Research article Special Issues

Bloch estimates in non-doubling generalized Orlicz spaces

  • Received: 01 July 2022 Revised: 09 August 2022 Accepted: 19 September 2022 Published: 23 September 2022
  • We study minimizers of non-autonomous functionals

    $ \begin{align*} \inf\limits_u \int_\Omega \varphi(x,|\nabla u|) \, dx \end{align*} $

    when $ \varphi $ has generalized Orlicz growth. We consider the case where the upper growth rate of $ \varphi $ is unbounded and prove the Harnack inequality for minimizers. Our technique is based on "truncating" the function $ \varphi $ to approximate the minimizer and Harnack estimates with uniform constants via a Bloch estimate for the approximating minimizers.

    Citation: Petteri Harjulehto, Peter Hästö, Jonne Juusti. Bloch estimates in non-doubling generalized Orlicz spaces[J]. Mathematics in Engineering, 2023, 5(3): 1-21. doi: 10.3934/mine.2023052

    Related Papers:

  • We study minimizers of non-autonomous functionals

    $ \begin{align*} \inf\limits_u \int_\Omega \varphi(x,|\nabla u|) \, dx \end{align*} $

    when $ \varphi $ has generalized Orlicz growth. We consider the case where the upper growth rate of $ \varphi $ is unbounded and prove the Harnack inequality for minimizers. Our technique is based on "truncating" the function $ \varphi $ to approximate the minimizer and Harnack estimates with uniform constants via a Bloch estimate for the approximating minimizers.



