• Previous Article
    On the lifespan of classical solutions to a non-local porous medium problem with nonlinear boundary conditions
  • DCDS-S Home
  • This Issue
  • Next Article
    Schrödinger–Kirchhoff–Hardy $ p $–fractional equations without the Ambrosetti–Rabinowitz condition
July  2020, 13(7): 2009-2031. doi: 10.3934/dcdss.2020155

Regularity under general and $ p,q- $ growth conditions

Dipartimento di Matematica e Informatica "U. Dini", Università di Firenze, Viale Morgagni 67/A, 50134 - Firenze, Italy

Received  July 2018 Revised  July 2018 Published  November 2019

Fund Project: The author is a member of GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica)

This paper deals with existence and regularity in variational problems related to partial differential equations and systems - both in the elliptic and in the parabolic contexts - and to calculus of variations, under general and $ p,q- $ growth conditions. The manuscript is dedicated to my friend and colleague Patrizia Pucci, with great esteem and sympathy.

Citation: Paolo Marcellini. Regularity under general and $ p,q- $ growth conditions. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 2009-2031. doi: 10.3934/dcdss.2020155
References:
[1]

G. Autuori and P. Pucci, Asymptotic stability for Kirchhoff systems in variable exponent Sobolev spaces, Complex Var. Elliptic Equ., 56 (2011), 715-753.  doi: 10.1080/17476931003786691.

[2]

J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal., 63 (1976/77), 337-403.  doi: 10.1007/BF00279992.

[3]

J. M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. Trans. R. Soc. Lond. Ser. A, 306 (1982), 557-611.  doi: 10.1098/rsta.1982.0095.

[4]

P. BaroniM. Colombo and G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal., 121 (2015), 206-222.  doi: 10.1016/j.na.2014.11.001.

[5]

P. BaroniM. Colombo and G. Mingione, Non-autonomous functionals, borderline cases and related function classes, St. Petersburg Math. J., 27 (2016), 347-379.  doi: 10.1090/spmj/1392.

[6]

P. Baroni, M. Colombo and G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations, 57 (2018), Art. 62, 48 pp. doi: 10.1007/s00526-018-1332-z.

[7]

L. Beck and G. Mingione, Lipschitz bounds and non-uniformly ellipticity, in preparation, 2018.

[8]

M. Bildhauer, Convex Variational Problems. Linear, Nearly Linear and Anisotropic Growth Conditions, Lecture Notes in Mathematics, 1818. Springer-Verlag, Berlin, 2003. doi: 10.1007/b12308.

[9]

I. Birindelli and F. Demengel, Fully nonlinear operators with Hamiltonian: Hölder regularity of the gradient, NoDEA Nonlinear Differential Equations Appl., 23 (2016), Art. 41, 17 pp. doi: 10.1007/s00030-016-0392-z.

[10]

L. BoccardoP. Marcellini and C. Sbordone, $L^\infty$-regularity for variational problems with sharp nonstandard growth conditions, Boll. Un. Mat. Ital. A, 4 (1990), 219-225.  doi: 10.1007/bf01934372.

[11]

L. BoccardoT. Gallouët and P. Marcellini, Anisotropic equations in $L^1$, Differential Integral Equations, 9 (1996), 209-212. 

[12]

V. BögeleinF. Duzaar and P. Marcellini, Parabolic equations with $p, q$-growth, Journal de Mathématiques Pures et Appliquées, 100 (2013), 535-563.  doi: 10.1016/j.matpur.2013.01.012.

[13]

V. BögeleinF. Duzaar and P. Marcellini, Parabolic systems with $p, q-$growth: A variational approach, Arch. Ration. Mech. Anal., 210 (2013), 219-267.  doi: 10.1007/s00205-013-0646-4.

[14]

V. BögeleinF. Duzaar and P. Marcellini, Existence of evolutionary variational solutions via the calculus of variations, J. Differential Equations, 256 (2014), 3912-3942.  doi: 10.1016/j.jde.2014.03.005.

[15]

V. BögeleinF. Duzaar and P. Marcellini, A time dependent variational approach to image restoration, SIAM J. on Imaging Sciences, 8 (2015), 968-1006.  doi: 10.1137/140992771.

[16]

V. BögeleinF. DuzaarP. Marcellini and C. Scheven, Doubly nonlinear equations of porous medium type, Arch. Ration. Mech. Anal., 229 (2018), 503-545.  doi: 10.1007/s00205-018-1221-9.

[17]

V. BögeleinF. DuzaarP. Marcellini and C. Scheven, A variational approach to doubly nonlinear equations, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 29 (2018), 739-772.  doi: 10.4171/RLM/832.

[18]

V. BögeleinF. DuzaarP. Marcellini and S. Signoriello, Nonlocal diffusion equations, J. Math. Anal. Appl., 432 (2015), 398-428.  doi: 10.1016/j.jmaa.2015.06.053.

