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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 & 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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