    加载中


    [1] R. A. Adams, J. J. F. Fournier, Sobolev spaces, 2 Eds., Amsterdam: Elsevier/Academic Press, 2003.
    [2] D. R. Adams, L. I. Hedberg, Function spaces and potential theory, Berlin, Heidelberg: Springer, 1996. https://doi.org/10.1007/978-3-662-03282-4
    [3] P. Baroni, M. Colombo, G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal. Theor., 121 (2015), 206–222. https://doi.org/10.1016/j.na.2014.11.001 doi: 10.1016/j.na.2014.11.001
    [4] P. Baroni, M. Colombo, G. Mingione, Nonautonomous functionals, borderline cases and related function classes, St. Petersburg Math. J., 27 (2016), 347–379. https://doi.org/10.1090/spmj/1392 doi: 10.1090/spmj/1392
    [5] P. Baroni, M. Colombo, G. Mingione, Regularity for general functionals with double phase, Calc. Var., 57 (2018), 62. https://doi.org/10.1007/s00526-018-1332-z doi: 10.1007/s00526-018-1332-z
    [6] S. Baasandorj, S.-S. Byun, Irregular obstacle problems for Orlicz double phase, J. Math. Anal. Appl., 507 (2022), 125791. https://doi.org/10.1016/j.jmaa.2021.125791 doi: 10.1016/j.jmaa.2021.125791
    [7] S. Baasandorj, S.-S. Byun, J. Oh, Gradient estimates for multi-phase problems, Calc. Var., 60 (2021), 104. https://doi.org/10.1007/s00526-021-01940-8 doi: 10.1007/s00526-021-01940-8
    [8] A. Benyaiche, P. Harjulehto, P. Hästö, A. Karppinen, The weak Harnack inequality for unbounded supersolutions of equations with generalized Orlicz growth, J. Differ. Equations, 275 (2021), 790–814. https://doi.org/10.1016/j.jde.2020.11.007 doi: 10.1016/j.jde.2020.11.007
    [9] A. Benyaiche, I. Khlifi, Harnack inequality for quasilinear elliptic equations in generalized Orlicz-Sobolev spaces, Potential Anal., 53 (2020), 631–643. https://doi.org/10.1007/s11118-019-09781-z doi: 10.1007/s11118-019-09781-z
    [10] S.-S. Byun, J. Oh, Regularity results for generalized double phase functionals, Anal. PDE, 13 (2020), 1269–1300. https://doi.org/10.2140/apde.2020.13.1269 doi: 10.2140/apde.2020.13.1269
    [11] I. Chlebicka, A pocket guide to nonlinear differential equations in Musielak–Orlicz spaces, Nonlinear Anal., 175 (2018), 1–27. https://doi.org/10.1016/j.na.2018.05.003 doi: 10.1016/j.na.2018.05.003
    [12] I. Chlebicka, A. Zatorska-Goldstein, Generalized superharmonic functions with strongly nonlinear operator, Potential Anal., 57 (2022), 379–400. https://doi.org/10.1007/s11118-021-09920-5 doi: 10.1007/s11118-021-09920-5
    [13] I. Chlebicka, F. Gianetti, A. Zatorska-Goldstein, Elliptic problems with growth in nonreflexive Orlicz spaces and with measure or $L^1$ data, J. Math. Anal. Appl., 479 (2019), 185–213. https://doi.org/10.1016/j.jmaa.2019.06.022 doi: 10.1016/j.jmaa.2019.06.022
    [14] I. Chlebicka, P. Gwiazda, A. Świerczewska-Gwiazda, A. Wróblewska-Kamińska, Partial differential equations in anisotropic Musielak-Orlicz spaces, Cham: Springer, 2021. https://doi.org/10.1007/978-3-030-88856-5
    [15] I. Chlebicka, P. Gwiazda, A. Zatorska-Goldstein, Renormalized solutions to parabolic equation in time and space dependent anisotropic Musielak–Orlicz spaces in absence of Lavrentiev's phenomenon, J. Differ. Equations, 267 (2019), 1129–1166. https://doi.org/10.1016/j.jde.2019.02.005 doi: 10.1016/j.jde.2019.02.005
    [16] 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
    [17] M. Colombo, G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Rational Mech. Anal., 218 (2015), 219–273. https://doi.org/10.1007/s00205-015-0859-9 doi: 10.1007/s00205-015-0859-9
    [18] Á. Crespo-Blanco, L. Gasiński, P. Harjulehto, P. Winkert, A new class of double phase variable exponent problems: existence and uniqueness, J. Differ. Equations, 323 (2022), 182–228. https://doi.org/10.1016/j.jde.2022.03.029 doi: 10.1016/j.jde.2022.03.029
    [19] C. De Filippis, G. Mingione, On the regularity of minima of non-autonomous functionals, J. Geom. Anal., 30 (2020), 1584–1626. https://doi.org/10.1007/s12220-019-00225-z doi: 10.1007/s12220-019-00225-z
    [20] C. De Filippis, G. Mingione, Lipschitz bounds and nonautonomous integrals, Arch. Rational Mech. Anal., 242 (2021), 973–1057. https://doi.org/10.1007/s00205-021-01698-5 doi: 10.1007/s00205-021-01698-5