[19]

V. BögeleinF. DuzaarP. Marcellini and S. Signoriello, Parabolic equations and the bounded slope condition, Ann. Inst. H. Poincare, Anal. Non Lineaire, 34 (2017), 355-379.  doi: 10.1016/j.anihpc.2015.12.005.

[20]

B. Botteron and P. Marcellini, A general approach to the existence of minimizers of one-dimensional non-coercive integrals of the calculus of variations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8 (1991), 197-223.  doi: 10.1016/S0294-1449(16)30272-4.

[21]

L. Brasco and G. Carlier, On certain anisotropic elliptic equations arising in congested optimal transport: Local gradient bounds, Adv. Calc. Var., 7 (2014), 379-407.  doi: 10.1515/acv-2013-0007.

[22]

P. Celada and S. Perrotta, Polyconvex energies and cavitation, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 295-321.  doi: 10.1007/s00030-012-0184-z.

[23]

I. Chlebicka, A pocket guide to nonlinear differential equations in Musielak-Orlicz spaces, Nonlinear Analysis, 175 (1918), 1-27.  doi: 10.1016/j.na.2018.05.003.

[24]

F. Colasuonno and P. Pucci, Multiplicity of solutions for $p(x)$-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal., 74 (2011), 5962-5974.  doi: 10.1016/j.na.2011.05.073.

[25]

M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Rat. Mech. Anal., 215 (2015), 443-496.  doi: 10.1007/s00205-014-0785-2.

[26]

M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Rat. Mech. Anal., 218 (2015), 219-273.  doi: 10.1007/s00205-015-0859-9.

[27]

G. CupiniF. GiannettiR. Giova and A. Passarelli di Napoli, Higher integrability for minimizers of asymptotically convex integrals with discontinuous coefficients, Nonlinear Anal., 154 (2017), 7-24.  doi: 10.1016/j.na.2016.02.017.

[28]

G. CupiniM. Guidorzi and E. Mascolo, Regularity of minimizers of vectorial integrals with $p-q$ growth, Nonlinear Anal., 54 (2003), 591-616.  doi: 10.1016/S0362-546X(03)00087-7.

[29]

G. CupiniP. Marcellini and E. Mascolo, Regularity under sharp anisotropic general growth conditions, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 66-86.  doi: 10.3934/dcdsb.2009.11.67.

[30]

G. CupiniP. Marcellini and E. Mascolo, Local boundedness of solutions to quasilinear elliptic systems, Manuscripta Math., 137 (2012), 287-315.  doi: 10.1007/s00229-011-0464-7.

[31]

G. CupiniP. Marcellini and E. Mascolo, Local boundedness of solutions to some anisotropic elliptic systems, Recent Trends in Nonlinear Partial Differential Equations. II. Stationary problems, Contemp. Math., Amer. Math. Soc., Providence, RI, 595 (2013), 169-186.  doi: 10.1090/conm/595/11803.

[32]

G. CupiniP. Marcellini and E. Mascolo, Existence and regularity for elliptic equations under $p, q$-growth, Adv. Differential Equations, 19 (2014), 693-724. 

[33]

G. CupiniP. Marcellini and E. Mascolo, Local boundedness of minimizers with limit growth conditions, J. Optim. Theory Appl., 166 (2015), 1-22.  doi: 10.1007/s10957-015-0722-z.

[34]

G. CupiniP. Marcellini and E. Mascolo, Regularity of minimizers under limit growth conditions, Nonlinear Analysis, 153 (2017), 294-310.  doi: 10.1016/j.na.2016.06.002.

[35]

G. CupiniP. Marcellini and E. Mascolo, Nonuniformly elliptic energy integrals with $p, q$-growth, Nonlinear Analysis, 177 (2018), 312-324.  doi: 10.1016/j.na.2018.03.018.

[36]

E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli, Mem. Accad. Sci. Torino, cl. Sci. Fis. Mat. Nat. (3), 3 (1957), 25-43. 

[37]

F. Demengel, Regularity properties of viscosity solutions for fully nonlinear equations on the model of the anisotropic $p$-Laplacian, Asymptot. Anal., 105 (2017), 27-43.  doi: 10.3233/ASY-171433.

[38]

E. DiBenedetto, $\mathcal{C}^{1+\alpha }$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850.  doi: 10.1016/0362-546X(83)90061-5.

[39]

E. DiBenedettoU. Gianazza and 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.  doi: 10.1007/BF03377399.

[40]

L. Diening, P. Harjulehto, P. Hästöand M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, 2017. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.

[41]

G. F. DüzgunP. Marcellini and V. Vespri, Space expansion for a solution of an anisotropic $p$-Laplacian equation by using a parabolic approach, Riv. Math. Univ. Parma (N.S.), 5 (2014), 93-111. 

[42]

G. DüzgunP. Marcellini and V. Vespri, An alternative approach to the Hölder continuity of solutions to some elliptic equations, Nonlinear Anal., 94 (2014), 133-141.  doi: 10.1016/j.na.2013.08.018.

