August  2017, 10(4): 647-659. doi: 10.3934/dcdss.2017032

Intrinsic geometry and De Giorgi classes for certain anisotropic problems

1. 

Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Campus -Parco Area delle Scienze 53/A, 43124 Parma, Italy

2. 

Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Via A. Scarpa 14, 00162 Roma, Italy

* Corresponding author: Giampiero Palatucci

Received  January 2016 Revised  July 2016 Published  April 2017

We analyze a natural approach to the regularity of solutions of problems related to some anisotropic Laplacian operators, and a subsequent extension of the usual De Giorgi classes, by investigating the relation of the functions in such classes with the weak solutions to some anisotropic elliptic equations as well as with the quasi-minima of the corresponding functionals with anisotropic polynomial growth.

Citation: Paolo Baroni, Agnese Di Castro, Giampiero Palatucci. Intrinsic geometry and De Giorgi classes for certain anisotropic problems. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 647-659. doi: 10.3934/dcdss.2017032
References:
[1]

E. Acerbi and N. Fusco, Partial regularity under anisotropic (p, q) growth conditions, J. Differential Equations, 107 (1994), 46-67.  doi: 10.1006/jdeq.1994.1002.

[2]

E. Acerbi and G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Rational Mech. Anal., 156 (2001), 121-140.  doi: 10.1007/s002050100117.

[3]

E. Acerbi and G. Mingione, Gradient estimates for the p(x)-Laplacean system, J. Reine Angew. Math., 584 (2005), 117-148. 

[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]

L. BoccardoP. Marcellini and C. Sbordone, L-regularity for variational problems with sharp non-standard growth conditions, Boll. Un. Mat. Ital. A, 4 (1990), 219-225. 

[7]

A. Cianchi, Local boundedness of minimizers of anisotropic functionals, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 147-168.  doi: 10.1016/S0294-1449(99)00107-9.

[8]

M. Colombo and G. Mingione, Calderón-Zygmund estimates and non-uniformly elliptic operators, J. Funct. Anal., 270 (2016), 1416-1478.  doi: 10.1016/j.jfa.2015.06.022.

[9]

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

[10]

G. CupiniP. Marcellini and E. Mascolo, Local boundedness of minimisers with limit growth conditions, J. Optim. Theory Appl., 166 (2015), 1-22. 

[11]

E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, Series Universitext, New York, 1993. doi: 10.1007/978-1-4612-0895-2.

[12]

E. DiBenedettoU. Gianazza and V. Vespri, Harnack estimates for quasi-linear degenerate parabolic differential equations, Acta Math., 200 (2008), 181-209.  doi: 10.1007/s11511-008-0026-3.

[13]

A. Di Castro, Existence and regularity results for anisotropic elliptic problems, Adv. Nonlinear Stud., 9 (2009), 367-393. 

[14]

A. Di CastroT. Kuusi and G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal., 267 (2014), 1807-1836.  doi: 10.1016/j.jfa.2014.05.023.

[15]

A. Di CastroT. Kuusi and G. Palatucci, Local behavior of fractional p-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1279-1299.  doi: 10.1016/j.anihpc.2015.04.003.

[16]

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

[17]

I. FragalàF. Gazzola and B. Kawohl, Existence and nonexistence results for anisotropic quasilinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 715-734. 

[18]

M. Giaquinta and E. Giusti, On the regularity of the minima of variational integrals, Acta Math., 148 (1982), 31-46.  doi: 10.1007/BF02392725.

[19]

L. EspositoF. Leonetti and G. Mingione, Regularity for minimizers of functionals with p-q growth, NoDEA Nonlinear Differential Equations Appl., 6 (1999), 133-148.  doi: 10.1007/s000300050069.

[20]

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.

[21]

N. Fusco and C. Sbordone, Some remarks on the regularity of minima of anisotropic integrals, Comm. Partial Differential Equations, 18 (1993), 153-167.  doi: 10.1080/03605309308820924.

[22]

J. Haskovec and C. Schmeiser, A note on the anisotropic generalizations of the Sobolev and Morrey embedding theorems, Monatsh. Math., 158 (2009), 71-79.  doi: 10.1007/s00605-008-0059-x.

[23]

A. Innamorati and F. Leonetti, Global integrability for weak solutions to some anisotropic elliptic equations, Nonlinear Anal., 113 (2015), 430-434.  doi: 10.1016/j.na.2014.09.027.

[24]

T. Kuusi, Harnack estimates for weak supersolutions to nonlinear degenerate parabolic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7 (2008), 673-716.  doi: 10.2422/2036-2145.2008.4.04.

[25]

T. Kuusi and G. Mingione, Universal potential estimates, J. Funct. Anal., 262 (2012), 4205-4269.  doi: 10.1016/j.jfa.2012.02.018.

[26]

F. Leonetti, Higher integrability for minimizers of integral functionals with nonstandard growth, J. Differential Equations, 112 (1994), 308-324.  doi: 10.1006/jdeq.1994.1106.

[27]

F. LeonettiE. Mascolo and F. Siepe, Everywhere regularity for a class of vectorial functionals under subquadratic general growth conditions, J. Math. Anal. Appl., 287 (2003), 593-608.  doi: 10.1016/S0022-247X(03)00584-5.

[28]

G. M. Lieberman, Gradient estimates for anisotropic elliptic equations, Adv. Differential Equations, 10 (2005), 767-812. 

[29]

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

[30]

P. Marcellini, Regularity of minimizers of integrals of the Calculus of Variations with non standard growth conditions, Arch. Rational Mech. Anal., 105 (1989), 267-284.  doi: 10.1007/BF00251503.

[31]

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.

[32]

M. Masson and J. Siljander, Hölder regularity for parabolic De Giorgi classes in metric measure spaces, Manuscripta Math., 142 (2013), 187-214.  doi: 10.1007/s00229-012-0598-2.

show all references

References:
[1]

E. Acerbi and N. Fusco, Partial regularity under anisotropic (p, q) growth conditions, J. Differential Equations, 107 (1994), 46-67.  doi: 10.1006/jdeq.1994.1002.

[2]

E. Acerbi and G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Rational Mech. Anal., 156 (2001), 121-140.  doi: 10.1007/s002050100117.

[3]

E. Acerbi and G. Mingione, Gradient estimates for the p(x)-Laplacean system, J. Reine Angew. Math., 584 (2005), 117-148. 

[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]

L. BoccardoP. Marcellini and C. Sbordone, L-regularity for variational problems with sharp non-standard growth conditions, Boll. Un. Mat. Ital. A, 4 (1990), 219-225. 

[7]

A. Cianchi, Local boundedness of minimizers of anisotropic functionals, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 147-168.  doi: 10.1016/S0294-1449(99)00107-9.

[8]

M. Colombo and G. Mingione, Calderón-Zygmund estimates and non-uniformly elliptic operators, J. Funct. Anal., 270 (2016), 1416-1478.  doi: 10.1016/j.jfa.2015.06.022.

[9]

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

[10]

G. CupiniP. Marcellini and E. Mascolo, Local boundedness of minimisers with limit growth conditions, J. Optim. Theory Appl., 166 (2015), 1-22. 

[11]

E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, Series Universitext, New York, 1993. doi: 10.1007/978-1-4612-0895-2.

[12]

E. DiBenedettoU. Gianazza and V. Vespri, Harnack estimates for quasi-linear degenerate parabolic differential equations, Acta Math., 200 (2008), 181-209.  doi: 10.1007/s11511-008-0026-3.

[13]

A. Di Castro, Existence and regularity results for anisotropic elliptic problems, Adv. Nonlinear Stud., 9 (2009), 367-393. 

[14]

A. Di CastroT. Kuusi and G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal., 267 (2014), 1807-1836.  doi: 10.1016/j.jfa.2014.05.023.

[15]

A. Di CastroT. Kuusi and G. Palatucci, Local behavior of fractional p-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1279-1299.  doi: 10.1016/j.anihpc.2015.04.003.

[16]

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

[17]

I. FragalàF. Gazzola and B. Kawohl, Existence and nonexistence results for anisotropic quasilinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 715-734. 

[18]

M. Giaquinta and E. Giusti, On the regularity of the minima of variational integrals, Acta Math., 148 (1982), 31-46.  doi: 10.1007/BF02392725.

[19]

L. EspositoF. Leonetti and G. Mingione, Regularity for minimizers of functionals with p-q growth, NoDEA Nonlinear Differential Equations Appl., 6 (1999), 133-148.  doi: 10.1007/s000300050069.

[20]

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.

[21]

N. Fusco and C. Sbordone, Some remarks on the regularity of minima of anisotropic integrals, Comm. Partial Differential Equations, 18 (1993), 153-167.  doi: 10.1080/03605309308820924.

[22]

J. Haskovec and C. Schmeiser, A note on the anisotropic generalizations of the Sobolev and Morrey embedding theorems, Monatsh. Math., 158 (2009), 71-79.  doi: 10.1007/s00605-008-0059-x.

[23]

A. Innamorati and F. Leonetti, Global integrability for weak solutions to some anisotropic elliptic equations, Nonlinear Anal., 113 (2015), 430-434.  doi: 10.1016/j.na.2014.09.027.

[24]

T. Kuusi, Harnack estimates for weak supersolutions to nonlinear degenerate parabolic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7 (2008), 673-716.  doi: 10.2422/2036-2145.2008.4.04.

[25]

T. Kuusi and G. Mingione, Universal potential estimates, J. Funct. Anal., 262 (2012), 4205-4269.  doi: 10.1016/j.jfa.2012.02.018.

[26]

F. Leonetti, Higher integrability for minimizers of integral functionals with nonstandard growth, J. Differential Equations, 112 (1994), 308-324.  doi: 10.1006/jdeq.1994.1106.

[27]

F. LeonettiE. Mascolo and F. Siepe, Everywhere regularity for a class of vectorial functionals under subquadratic general growth conditions, J. Math. Anal. Appl., 287 (2003), 593-608.  doi: 10.1016/S0022-247X(03)00584-5.

[28]

G. M. Lieberman, Gradient estimates for anisotropic elliptic equations, Adv. Differential Equations, 10 (2005), 767-812. 

[29]

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

[30]

P. Marcellini, Regularity of minimizers of integrals of the Calculus of Variations with non standard growth conditions, Arch. Rational Mech. Anal., 105 (1989), 267-284.  doi: 10.1007/BF00251503.

[31]

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.

[32]

M. Masson and J. Siljander, Hölder regularity for parabolic De Giorgi classes in metric measure spaces, Manuscripta Math., 142 (2013), 187-214.  doi: 10.1007/s00229-012-0598-2.

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