (2017), 46. https://doi.org/10.1007/s00526-017-1148-2 doi: 10.1007/s00526-017-1148-2
[9]	S. S. Byun, L. Wang, Elliptic equations with BMO coefficients in Reifenberg domains, Comm. Pure Appl. Math., 57 (2004), 1283–1310. https://doi.org/10.1002/cpa.20037 doi: 10.1002/cpa.20037
[10]	M. Colombo, G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal., 218 (2015), 219–273. https://doi.org/10.1007/s00205-015-0859-9 doi: 10.1007/s00205-015-0859-9
[11]	M. Colombo, G. Mingione, Calderón-Zygmund estimates and non-uniformly elliptic operators, J. Funct. Anal., 270 (2016), 1416–1478. https://doi.org/10.1016/j.jfa.2015.06.022 doi: 10.1016/j.jfa.2015.06.022
[12]	M. Colombo, G. Mingione, Regularity for double phase variational problems, Arch. Rational Mech. Anal., 215 (2015), 443–496. https://doi.org/10.1007/s00205-014-0785-2 doi: 10.1007/s00205-014-0785-2
[13]	C. De Filippis, G. Mingione, A borderline case of Calderón-Zygmund estimates for nonuniformly elliptic problems, St. Petersbg. Math. J., 31 (2020), 455–477. https://doi.org/10.1090/spmj/1608 doi: 10.1090/spmj/1608
[14]	C. De Filippis, G. Mingione, The sharp growth rate in nonuniformly elliptic Schauder theory, Duke Math. J., 174 (2025), 1775–1848. https://doi.org/10.1215/00127094-2024-0075 doi: 10.1215/00127094-2024-0075
[15]	L. Esposito, F. Leonetti, G. Mingione, Sharp regularity for functionals with $(p, q)$-growth, J. Differential Equations, 204 (2004), 5–55. https://doi.org/10.1016/j.jde.2003.11.007 doi: 10.1016/j.jde.2003.11.007
[16]	Y. Fang, V. D. Rădulescu, C. Zhang, Equivalence of weak and viscosity solutions for the nonhomogeneous double phase equation, Math. Ann., 388 (2024), 2519–2559. https://doi.org/10.1007/s00208-023-02593-y doi: 10.1007/s00208-023-02593-y
[17]	Y. Fang, V. D. Rădulescu, C. Zhang, Regularity for a class of degenerate fully nonlinear nonlocal elliptic equations, Calc. Var. Partial Differential Equations, 64 (2025), 159. https://doi.org/10.1007/s00526-025-03023-4 doi: 10.1007/s00526-025-03023-4
[18]	Y. Fang, C. Zhang, Equivalence between distributional and viscosity solutions for the double-phase equation, Adv. Calc. Var., 15 (2022), 811–829. https://doi.org/10.1515/acv-2020-0059 doi: 10.1515/acv-2020-0059
[19]	L. Fonseca, J. Maly, G. Mingione, Scalar minimizers with fractal singular sets, Arch. Rational Mech. Anal., 172 (2004), 295–307. https://doi.org/10.1007/s00205-003-0301-6 doi: 10.1007/s00205-003-0301-6
[20]	B. Ge, B. Zhang, Campanato type estimates for the multi-phase problems with irregular obstacles, Mediterr. J. Math., 22 (2025), 81. https://doi.org/10.1007/s00009-025-02849-8 doi: 10.1007/s00009-025-02849-8
[21]	G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations, Comm. Partial Differential Equations, 16 (1991), 311–361. https://doi.org/10.1080/03605309108820761 doi: 10.1080/03605309108820761
[22]	P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch. Rational Mech. Anal., 105 (1989), 267–284. https://doi.org/10.1007/BF00251503 doi: 10.1007/BF00251503
[23]	P. Marcellini, Regularity and existence of solutions of elliptic equations with $p, q$-growth conditions, J. Differential Equations, 90 (1991), 1–30. https://doi.org/10.1016/0022-0396(91)90158-6 doi: 10.1016/0022-0396(91)90158-6
[24]	G. Mingione, V. Rădulescu, Recent developments in problems with nonstandard growth and nonuniform ellipticity, J. Math. Anal. Appl., 501 (2021), 125197. https://doi.org/10.1016/j.jmaa.2021.125197 doi: 10.1016/j.jmaa.2021.125197
[25]	J. Ok, Regularity of $\omega$-minimizers for a class of functionals with non-standard growth, Calc. Var. Partial Differential Equations, 56 (2017), 48. https://doi.org/10.1007/s00526-017-1137-5 doi: 10.1007/s00526-017-1137-5
[26]	J. Ok, G. Scilla, B. Stroffolini, Partial regularity for degenerate systems of double phase type, J. Differential Equations, 432 (2025), 113207. https://doi.org/10.1016/j.jde.2025.02.078 doi: 10.1016/j.jde.2025.02.078
[27]	P. Shin, Calderón-Zygmund estimates for general elliptic operators with double phase, Nonlinear Anal., 194 (2020), 111409. https://doi.org/10.1016/j.na.2018.12.020 doi: 10.1016/j.na.2018.12.020
[28]	B. Zhang, B. Ge, Gradient estimates in the whole space for the double phase problems by the maximal function method, Complex Anal. Oper. Theory, 18 (2024), 136. https://doi.org/10.1007/s11785-024-01579-1 doi: 10.1007/s11785-024-01579-1
[29]	Q. Zhang, V. D. Rădulescu, Double phase anisotropic variational problems and combined effects of reaction and absorption terms, J. Math. Pures Appl., 118 (2018), 159–203. https://doi.org/10.1016/j.matpur.2018.06.015 doi: 10.1016/j.matpur.2018.06.015
[30]	V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR Izv., 29 (1987), 33–66. https://doi.org/10.1070/IM1987v029n01ABEH000958 doi: 10.1070/IM1987v029n01ABEH000958
[31]	V. V. Zhikov, On Lavrentiev's Phenomenon, Russian J. Math. Phys., 3 (1995), 249–269.

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)