[43]

M. EleuteriP. Marcellini and E. Mascolo, Lipschitz estimates for systems with ellipticity conditions at infinity, Ann. Mat. Pura Appl., 195 (2016), 1575-1603.  doi: 10.1007/s10231-015-0529-4.

[44]

M. EleuteriP. Marcellini and E. Mascolo, Lipschitz continuity for functionals with variable exponents, Rend. Lincei Mat. Appl., 27 (2016), 61-87.  doi: 10.4171/RLM/723.

[45]

M. Eleuteri, P. Marcellini and E. Mascolo, Regularity for scalar integrals without structure conditions, Advances in Calculus of Variations, (2018). doi: 10.1515/acv-2017-0037.

[46]

M. EleuteriP. Marcellini and E. Mascolo, Local Lipschitz continuity of minimizers with mild assumptions on the $x$-dependence, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 251-265.  doi: 10.3934/dcdss.2019018.

[47]

M. Eleuteri and A. Passarelli di Napoli, Higher differentiability for solutions to a class of obstacle problems, Calc. Var. Partial Differential Equations, 57 (2018), Art. 115, 29 pp. doi: 10.1007/s00526-018-1387-x.

[48]

L. EspositoF. Leonetti and G. Mingione, Regularity results for minimizers of irregular integrals with $(p, q)$ growth, Forum Math., 14 (2002), 245-272.  doi: 10.1515/form.2002.011.

[49]

L. EspositoF. Leonetti and G. Mingione, Sharp regularity for functionals with $(p, q)$ growth, J. Differential Equations, 204 (2004), 5-55.  doi: 10.1016/j.jde.2003.11.007.

[50]

A. EspositoF. Leonetti and P. Vincenzo Petricca, Absence of Lavrentiev gap for non-autonomous functionals with $(p, q)$-growth, Adv. Nonlinear Anal., 8 (2019), 73-78.  doi: 10.1515/anona-2016-0198.

[51]

M. Focardi and E. Mascolo, Lower semicontinuity of quasi-convex functionals with non-standard growth, J. Convex Anal., 8 (2001), 327-347. 

[52]

I. FonsecaN. Fusco and P. Marcellini, An existence result for a nonconvex variational problem via regularity, ESAIM: Control, Optimisation and Calculus of Variations, 7 (2002), 69-95.  doi: 10.1051/cocv:2002004.

[53]

R. FortiniD. Mugnai and P. Pucci, Maximum principles for anisotropic elliptic inequalities, Nonlinear Anal., 70 (2009), 2917-2929.  doi: 10.1016/j.na.2008.12.030.

[54]

G. Fragnelli, Positive periodic solutions for a system of anisotropic parabolic equations, J. Math. Anal. Appl., 367 (2010), 204-228.  doi: 10.1016/j.jmaa.2009.12.039.

[55]

N. FuscoP. Marcellini and A. Ornelas, Existence of minimizers for some non convex one-dimensional integrals, Portugaliae Mathematica, 55 (1998), 167-185. 

[56] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies, 105. Princeton University Press, Princeton, NJ, 1983. 
[57]

M. Giaquinta, Growth conditions and regularity, a counterexample, Manuscripta Math., 59 (1987), 245-248.  doi: 10.1007/BF01158049.

[58]

E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/9789812795557.

[59]

P. Harjulehto and P. Hästö, Orlicz Spaces and Generalized Orlicz Spaces, Lecture Notes in Mathematics, 2236. Springer, Cham, 2019. doi: 10.1007/978-3-030-15100-3.

[60]

M. C. Hong, Some remarks on the minimizers of variational integrals wtih non standard growth conditions, Boll. Un. Mat. Ital. A, 6 (1992), 91-101. 

[61]

M. A. Krasnosel'skij and J. B. Ruticki$\check{{\rm i}}$, Convex Functions and Orlicz Spaces, P. Noordhoff Ltd., Groningen, 1961.

[62] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968. 
[63]

V. Liskevich and I. I. Skrypnik, Hölder continuity of solutions to an anisotropic elliptic equation, Nonlinear Analysis, 71 (2009), 1699-1708.  doi: 10.1016/j.na.2009.01.007.

[64]

J. J. LiuP. PucciH. T. Wu and Q. H. Zhang, Existence and blow-up rate of large solutions of $p(x)$-Laplacian equations with gradient terms, J. Math. Anal. Appl., 457 (2018), 944-977.  doi: 10.1016/j.jmaa.2017.08.038.

[65]

P. Marcellini, Quasiconvex quadratic forms in two dimensions, Applied Mathematics and Optimization, 11 (1984), 183-189.  doi: 10.1007/BF01442177.

[66]

P. Marcellini, Un example de solution discontinue d'un problème variationnel dans le cas scalaire, Preprint 11, Istituto Matematico, "U.Dini", Università di Firenze, 1987. http://web.math.unifi.it/users/marcellini/lavori/reprints/

[67]

P. Marcellini, The stored-energy for some discontinuous deformations in nonlinear elasticity, Partial Differential Equations and the Calculus of Variations, Vol. II, Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 2 (1989), 767-786. 

