Abstract
In this work we prove that the non-negative functions
under proper assumptions on Λ, satisfy a weak Harnack inequality with a constant depending on the
Funding source: Fundação para a Ciência e a Tecnologia
Award Identifier / Grant number: UIDB/00013/2020 and UIDP/00013/2020
Funding source: Simons Foundation
Award Identifier / Grant number: 1160640
Funding statement: Eurica Henriques was financed by Portuguese Funds through FCT–Fundação para a Ciência e a Tecnologia – within the Projects UIDB/00013/2020 and UIDP/00013/2020. Igor I. Skrypnik is partial supported by a grant from the Simons Foundation (Award 1160640, Presidential Discretionary-Ukraine Support Grants, Skrypnik I. I.)
References
[1] Y. A. Alkhutov, The Harnack inequality and the Hölder property of solutions of nonlinear elliptic equations with a nonstandard growth condition (in Russian), Differ. Uravn. 33 (1997), no. 12, 1651–1660; translation in Differential Equations 33 (1997), no. 12, 1653–1663. Suche in Google Scholar
[2] Y. A. Alkhutov and O. V. Krasheninnikova, On the continuity of solutions of elliptic equations with a variable order of nonlinearity (in Russian, Tr. Mat. Inst. Steklova 261 (2008), 7–15; translation in Proc. Steklov Inst. Math. 261 (2008), 1–10. Suche in Google Scholar
[3]
Y. A. Alkhutov and M. D. Surnachev,
A Harnack inequality for a transmission problem with
[4]
Y. A. Alkhutov and M. D. Surnachev,
Harnack’s inequality for the
[5]
Y. A. Alkhutov and M. D. Surnachev,
Hölder continuity and Harnack’s inequality for
[6] W. Arriagada and J. Huentutripay, A Harnack inequality in Orlicz–Sobolev spaces, Studia Math. 243 (2018), no. 2, 117–137. 10.4064/sm8764-9-2017Suche in Google Scholar
[7] P. Baroni, M. Colombo and G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal. 121 (2015), 206–222. 10.1016/j.na.2014.11.001Suche in Google Scholar
[8] P. Baroni, M. Colombo and G. Mingione, Nonautonomous functionals, borderline cases and related function classes, St. Petersburg Math. J. 27 (2016), 347–379. 10.1090/spmj/1392Suche in Google Scholar
[9] 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-zSuche in Google Scholar
[10] 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.21876Suche in Google Scholar
[11] A. Benyaiche, P. Harjulehto, P. Hästö and A. Karppinen, The weak Harnack inequality for unbounded supersolutions of equations with generalized Orlicz growth, J. Differential Equations 275 (2021), 790–814. 10.1016/j.jde.2020.11.007Suche in Google Scholar
[12] E. Bombieri and E. Giusti, Harnack’s inequality for elliptic differential equations on minimal surfaces, Invent. Math. 15 (1972), 24–46. 10.1007/BF01418640Suche in Google Scholar
[13] K. O. Buryachenko and I. I. Skrypnik, Local continuity and Harnack’s inequality for double-phase parabolic equations, Potential Anal. 56 (2022), no. 1, 137–164. 10.1007/s11118-020-09879-9Suche in Google Scholar
[14] M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal. 218 (2015), no. 1, 219–273. 10.1007/s00205-015-0859-9Suche in Google Scholar
[15] M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal. 215 (2015), no. 2, 443–496. 10.1007/s00205-014-0785-2Suche in Google Scholar
[16] M. Colombo and G. Mingione, Calderón–Zygmund estimates and non-uniformly elliptic operators, J. Funct. Anal. 270 (2016), no. 4, 1416–1478. 10.1016/j.jfa.2015.06.022Suche in Google Scholar
[17] E. De Giorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25–43. Suche in Google Scholar
[18]
E. DiBenedetto, U. Gianazza and V. Vespri,
Local clustering of the non-zero set of functions in
[19] E. DiBenedetto and N. S. Trudinger, Harnack inequalities for quasiminima of variational integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 295–308. 10.1016/s0294-1449(16)30424-3Suche in Google Scholar
[20]
X. Fan,
A Class of De Giorgi Type and Hölder Continuity of Minimizers of Variational with