    [21] C. De Filippis, G. Mingione, Interpolative gap bounds for nonautonomous integrals, Anal. Math. Phys., 11 (2021), 117. https://doi.org/10.1007/s13324-021-00534-z doi: 10.1007/s13324-021-00534-z
    [22] C. De Filippis, J. Oh, Regularity for multi-phase variational problems, J. Differ. Equations, 267 (2019), 1631–1670. https://doi.org/10.1016/j.jde.2019.02.015 doi: 10.1016/j.jde.2019.02.015
    [23] L. Diening, P. Harjulehto, P. Hästö, M. Růžička, Lebesgue and Sobolev spaces with variable exponents, Berlin, Heidelberg: Springer, 2011. https://doi.org/10.1007/978-3-642-18363-8
    [24] M. Eleuteri, A. Passarelli di Napoli, On the validity of variational inequalities for obstacle problems with non-standard growth, Ann. Fenn. Math., 47 (2022), 395–416. https://doi.org/10.54330/afm.114655 doi: 10.54330/afm.114655
    [25] L. C. Evans, R. F. Gariepy, Measure theory and fine properties of functions, Boca Raton: CRC Press, 2015. https://doi.org/10.1201/b18333
    [26] F. W. Gehring, Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc., 103 (1962), 353–393. https://doi.org/10.1090/S0002-9947-1962-0139735-8 doi: 10.1090/S0002-9947-1962-0139735-8
    [27] F. Giannetti, A. Passarelli di Napoli, M. A. Ragusa, A. Tachikawa, Partial regularity for minimizers of a class of non autonomous functionals with nonstandard growth, Calc. Var., 56 (2017), 153. https://doi.org/10.1007/s00526-017-1248-z doi: 10.1007/s00526-017-1248-z
    [28] P. Gwiazda, I. Skrzypczak, A. Zatorska-Goldstein, Existence of renormalized solutions to elliptic equation in Musielak–Orlicz space, J. Differ. Equations, 264 (2018), 341–377. https://doi.org/10.1016/j.jde.2017.09.007 doi: 10.1016/j.jde.2017.09.007
    [29] P. Harjulehto, P. Hästö, Boundary regularity under generalized growth conditions, Z. Anal. Anwend., 38 (2019), 73–96. https://doi.org/10.4171/zaa/1628 doi: 10.4171/zaa/1628
    [30] P. Harjulehto, P. Hästö, Orlicz spaces and generalized Orlicz spaces, Cham: Springer, 2019. https://doi.org/10.1007/978-3-030-15100-3
    [31] P. Harjulehto, P. Hästö, Double phase image restoration, J. Math. Anal. Appl., 501 (2021), 123832. https://doi.org/10.1016/j.jmaa.2019.123832 doi: 10.1016/j.jmaa.2019.123832
    [32] P. Harjulehto, P. Hästö, A. Karppinen, Local higher integrability of the gradient of a quasiminimizer under generalized Orlicz growth conditions, Nonlinear Anal., 177 (2018), 543–552. https://doi.org/10.1016/j.na.2017.09.010 doi: 10.1016/j.na.2017.09.010
    [33] P. Harjulehto, P. Hästö, R. Klén, Generalized Orlicz spaces and related PDE, Nonlinear Anal. Theor., 143 (2016), 155–173. https://doi.org/10.1016/j.na.2016.05.002 doi: 10.1016/j.na.2016.05.002
    [34] P. Harjulehto, P. Hästö, V. Latvala, Minimizers of the variable exponent, non-uniformly convex Dirichlet energy, J. Math. Pure. Appl., 89 (2008), 174–197. https://doi.org/10.1016/j.matpur.2007.10.006 doi: 10.1016/j.matpur.2007.10.006
    [35] P. Harjulehto, P. Hästö, V. Latvala, Harnack's inequality for $p(\cdot)$-harmonic functions with unbounded exponent $p$, J. Math. Anal. Appl., 352 (2009), 345–359. https://doi.org/10.1016/j.jmaa.2008.05.090 doi: 10.1016/j.jmaa.2008.05.090
    [36] P. Harjulehto, P. Hästö, M. Lee, Hölder continuity of $\omega$-minimizers of functionals with generalized Orlicz growth, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), XXII (2021), 549–582. https://doi.org/10.2422/2036-2145.201908_015 doi: 10.2422/2036-2145.201908_015
    [37] P. Harjulehto, P. Hästö, O. Toivanen, Hölder regularity of quasiminimizers under generalized growth conditions, Calc. Var., 56 (2017), 22. https://doi.org/10.1007/s00526-017-1114-z doi: 10.1007/s00526-017-1114-z
    [38] P. Hästö, J. Ok, Maximal regularity for local minimizers of non-autonomous functionals, J. Eur. Math. Soc., 24 (2022), 1285–1334. https://doi.org/10.4171/jems/1118 doi: 10.4171/jems/1118
    [39] P. Hästö, J. Ok, Regularity theory for non-autonomous partial differential equations without Uhlenbeck structure, Arch. Rational Mech. Anal., 245 (2022), 1401–1436. https://doi.org/10.1007/s00205-022-01807-y doi: 10.1007/s00205-022-01807-y