[68]

P. Marcellini, Nonconvex integrals of the calculus of variations, Methods of Nonconvex Analysis (Varenna, 1989), Lecture Notes in Math., Springer, Berlin, 1446 (1990), 16-57.  doi: 10.1007/BFb0084930.

[69]

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.  doi: 10.1007/BF00251503.

[70]

P. Marcellini, Regularity and existence of solutions of elliptic equations with $p, q$-growth conditions, J. Differential Equations, 90 (1991), 1-30.  doi: 10.1016/0022-0396(91)90158-6.

[71]

P. Marcellini, Regularity for elliptic equations with general growth conditions, J. Differential Equations, 105 (1993), 296-333.  doi: 10.1006/jdeq.1993.1091.

[72]

P. Marcellini, Regularity for some scalar variational problems under general growth conditions, J. Optim. Theory Appl., 90 (1996), 161-181.  doi: 10.1007/BF02192251.

[73]

P. Marcellini, Everywhere regularity for a class of elliptic systems without growth conditions, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23 (1996), 1-25. 

[74]

P. Marcellini, Some recent developments in the study of Hilbert's 19th and 20th problems, Boll. Un. Mat. Ital. A (7), 11 (1997), 323-352. 

[75]

P. Marcellini, A variational approach to parabolic equations under general and $p, q$-growth conditions, Nonlinear Anal., (2019). doi: 10.1016/j.na.2019.02.010.

[76]

P. Marcellini and G. Papi, Nonlinear elliptic systems with general growth, J. Differential Equations, 221 (2006), 412-443.  doi: 10.1016/j.jde.2004.11.011.

[77]

M. MihailescuP. Pucci and V. Rǎdulescu, Nonhomogeneous boundary value problems in anisotropic Sobolev spaces, C. R. Math. Acad. Sci. Paris, 345 (2007), 561-566.  doi: 10.1016/j.crma.2007.10.012.

[78]

M. MihailescuP. Pucci and V. Radulescu, Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J. Math. Anal. Appl., 340 (2008), 687-698.  doi: 10.1016/j.jmaa.2007.09.015.

[79]

G. Mingione, Regularity of minima: An invitation to the dark side of the calculus of variations, Appl. Math., 51 (2006), 355-426.  doi: 10.1007/s10778-006-0110-3.

[80]

A. Passarelli di Napoli, Higher differentiability of minimizers of variational integrals with Sobolev coefficients, Advances in Calculus of Variations, 7 (2014), 59-89.  doi: 10.1515/acv-2012-0006.

[81]

S. Piro VernierF. Ragnedda and V. Vespri, Hölder regularity for bounded solutions to a class of anisotropic operators, Manuscripta Math., 158 (2019), 421-439.  doi: 10.1007/s00229-018-1034-z.

[82]

P. Pucci and Q. H. Zhang, Existence of entire solutions for a class of variable exponent elliptic equations, J. Differential Equations, 257 (2014), 1529-1566.  doi: 10.1016/j.jde.2014.05.023.

[83]

K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math., 138 (1977), 219-240.  doi: 10.1007/BF02392316.

[84]

V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675–710,877.

[85]

V. V. Zhikov, On Lavrentiev phenomenon, Russian J. Math. Phys., 3 (1995), 249-269. 

[86]

V. V. Zhikov, On some variational problems, Russian J. Math. Phys., 5 (1997), 105-116. 

show all references

References:
[1]

G. Autuori and P. Pucci, Asymptotic stability for Kirchhoff systems in variable exponent Sobolev spaces, Complex Var. Elliptic Equ., 56 (2011), 715-753.  doi: 10.1080/17476931003786691.

[2]

J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal., 63 (1976/77), 337-403.  doi: 10.1007/BF00279992.

[3]

J. M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. Trans. R. Soc. Lond. Ser. A, 306 (1982), 557-611.  doi: 10.1098/rsta.1982.0095.

[4]

P. BaroniM. Colombo and G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal., 121 (2015), 206-222.  doi: 10.1016/j.na.2014.11.001.

[5]

P. BaroniM. Colombo and G. Mingione, Non-autonomous functionals, borderline cases and related function classes, St. Petersburg Math. J., 27 (2016), 347-379.  doi: 10.1090/spmj/1392.

[6]

P. Baroni, M. Colombo and G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations, 57 (2018), Art. 62, 48 pp. doi: 10.1007/s00526-018-1332-z.

[7]

L. Beck and G. Mingione, Lipschitz bounds and non-uniformly ellipticity, in preparation, 2018.

[8]

M. Bildhauer, Convex Variational Problems. Linear, Nearly Linear and Anisotropic Growth Conditions, Lecture Notes in Mathematics, 1818. Springer-Verlag, Berlin, 2003. doi: 10.1007/b12308.

[9]

I. Birindelli and F. Demengel, Fully nonlinear operators with Hamiltonian: Hölder regularity of the gradient, NoDEA Nonlinear Differential Equations Appl., 23 (2016), Art. 41, 17 pp. doi: 10.1007/s00030-016-0392-z.

