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 & 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.  Google Scholar

[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.  Google Scholar

[3]

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

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[11]

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

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[11]

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

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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