[21] X. Fan and D. Zhao, A class of De Giorgi type and Hölder continuity, Nonlinear Anal. 36 (1999), no. 3, 295–318. 10.1016/S0362-546X(97)00628-7Suche in Google Scholar
[22] O. V. Hadzhy, I. I. Skrypnik and M. V. Voitovych, Interior continuity, continuity up to the boundary, and Harnack’s inequality for double-phase elliptic equations with nonlogarithmic conditions, Math. Nachr. 296 (2023), no. 9, 3892–3914. 10.1002/mana.202000574Suche in Google Scholar
[23] P. Harjulehto and P. Hästö, Boundary regularity under generalized growth conditions, Z. Anal. Anwend. 38 (2019), no. 1, 73–96. 10.4171/zaa/1628Suche in Google Scholar
[24] P. Harjulehto, P. Hästö and M. Lee, Hölder continuity of ω-minimizers of functionals with generalized Orlicz growth, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 22 (2021), no. 2, 549–582. 10.2422/2036-2145.201908_015Suche in Google Scholar
[25] 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-zSuche in Google Scholar
[26] P. Harjulehto, J. Kinnunen and T. Lukkari, Unbounded supersolutions of nonlinear equations with nonstandard growth, Bound. Value Probl. 2007 (2007), Article ID 48348. 10.1155/2007/48348Suche in Google Scholar
[27] P. Harjulehto, T. Kuusi, T. Lukkari, N. Marola and M. Parviainen, Harnack’s inequality for quasiminimizers with nonstandard growth conditions, J. Math. Anal. Appl. 344 (2008), no. 1, 504–520. 10.1016/j.jmaa.2008.03.018Suche in Google Scholar
[28] P. Hästö and J. Ok, Regularity theory for non-autonomous problems with a priori assumptions, Calc. Var. Partial Differential Equations 62 (2023), no. 9, Paper No. 251. 10.1007/s00526-023-02587-3Suche in Google Scholar
[29] O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations, Nauka, Moscow, 1973. Suche in Google Scholar
[30] G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations, Comm. Partial Differential Equations 16 (1991), no. 2–3, 311–361. 10.1080/03605309108820761Suche in Google Scholar
[31] V. Liskevich and I. I. Skrypnik, Harnack inequality and continuity of solutions to elliptic equations with nonstandard growth conditions and lower order terms, Ann. Mat. Pura Appl. (4) 189 (2010), no. 2, 333–356. 10.1007/s10231-009-0111-zSuche in Google Scholar
[32] 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/BF00251503Suche in Google Scholar
[33]
P. Marcellini,
Regularity and existence of solutions of elliptic equations with
[34] Y. Mizuta, T. Ohno and T. Shimomura, Sobolev’s theorem for double phase functionals, Math. Inequal. Appl. 23 (2020), no. 1, 17–33. 10.7153/mia-2020-23-02Suche in Google Scholar
[35] J. Moser, On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577–591. 10.1002/cpa.3160140329Suche in Google Scholar
[36] J. Ok, Regularity for double phase problems under additional integrability assumptions, Nonlinear Anal. 194 (2020), Article ID 111408. 10.1016/j.na.2018.12.019Suche in Google Scholar
[37] M. A. Ragusa and A. Tachikawa, Regularity for minimizers for functionals of double phase with variable exponents, Adv. Nonlinear Anal. 9 (2020), no. 1, 710–728. 10.1515/anona-2020-0022Suche in Google Scholar
[38] M. A. Savchenko, I. I. Skrypnik and Y. A. Yevgenieva, A note on the weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth, preprint (2023), https://arxiv.org/abs/2304.04499. Suche in Google Scholar
[39] M. O. Savchenko, I. I. Skrypnik and Y. A. Yevgenieva, Continuity and Harnack inequalities for local minimizers of non-uniformly elliptic functionals with generalized Orlicz growth under the non-logarithmic conditions, Nonlinear Anal. 230 (2023), Article ID 113221. 10.1016/j.na.2023.113221Suche in Google Scholar
[40] M. A. Shan, I. I. Skrypnik and M. V. Voitovych, Harnack’s inequality for quasilinear elliptic equations with generalized Orlicz growth, Electron. J. Differential Equations 2021 (2021), Paper No. 27. 10.58997/ejde.2021.27Suche in Google Scholar