[10]

L. BoccardoP. Marcellini and C. Sbordone, $L^\infty$-regularity for variational problems with sharp nonstandard growth conditions, Boll. Un. Mat. Ital. A, 4 (1990), 219-225.  doi: 10.1007/bf01934372.

[11]

L. BoccardoT. Gallouët and P. Marcellini, Anisotropic equations in $L^1$, Differential Integral Equations, 9 (1996), 209-212. 

[12]

V. BögeleinF. Duzaar and P. Marcellini, Parabolic equations with $p, q$-growth, Journal de Mathématiques Pures et Appliquées, 100 (2013), 535-563.  doi: 10.1016/j.matpur.2013.01.012.

[13]

V. BögeleinF. Duzaar and P. Marcellini, Parabolic systems with $p, q-$growth: A variational approach, Arch. Ration. Mech. Anal., 210 (2013), 219-267.  doi: 10.1007/s00205-013-0646-4.

[14]

V. BögeleinF. Duzaar and P. Marcellini, Existence of evolutionary variational solutions via the calculus of variations, J. Differential Equations, 256 (2014), 3912-3942.  doi: 10.1016/j.jde.2014.03.005.

[15]

V. BögeleinF. Duzaar and P. Marcellini, A time dependent variational approach to image restoration, SIAM J. on Imaging Sciences, 8 (2015), 968-1006.  doi: 10.1137/140992771.

[16]

V. BögeleinF. DuzaarP. Marcellini and C. Scheven, Doubly nonlinear equations of porous medium type, Arch. Ration. Mech. Anal., 229 (2018), 503-545.  doi: 10.1007/s00205-018-1221-9.

[17]

V. BögeleinF. DuzaarP. Marcellini and C. Scheven, A variational approach to doubly nonlinear equations, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 29 (2018), 739-772.  doi: 10.4171/RLM/832.

[18]

V. BögeleinF. DuzaarP. Marcellini and S. Signoriello, Nonlocal diffusion equations, J. Math. Anal. Appl., 432 (2015), 398-428.  doi: 10.1016/j.jmaa.2015.06.053.

[19]

V. BögeleinF. DuzaarP. Marcellini and S. Signoriello, Parabolic equations and the bounded slope condition, Ann. Inst. H. Poincare, Anal. Non Lineaire, 34 (2017), 355-379.  doi: 10.1016/j.anihpc.2015.12.005.

[20]

B. Botteron and P. Marcellini, A general approach to the existence of minimizers of one-dimensional non-coercive integrals of the calculus of variations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8 (1991), 197-223.  doi: 10.1016/S0294-1449(16)30272-4.

[21]

L. Brasco and G. Carlier, On certain anisotropic elliptic equations arising in congested optimal transport: Local gradient bounds, Adv. Calc. Var., 7 (2014), 379-407.  doi: 10.1515/acv-2013-0007.

[22]

P. Celada and S. Perrotta, Polyconvex energies and cavitation, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 295-321.  doi: 10.1007/s00030-012-0184-z.

[23]

I. Chlebicka, A pocket guide to nonlinear differential equations in Musielak-Orlicz spaces, Nonlinear Analysis, 175 (1918), 1-27.  doi: 10.1016/j.na.2018.05.003.

[24]

F. Colasuonno and P. Pucci, Multiplicity of solutions for $p(x)$-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal., 74 (2011), 5962-5974.  doi: 10.1016/j.na.2011.05.073.

[25]

M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Rat. Mech. Anal., 215 (2015), 443-496.  doi: 10.1007/s00205-014-0785-2.

[26]

M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Rat. Mech. Anal., 218 (2015), 219-273.  doi: 10.1007/s00205-015-0859-9.

[27]

G. CupiniF. GiannettiR. Giova and A. Passarelli di Napoli, Higher integrability for minimizers of asymptotically convex integrals with discontinuous coefficients, Nonlinear Anal., 154 (2017), 7-24.  doi: 10.1016/j.na.2016.02.017.

[28]

G. CupiniM. Guidorzi and E. Mascolo, Regularity of minimizers of vectorial integrals with $p-q$ growth, Nonlinear Anal., 54 (2003), 591-616.  doi: 10.1016/S0362-546X(03)00087-7.

[29]

G. CupiniP. Marcellini and E. Mascolo, Regularity under sharp anisotropic general growth conditions, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 66-86.  doi: 10.3934/dcdsb.2009.11.67.

[30]

G. CupiniP. Marcellini and E. Mascolo, Local boundedness of solutions to quasilinear elliptic systems, Manuscripta Math., 137 (2012), 287-315.  doi: 10.1007/s00229-011-0464-7.

[31]

G. CupiniP. Marcellini and E. Mascolo, Local boundedness of solutions to some anisotropic elliptic systems, Recent Trends in Nonlinear Partial Differential Equations. II. Stationary problems, Contemp. Math., Amer. Math. Soc., Providence, RI, 595 (2013), 169-186.  doi: 10.1090/conm/595/11803.

[32]

G. CupiniP. Marcellini and E. Mascolo, Existence and regularity for elliptic equations under $p, q$-growth, Adv. Differential Equations, 19 (2014), 693-724. 

[33]

G. CupiniP. Marcellini and E. Mascolo, Local boundedness of minimizers with limit growth conditions, J. Optim. Theory Appl., 166 (2015), 1-22.  doi: 10.1007/s10957-015-0722-z.

[34]

G. CupiniP. Marcellini and E. Mascolo, Regularity of minimizers under limit growth conditions, Nonlinear Analysis, 153 (2017), 294-310.  doi: 10.1016/j.na.2016.06.002.

[35]

G. CupiniP. Marcellini and E. Mascolo, Nonuniformly elliptic energy integrals with $p, q$-growth, Nonlinear Analysis, 177 (2018), 312-324.  doi: 10.1016/j.na.2018.03.018.

[36]

E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli, Mem. Accad. Sci. Torino, cl. Sci. Fis. Mat. Nat. (3), 3 (1957), 25-43. 

[37]

F. Demengel, Regularity properties of viscosity solutions for fully nonlinear equations on the model of the anisotropic $p$-Laplacian, Asymptot. Anal., 105 (2017), 27-43.  doi: 10.3233/ASY-171433.

[38]

E. DiBenedetto, $\mathcal{C}^{1+\alpha }$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850.  doi: 10.1016/0362-546X(83)90061-5.

[39]

E. DiBenedettoU. Gianazza and 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.  doi: 10.1007/BF03377399.

[40]

L. Diening, P. Harjulehto, P. Hästöand M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, 2017. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.

[41]

G. F. DüzgunP. Marcellini and V. Vespri, Space expansion for a solution of an anisotropic $p$-Laplacian equation by using a parabolic approach, Riv. Math. Univ. Parma (N.S.), 5 (2014), 93-111. 

[42]

G. DüzgunP. Marcellini and V. Vespri, An alternative approach to the Hölder continuity of solutions to some elliptic equations, Nonlinear Anal., 94 (2014), 133-141.  doi: 10.1016/j.na.2013.08.018.

[43]

M. EleuteriP. Marcellini and E. Mascolo, Lipschitz estimates for systems with ellipticity conditions at infinity, Ann. Mat. Pura Appl., 195 (2016), 1575-1603.  doi: 10.1007/s10231-015-0529-4.

[44]

M. EleuteriP. Marcellini and E. Mascolo, Lipschitz continuity for functionals with variable exponents, Rend. Lincei Mat. Appl., 27 (2016), 61-87.  doi: 10.4171/RLM/723.

[45]

M. Eleuteri, P. Marcellini and E. Mascolo, Regularity for scalar integrals without structure conditions, Advances in Calculus of Variations, (2018). doi: 10.1515/acv-2017-0037.

[46]

M. EleuteriP. Marcellini and E. Mascolo, Local Lipschitz continuity of minimizers with mild assumptions on the $x$-dependence, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 251-265.  doi: 10.3934/dcdss.2019018.

[47]

M. Eleuteri and A. Passarelli di Napoli, Higher differentiability for solutions to a class of obstacle problems, Calc. Var. Partial Differential Equations, 57 (2018), Art. 115, 29 pp. doi: 10.1007/s00526-018-1387-x.

[48]

L. EspositoF. Leonetti and G. Mingione, Regularity results for minimizers of irregular integrals with $(p, q)$ growth, Forum Math., 14 (2002), 245-272.  doi: 10.1515/form.2002.011.

[49]

L. EspositoF. Leonetti and G. Mingione, Sharp regularity for functionals with $(p, q)$ growth, J. Differential Equations, 204 (2004), 5-55.  doi: 10.1016/j.jde.2003.11.007.

[50]

A. EspositoF. Leonetti and P. Vincenzo Petricca, Absence of Lavrentiev gap for non-autonomous functionals with $(p, q)$-growth, Adv. Nonlinear Anal., 8 (2019), 73-78.  doi: 10.1515/anona-2016-0198.

[51]

M. Focardi and E. Mascolo, Lower semicontinuity of quasi-convex functionals with non-standard growth, J. Convex Anal., 8 (2001), 327-347. 

[52]

I. FonsecaN. Fusco and P. Marcellini, An existence result for a nonconvex variational problem via regularity, ESAIM: Control, Optimisation and Calculus of Variations, 7 (2002), 69-95.  doi: 10.1051/cocv:2002004.

[53]

R. FortiniD. Mugnai and P. Pucci, Maximum principles for anisotropic elliptic inequalities, Nonlinear Anal., 70 (2009), 2917-2929.  doi: 10.1016/j.na.2008.12.030.

[54]

G. Fragnelli, Positive periodic solutions for a system of anisotropic parabolic equations, J. Math. Anal. Appl., 367 (2010), 204-228.  doi: 10.1016/j.jmaa.2009.12.039.

[55]

N. FuscoP. Marcellini and A. Ornelas, Existence of minimizers for some non convex one-dimensional integrals, Portugaliae Mathematica, 55 (1998), 167-185. 

[56] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies, 105. Princeton University Press, Princeton, NJ, 1983. 
[57]

M. Giaquinta, Growth conditions and regularity, a counterexample, Manuscripta Math., 59 (1987), 245-248.  doi: 10.1007/BF01158049.

[58]

E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/9789812795557.

[59]

P. Harjulehto and P. Hästö, Orlicz Spaces and Generalized Orlicz Spaces, Lecture Notes in Mathematics, 2236. Springer, Cham, 2019. doi: 10.1007/978-3-030-15100-3.

[60]

M. C. Hong, Some remarks on the minimizers of variational integrals wtih non standard growth conditions, Boll. Un. Mat. Ital. A, 6 (1992), 91-101. 

[61]

M. A. Krasnosel'skij and J. B. Ruticki$\check{{\rm i}}$, Convex Functions and Orlicz Spaces, P. Noordhoff Ltd., Groningen, 1961.

[62] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968. 
[63]

V. Liskevich and I. I. Skrypnik, Hölder continuity of solutions to an anisotropic elliptic equation, Nonlinear Analysis, 71 (2009), 1699-1708.  doi: 10.1016/j.na.2009.01.007.

[64]

J. J. LiuP. PucciH. T. Wu and Q. H. Zhang, Existence and blow-up rate of large solutions of $p(x)$-Laplacian equations with gradient terms, J. Math. Anal. Appl., 457 (2018), 944-977.  doi: 10.1016/j.jmaa.2017.08.038.

[65]

P. Marcellini, Quasiconvex quadratic forms in two dimensions, Applied Mathematics and Optimization, 11 (1984), 183-189.  doi: 10.1007/BF01442177.

[66]

P. Marcellini, Un example de solution discontinue d'un problème variationnel dans le cas scalaire, Preprint 11, Istituto Matematico, "U.Dini", Università di Firenze, 1987. http://web.math.unifi.it/users/marcellini/lavori/reprints/

[67]

P. Marcellini, The stored-energy for some discontinuous deformations in nonlinear elasticity, Partial Differential Equations and the Calculus of Variations, Vol. II, Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 2 (1989), 767-786. 

[68]

P. Marcellini, Nonconvex integrals of the calculus of variations, Methods of Nonconvex Analysis (Varenna, 1989), Lecture Notes in Math., Springer, Berlin, 1446 (1990), 16-57.  doi: 10.1007/BFb0084930.

[69]

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.  doi: 10.1007/BF00251503.

[70]

P. Marcellini, Regularity and existence of solutions of elliptic equations with $p, q$-growth conditions, J. Differential Equations, 90 (1991), 1-30.  doi: 10.1016/0022-0396(91)90158-6.

[71]

P. Marcellini, Regularity for elliptic equations with general growth conditions, J. Differential Equations, 105 (1993), 296-333.  doi: 10.1006/jdeq.1993.1091.

[72]

P. Marcellini, Regularity for some scalar variational problems under general growth conditions, J. Optim. Theory Appl., 90 (1996), 161-181.  doi: 10.1007/BF02192251.

[73]

P. Marcellini, Everywhere regularity for a class of elliptic systems without growth conditions, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23 (1996), 1-25. 

[74]

P. Marcellini, Some recent developments in the study of Hilbert's 19th and 20th problems, Boll. Un. Mat. Ital. A (7), 11 (1997), 323-352. 

[75]

P. Marcellini, A variational approach to parabolic equations under general and $p, q$-growth conditions, Nonlinear Anal., (2019). doi: 10.1016/j.na.2019.02.010.

[76]

P. Marcellini and G. Papi, Nonlinear elliptic systems with general growth, J. Differential Equations, 221 (2006), 412-443.  doi: 10.1016/j.jde.2004.11.011.

[77]

M. MihailescuP. Pucci and V. Rǎdulescu, Nonhomogeneous boundary value problems in anisotropic Sobolev spaces, C. R. Math. Acad. Sci. Paris, 345 (2007), 561-566.  doi: 10.1016/j.crma.2007.10.012.

[78]

M. MihailescuP. Pucci and V. Radulescu, Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J. Math. Anal. Appl., 340 (2008), 687-698.  doi: 10.1016/j.jmaa.2007.09.015.

[79]

G. Mingione, Regularity of minima: An invitation to the dark side of the calculus of variations, Appl. Math., 51 (2006), 355-426.  doi: 10.1007/s10778-006-0110-3.

[80]

A. Passarelli di Napoli, Higher differentiability of minimizers of variational integrals with Sobolev coefficients, Advances in Calculus of Variations, 7 (2014), 59-89.  doi: 10.1515/acv-2012-0006.

[81]

S. Piro VernierF. Ragnedda and V. Vespri, Hölder regularity for bounded solutions to a class of anisotropic operators, Manuscripta Math., 158 (2019), 421-439.  doi: 10.1007/s00229-018-1034-z.

[82]

P. Pucci and Q. H. Zhang, Existence of entire solutions for a class of variable exponent elliptic equations, J. Differential Equations, 257 (2014), 1529-1566.  doi: 10.1016/j.jde.2014.05.023.

[83]

K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math., 138 (1977), 219-240.  doi: 10.1007/BF02392316.

[84]

V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675–710,877.

[85]

V. V. Zhikov, On Lavrentiev phenomenon, Russian J. Math. Phys., 3 (1995), 249-269. 

[86]

V. V. Zhikov, On some variational problems, Russian J. Math. Phys., 5 (1997), 105-116. 

[1]

Pablo Amster, Alberto Déboli, Manuel Pinto. Hartman and Nirenberg type results for systems of delay differential equations under $ (\omega,Q) $-periodic conditions. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3019-3037. doi: 10.3934/dcdsb.2021171

[2]

Junjie Zhang, Shenzhou Zheng, Haiyan Yu. $ L^{p(\cdot)} $-regularity of Hessian for nondivergence parabolic and elliptic equations with measurable coefficients. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2777-2796. doi: 10.3934/cpaa.2020121

[3]

Ildoo Kim. An $L_p$-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2751-2771. doi: 10.3934/cpaa.2018130

[4]

Junyong Eom, Ryuichi Sato. Large time behavior of ODE type solutions to parabolic $ p $-Laplacian type equations. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4373-4386. doi: 10.3934/cpaa.2020199

[5]

Gabriele Bonanno, Giuseppina D'Aguì. Mixed elliptic problems involving the $p-$Laplacian with nonhomogeneous boundary conditions. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5797-5817. doi: 10.3934/dcds.2017252

[6]

Lun Guo, Wentao Huang, Huifang Jia. Ground state solutions for the fractional Schrödinger-Poisson systems involving critical growth in $ \mathbb{R} ^{3} $. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1663-1693. doi: 10.3934/cpaa.2019079

[7]

Mihai Mihăilescu, Julio D. Rossi. Monotonicity with respect to $ p $ of the First Nontrivial Eigenvalue of the $ p $-Laplacian with Homogeneous Neumann Boundary Conditions. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4363-4371. doi: 10.3934/cpaa.2020198

[8]

Andrzej Świȩch. Pointwise properties of $ L^p $-viscosity solutions of uniformly elliptic equations with quadratically growing gradient terms. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2945-2962. doi: 10.3934/dcds.2020156

[9]

Giovanni Cupini, Paolo Marcellini, Elvira Mascolo. Regularity under sharp anisotropic general growth conditions. Discrete and Continuous Dynamical Systems - B, 2009, 11 (1) : 67-86. doi: 10.3934/dcdsb.2009.11.67

[10]

Brahim Allal, Abdelkarim Hajjaj, Lahcen Maniar, Jawad Salhi. Null controllability for singular cascade systems of $ n $-coupled degenerate parabolic equations by one control force. Evolution Equations and Control Theory, 2021, 10 (3) : 545-573. doi: 10.3934/eect.2020080

[11]

VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with $ p(x) $-growth and generalized Robin boundary value condition. Communications on Pure and Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003

[12]

Jian-Guo Liu, Zhaoyun Zhang. Existence of global weak solutions of $ p $-Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 469-486. doi: 10.3934/dcdsb.2021051

[13]

Teresa Isernia, Chiara Leone, Anna Verde. Partial regularity result for non-autonomous elliptic systems with general growth. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4271-4305. doi: 10.3934/cpaa.2021160

[14]

Li Wang, Qiang Xu, Shulin Zhou. $ L^p $ Neumann problems in homogenization of general elliptic operators. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 5019-5045. doi: 10.3934/dcds.2020210

[15]

Volodymyr O. Kapustyan, Ivan O. Pyshnograiev, Olena A. Kapustian. Quasi-optimal control with a general quadratic criterion in a special norm for systems described by parabolic-hyperbolic equations with non-local boundary conditions. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1243-1258. doi: 10.3934/dcdsb.2019014

[16]

Joackim Bernier. Bounds on the growth of high discrete Sobolev norms for the cubic discrete nonlinear Schrödinger equations on $ h\mathbb{Z} $. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3179-3195. doi: 10.3934/dcds.2019131

[17]

Niklas Sapountzoglou, Aleksandra Zimmermann. Renormalized solutions for stochastic $ p $-Laplace equations with $ L^1 $-initial data: The case of multiplicative noise. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022041

[18]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[19]

Nicholas J. Kass, Mohammad A. Rammaha. Local and global existence of solutions to a strongly damped wave equation of the $ p $-Laplacian type. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1449-1478. doi: 10.3934/cpaa.2018070

[20]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3351-3386. doi: 10.3934/dcdss.2020440

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (490)
  • HTML views (286)
  • Cited by (1)

Other articles
by authors

[Back to Top]