    [40] J. Heinonen, T. Kilpeläinen, O. Martio, Nonlinear potential theory of degenerate elliptic equations, Mineola, NY: Dover Publications Inc., 2006.
    [41] A. Karppinen, Global continuity and higher integrability of a minimizer of an obstacle problem under generalized Orlicz growth conditions, Manuscripta Math., 164 (2021), 67–94. https://doi.org/10.1007/s00229-019-01173-2 doi: 10.1007/s00229-019-01173-2
    [42] O. Mendez, J. Lang, Analysis on function spaces of Musielak-Orlicz type, Chapman & Hall/CRC, 2019. https://doi.org/10.1201/9781498762618
    [43] V. Latvala, BMO-invariance of quasiminimizers, Ann. Acad. Sci. Fenn. Math., 29 (2004), 407–418.
    [44] Q.-R. Li, W. Sheng, D. Ye, C. Yi, A flow approach to the Musielak-Orlicz-Gauss image problem, Adv. Math., 403 (2022), 108379. https://doi.org/10.1016/j.aim.2022.108379 doi: 10.1016/j.aim.2022.108379
    [45] P. Marcellini, Regularity under general $(p, q)$-conditions, Discrete Cont. Dyn. Syst. S, 13 (2020), 2009–2031. https://doi.org/10.3934/dcdss.2020155 doi: 10.3934/dcdss.2020155
    [46] P. Marcellini, Growth conditions and regularity for weak solutions to nonlinear elliptic pdes, J. Math. Anal. Appl., 501 (2021), 124408. https://doi.org/10.1016/j.jmaa.2020.124408 doi: 10.1016/j.jmaa.2020.124408
    [47] G. Mingione, G. Palatucci, Developments and perspectives in nonlinear potential theory, Nonlinear Anal., 194 (2020), 111452. https://doi.org/10.1016/j.na.2019.02.006 doi: 10.1016/j.na.2019.02.006
    [48] G. Mingione, V. Radulescu, 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
    [49] Y. Mizuta, E. Nakai, T. Ohno, T. Shimomura, Campanato–Morrey spaces for the double phase functionals with variable exponents, Nonlinear Anal., 197 (2020), 111827. https://doi.org/10.1016/j.na.2020.111827 doi: 10.1016/j.na.2020.111827
    [50] Y. Mizuta, T. Ohno, T. Shimomura, Boundedness of fractional maximal operators for double phase functionals with variable exponents, J. Math. Anal. Appl., 501 (2021), 124360. https://doi.org/10.1016/j.jmaa.2020.124360 doi: 10.1016/j.jmaa.2020.124360
    [51] C. P. Niculescu, L.-E. Persson, Convex functions and their applications: A contemporary approach, New York: Springer, 2006. https://doi.org/10.1007/0-387-31077-0
    [52] J. Ok, Gradient estimates for elliptic equations with $L^{p(\cdot)}\log L$ growth, Calc. Var., 55 (2016), 26. https://doi.org/10.1007/s00526-016-0965-z doi: 10.1007/s00526-016-0965-z
    [53] N. S. Papageorgiou, Double phase problems: a survey of some recent results, Opuscula Math., 42 (2022), 257–278. https://doi.org/10.7494/OpMath.2022.42.2.257 doi: 10.7494/OpMath.2022.42.2.257
    [54] I. I. Skrypnik, M. V. Voitovych, On the continuity of solutions of quasilinear parabolic equations with generalized Orlicz growth under non-logarithmic conditions, Annali di Matematica, 201 (2021), 1381–1416. https://doi.org/10.1007/s10231-021-01161-y doi: 10.1007/s10231-021-01161-y
    [55] B. Wang, D. Liu, P. Zhao, Hölder continuity for nonlinear elliptic problem in Musielak–Orlicz–Sobolev space, J. Differ. Equations, 266 (2019), 4835–4863. https://doi.org/10.1016/j.jde.2018.10.013 doi: 10.1016/j.jde.2018.10.013
    [56] Q. Zhang, V. Rădulescu, Double phase anisotropic variational problems and combined effects of reaction and absorption terms, J. Math. Pure Appl., 118 (2018), 159–203. https://doi.org/10.1016/j.matpur.2018.06.015 doi: 10.1016/j.matpur.2018.06.015
    [57] V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Mathematics of the USSR-Izvestiya, 29 (1987), 33. https://doi.org/10.1070/IM1987v029n01ABEH000958 doi: 10.1070/IM1987v029n01ABEH000958
    [58] W. P. Ziemer, Weakly differentiable functions, New York: Springer, 1989. https://doi.org/10.1007/978-1-4612-1015-3
  • Reader Comments
  • © 2023 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)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1530) PDF downloads(193) Cited by(1)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog