Local boundedness of weak solutions to elliptic equations with $ p, q- $growth

Giovanni Cupini; Paolo Marcellini; Elvira Mascolo; Giovanni Cupini; Paolo Marcellini; Elvira Mascolo

doi:10.3934/mine.2023065

Mathematics in Engineering

2023, Volume 5, Issue 3: 1-28. doi: 10.3934/mine.2023065

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Local boundedness of weak solutions to elliptic equations with $ p, q- $growth

1.
Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 - Bologna, Italy
2.
Dipartimento di Matematica e Informatica "U. Dini", Università di Firenze, Viale Morgagni 67/A, 50134 - Firenze, Italy

Received: 27 September 2022 Revised: 13 November 2022 Accepted: 13 November 2022 Published: 29 November 2022

This article is dedicated to Giuseppe Mingione for his $ 50^{th} $ birthday, a leading expert in the regularity theory and in particular in the subject of this manuscript. In this paper we give conditions for the local boundedness of weak solutions to a class of nonlinear elliptic partial differential equations in divergence form of the type considered below in (1.1), under $ p, q- $growth assumptions. The novelties with respect to the mathematical literature on this topic are the general growth conditions and the explicit dependence of the differential equation on $ u $, other than on its gradient $ Du $ and on the $ x $ variable.
- regularity,
- local boundedness,
- weak solutions,
- elliptic equations,
- $ p, q- $growth
Citation: Giovanni Cupini, Paolo Marcellini, Elvira Mascolo. Local boundedness of weak solutions to elliptic equations with $ p, q- $growth[J]. Mathematics in Engineering, 2023, 5(3): 1-28. doi: 10.3934/mine.2023065

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Abstract

This article is dedicated to Giuseppe Mingione for his $ 50^{th} $ birthday, a leading expert in the regularity theory and in particular in the subject of this manuscript. In this paper we give conditions for the local boundedness of weak solutions to a class of nonlinear elliptic partial differential equations in divergence form of the type considered below in (1.1), under $ p, q- $growth assumptions. The novelties with respect to the mathematical literature on this topic are the general growth conditions and the explicit dependence of the differential equation on $ u $, other than on its gradient $ Du $ and on the $ x $ variable.

References

[1]	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
[2]	L. Beck, G. Mingione, Lipschitz bounds and non-uniform ellipticity, Commun. Pure Appl. Math., 73 (2020), 944–1034. https://doi.org/10.1002/cpa.21880 doi: 10.1002/cpa.21880
[3]	P. Bella, M. Schäffner, On the regularity of minimizers for scalar integral functionals with $(p, q)-$ growth, Anal. PDE, 13 (2020), 2241–2257. https://doi.org/10.2140/apde.2020.13.2241 doi: 10.2140/apde.2020.13.2241
[4]	P. Bella, M. Schäffner, Local boundedness and Harnack inequality for solutions of linear nonuniformly elliptic equations, Commun. Pure Appl. Math., 74 (2021), 453–477. https://doi.org/10.1002/cpa.21876 doi: 10.1002/cpa.21876
[5]	S. Biagi, G. Cupini, E. Mascolo, Regularity of quasi-minimizers for non-uniformly elliptic integrals, J. Math. Anal. Appl., 485 (2020), 123838. https://doi.org/10.1016/j.jmaa.2019.123838 doi: 10.1016/j.jmaa.2019.123838
[6]	M. Bildhauer, M. Fuchs, $C^{1, \alpha}$-solutions to non-autonomous anisotropic variational problems, Calc. Var., 24 (2005), 309–340. https://doi.org/10.1007/s00526-005-0327-8 doi: 10.1007/s00526-005-0327-8
[7]	L. Boccardo, P. Marcellini, C. Sbordone, $L^{\infty}$- regularity for variational problems with sharp nonstandard growth conditions, Boll. Un. Mat. Ital. A (7), 4 (1990), 219–225.
[8]	V. Bögelein, F. Duzaar, P. Marcellini, C. Scheven, Boundary regularity for elliptic systems with $p, q-$growth, J. Math. Pure. Appl., 159 (2022), 250–293. https://doi.org/10.1016/j.matpur.2021.12.004 doi: 10.1016/j.matpur.2021.12.004
[9]	P. Bousquet, L. Brasco, Lipschitz regularity for orthotropic functionals with nonstandard growth conditions, Rev. Mat. Iberoam., 36 (2020), 1989–2032. https://doi.org/10.4171/RMI/1189 doi: 10.4171/RMI/1189
[10]	S. S. Byun, J. Oh, Global gradient estimates for non-uniformly elliptic equations, Calc. Var., 56 (2017), 46. https://doi.org/10.1007/s00526-017-1148-2 doi: 10.1007/s00526-017-1148-2
[11]	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
[12]	M. Carozza, H. Gao, R. Giova, F. Leonetti, A boundedness result for minimizers of some polyconvex integrals, J. Optim. Theory Appl., 178 (2018), 699–725. https://doi.org/10.1007/s10957-018-1335-0 doi: 10.1007/s10957-018-1335-0
[13]	M. Caselli, M. Eleuteri, A. Passarelli di Napoli, Regularity results for a class of obstacle problems with $p, q-$growth conditions, ESAIM: COCV, 27 (2021), 19. https://doi.org/10.1051/cocv/2021017 doi: 10.1051/cocv/2021017
[14]	A. Cianchi, Local boundedness of minimizers of anisotropic functionals, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 17 (2000), 147–168. https://doi.org/10.1016/S0294-1449(99)00107-9 doi: 10.1016/S0294-1449(99)00107-9
[15]	A. Cianchi, V. G. Maz'ya, Global boundedness of the gradient for a class of nonlinear elliptic systems, Arch. Rational Mech. Anal., 212 (2014), 129–177. https://doi.org/10.1007/s00205-013-0705-x doi: 10.1007/s00205-013-0705-x
[16]	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
[17]	G. Cupini, F. Leonetti, E. Mascolo, Local boundedness for minimizers of some polyconvex integrals, Arch. Rational Mech. Anal., 224 (2017), 269–289. https://doi.org/10.1007/s00205-017-1074-7 doi: 10.1007/s00205-017-1074-7
[18]	G. Cupini, P. Marcellini, E. Mascolo, Regularity under sharp anisotropic general growth conditions, Discrete Contin. Dyn. Syst. B, 11 (2009), 67–86. https://doi.org/10.3934/dcdsb.2009.11.67 doi: 10.3934/dcdsb.2009.11.67
[19]	G. Cupini, P. Marcellini, E. Mascolo, Local boundedness of solutions to quasilinear elliptic systems, Manuscripta Math., 137 (2012), 287–315. https://doi.org/10.1007/s00229-011-0464-7
[20]	G. Cupini, P. Marcellini, E. Mascolo, Local boundedness of solutions to some anisotropic elliptic systems, In: Recent trends in nonlinear partial differential equations. II. Stationary problems, Providence, RI: Amer. Math. Soc., 2013,169–186. http://doi.org/10.1090/conm/595/11803
[21]	G. Cupini, P. Marcellini, E. Mascolo, Existence and regularity for elliptic equations under $p, q-$growth, Adv. Differential Equations, 19 (2014), 693–724.
[22]	G. Cupini, P. Marcellini, E. Mascolo, Local boundedness of minimizers with limit growth condition, J. Optim. Theory Appl., 166 (2015), 1–22. https://doi.org/10.1007/s10957-015-0722-z
[23]	G. Cupini, P. Marcellini, E. Mascolo, Regularity of minimizers under limit growth conditions, Nonlinear Anal., 153 (2017), 294–310. https://doi.org/10.1016/j.na.2016.06.002 doi: 10.1016/j.na.2016.06.002
[24]	G. Cupini, P. Marcellini, E. Mascolo, Nonuniformly elliptic energy integrals with $p, q-$growth, Nonlinear Anal., 177, Part A (2018), 312–324. https://doi.org/10.1016/j.na.2018.03.018 doi: 10.1016/j.na.2018.03.018
[25]	G. Cupini, P. Marcellini, E. Mascolo, A. Passarelli di Napoli, Lipschitz regularity for degenerate elliptic integrals with $p, q-$growth, Adv. Calc. Var., in press. https://doi.org/10.1515/acv-2020-0120
[26]	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
[27]	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
[28]	C. De Filippis, M. Piccinini, Borderline global regularity for nonuniformly elliptic systems, Int. Math. Res. Notices, in press. https://doi.org/10.1093/imrn/rnac283
[29]	E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat., 3 (1957), 25–43.
[30]	M. De Rosa, A. G. Grimaldi, A local boundedness result for a class of obstacle problems with non-standard growth conditions, J. Optim. Theory Appl., 195 (2022), 282–296. https://doi.org/10.1007/s10957-022-02084-1 doi: 10.1007/s10957-022-02084-1
[31]	E. Di Benedetto, U. Gianazza, V. Vespri, Remarks on local boundedness and local Hölder continuity of local weak solutions to anisotropic $p-$Laplacian type equations, J. Elliptic Parabol. Equ., 2 (2016), 157–169. https://doi.org/10.1007/BF03377399 doi: 10.1007/BF03377399
[32]	T. Di Marco, P. Marcellini, A-priori gradient bound for elliptic systems under either slow or fast growth conditions, Calc. Var., 59 (2020), 120. https://doi.org/10.1007/s00526-020-01769-7 doi: 10.1007/s00526-020-01769-7
[33]	F. G. Düzgun, P. Marcellini, V. Vespri, An alternative approach to the Hölder continuity of solutions to some elliptic equations, Nonlinear Anal. Theor., 94 (2014), 133–141. https://doi.org/10.1016/j.na.2013.08.018 doi: 10.1016/j.na.2013.08.018
[34]	M. Eleuteri, P. Marcellini, E. Mascolo, Local Lipschitz continuity of minimizers with mild assumptions on the $x$-dependence, Discrete Contin. Dyn. Syst. S, 12 (2019), 251–265. https://doi.org/10.3934/dcdss.2019018 doi: 10.3934/dcdss.2019018
[35]	M. Eleuteri, P. Marcellini, E. Mascolo, Regularity for scalar integrals without structure conditions, Adv. Calc. Var., 13 (2020), 279–300. https://doi.org/10.1515/acv-2017-0037 doi: 10.1515/acv-2017-0037
[36]	M. Eleuteri, P. Marcellini, E. Mascolo, S. Perrotta, Local Lipschitz continuity for energy integrals with slow growth, Annali di Matematica, 201 (2022), 1005–1032. https://doi.org/10.1007/s10231-021-01147-w doi: 10.1007/s10231-021-01147-w
[37]	N. Fusco, C. Sbordone, Local boundedness of minimizers in a limit case, Manuscripta Math., 69 (1990), 19–25. https://doi.org/10.1007/BF02567909 doi: 10.1007/BF02567909
[38]	N. Fusco, C. Sbordone, Some remarks on the regularity of minima of anisotropic integrals, Commun. Part. Diff. Eq., 18 (1993), 153–167. https://doi.org/10.1080/03605309308820924 doi: 10.1080/03605309308820924
[39]	A. Gentile, Regularity for minimizers of a class of non-autonomous functionals with sub-quadratic growth, Adv. Calc. Var., 15 (2022), 385–399. https://doi.org/10.1515/acv-2019-0092 doi: 10.1515/acv-2019-0092
[40]	M. Giaquinta, E. Giusti, Quasi-minima, Ann. Inst. H. Poincaré C Analyse non-linéaire, 1 (1984), 79–107. https://doi.org/10.1016/S0294-1449(16)30429-2
[41]	E. Giusti, Direct methods in the calculus of variations, River Edge, NJ: World Scientific Publishing Co. Inc., 2003. https://doi.org/10.1142/5002
[42]	T. Granucci, M. Randolfi, Local boundedness of Quasi-minimizers of fully anisotropic scalar variational problems, Manuscripta Math., 160 (2019), 99–152. https://doi.org/10.1007/s00229-018-1055-7 doi: 10.1007/s00229-018-1055-7
[43]	J. Hirsch, M. Schäffner, Growth conditions and regularity, an optimal local boundedness result, Commun. Contemp. Math., 23 (2021), 2050029. https://doi.org/10.1142/S0219199720500297 doi: 10.1142/S0219199720500297
[44]	Ī. M. Kolodīĭ, The boundedness of generalized solutions of elliptic differential equations, Vestnik Moskov. Univ. Ser. I Mat. Meh., 25 (1970), 44–52.
[45]	O. Ladyzhenskaya, N. Ural'tseva, Linear and quasilinear elliptic equations, New York-London: Academic Press, 1968.
[46]	P. Marcellini, Regularity of minimizers of integrals in the calculus of variations with non standard growth conditions, Arch. Rational Mech. Anal., 105 (1989), 267–284. https://doi.org/10.1007/BF00251503 doi: 10.1007/BF00251503
[47]	P. Marcellini, Regularity and existence of solutions of elliptic equations with $p, q-$growth conditions, J. Differ. Equations, 90 (1991), 1–30. https://doi.org/10.1016/0022-0396(91)90158-6 doi: 10.1016/0022-0396(91)90158-6
[48]	P. Marcellini, Regularity for elliptic equations with general growth conditions, J. Differ. Equations, 105 (1993), 296–333. https://doi.org/10.1006/jdeq.1993.1091 doi: 10.1006/jdeq.1993.1091
[49]	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.
[50]	P. Marcellini, Regularity for some scalar variational problems under general growth conditions, J. Optim. Theory Appl., 90 (1996), 161–181. https://doi.org/10.1007/BF02192251 doi: 10.1007/BF02192251
[51]	P. Marcellini, Regularity under general and $p, q-$growth conditions, Discrete Contin. Dyn. Syst. S, 13 (2020), 2009–2031. https://doi.org/10.3934/dcdss.2020155 doi: 10.3934/dcdss.2020155
[52]	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
[53]	P. Marcellini, Local Lipschitz continuity for $p, q-$PDEs with explicit $u-$dependence, Nonlinear Anal., 226 (2023), 113066. https://doi.org/10.1016/j.na.2022.113066 doi: 10.1016/j.na.2022.113066
[54]	G. Mingione, Regularity of minima: an invitation to the dark side of the Calculus of Variations, Appl. Math., 51 (2006), 355–426. https://doi.org/10.1007/s10778-006-0110-3 doi: 10.1007/s10778-006-0110-3
[55]	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
[56]	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
[57]	P. Pucci, R. Servadei, Regularity of weak solutions of homogeneous or inhomogeneous quasilinear elliptic equations, Indiana Univ. Math. J., 57 (2008), 3329–3363. https://doi.org/10.1512/iumj.2008.57.3525 doi: 10.1512/iumj.2008.57.3525
[58]	M. Schäffner, Higher integrability for variational integrals with non-standard growth, Calc. Var., 60 (2021), 77. https://doi.org/10.1007/s00526-020-01907-1 doi: 10.1007/s00526-020-01907-1
[59]	B. Stroffolini, Global boundedness of solutions of anisotropic variational problems, Boll. Un. Mat. Ital. A (7), 5 (1991), 345–352.
[60]	P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differ. Equations, 51 (1984), 126–150. https://doi.org/10.1016/0022-0396(84)90105-0 doi: 10.1016/0022-0396(84)90105-0

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