[41]
I. Skrypnik and Y. Yevgenieva,
Harnack inequality for solutions of the
[42]
I. I. Skrypnik and M. V. Voitovych,
[43] I. I. Skrypnik and M. V. Voitovych, On the continuity of solutions of quasilinear parabolic equations with generalized Orlicz growth under non-logarithmic conditions, Ann. Mat. Pura Appl. (4) 201 (2022), no. 3, 1381–1416. 10.1007/s10231-021-01161-ySuche in Google Scholar
[44]
M. Surnachev,
On the weak Harnack inequality for the parabolic
[45]
M. D. Surnachev,
On Harnack’s inequality for
[46] N. S. Trudinger, On the regularity of generalized solutions of linear, non-uniformly elliptic equations, Arch. Ration. Mech. Anal. 42 (1971), 50–62. 10.1007/BF00282317Suche in Google Scholar
[47] B. Wang, D. Liu and P. Zhao, Hölder continuity for nonlinear elliptic problem in Musielak–Orlicz–Sobolev space, J. Differential Equations 266 (2019), no. 8, 4835–4863. 10.1016/j.jde.2018.10.013Suche in Google Scholar
[48] V. V. Zhikov, Questions of convergence, duality and averaging for functionals of the calculus of variations, Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), no. 5, 961–998. Suche in Google Scholar
[49] V. V. Zhikov, On Lavrentiev’s phenomenon, Russian J. Math. Phys. 3 (1995), no. 2, 249–269. Suche in Google Scholar
[50] V. V. Zhikov, On the density of smooth functions in Sobolev–Orlicz spaces (in Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 310 (2004), 67–81, 226; translation in J. Math. Sci. (N. Y.) 132 (2006), no. 3, 285–294. Suche in Google Scholar
© 2024 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- A characterization of ℓ1 double bubbles with general interface interaction
- The weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth
- On the behavior in time of the solutions to total variation flow
- Zaremba problem with degenerate weights
- Point-wise characterizations of limits of planar Sobolev homeomorphisms and their quasi-monotonicity
- Struwe’s global compactness and energy approximation of the critical Sobolev embedding in the Heisenberg group
- On the variational nature of the Anzellotti pairing
- Strongly nonlinear Robin problems for harmonic and polyharmonic functions in the half-space
- Upper semicontinuity of index plus nullity for minimal and CMC hypersurfaces
- A variational approach to the Navier–Stokes equations with shear-dependent viscosity
- Hölder regularity for the fractional p-Laplacian, revisited
- Poincaré inequality and energy of separating sets
- On a class of obstacle problems with (p, q)-growth and explicit u-dependence
- Stepanov differentiability theorem for intrinsic graphs in Heisenberg groups
- Parabolic Lipschitz truncation for multi-phase problems: The degenerate case
Artikel in diesem Heft
- Frontmatter
- A characterization of ℓ1 double bubbles with general interface interaction
- The weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth
- On the behavior in time of the solutions to total variation flow
- Zaremba problem with degenerate weights
- Point-wise characterizations of limits of planar Sobolev homeomorphisms and their quasi-monotonicity
- Struwe’s global compactness and energy approximation of the critical Sobolev embedding in the Heisenberg group
- On the variational nature of the Anzellotti pairing
- Strongly nonlinear Robin problems for harmonic and polyharmonic functions in the half-space
- Upper semicontinuity of index plus nullity for minimal and CMC hypersurfaces
- A variational approach to the Navier–Stokes equations with shear-dependent viscosity
- Hölder regularity for the fractional p-Laplacian, revisited
- Poincaré inequality and energy of separating sets
- On a class of obstacle problems with (p, q)-growth and explicit u-dependence
- Stepanov differentiability theorem for intrinsic graphs in Heisenberg groups
- Parabolic Lipschitz truncation for multi-phase problems: The degenerate case