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November  2015, 14(6): 2363-2376. doi: 10.3934/cpaa.2015.14.2363

Harnack inequality for degenerate elliptic equations and sum operators

Giuseppe Di Fazio 1, , Maria Stella Fanciullo 1, and  Pietro Zamboni 1,

1. 

Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Italy

Received  January 2015 Revised  July 2015 Published  September 2015

We define Stummel-Kato type classes in a quasimetric homogeneous setting using sum operators introduced in [13] by Franchi, Perez and Wheeden. Then we prove a Harnack inequality for positive solutions of some linear subelliptic equations.
Keywords: Harnack inequality., Stummel-Kato type classes, Fefferman-Phong inequality.
Mathematics Subject Classification: Primary: 35B45; Secondary: 35B6.
Citation: Giuseppe Di Fazio, Maria Stella Fanciullo, Pietro Zamboni. Harnack inequality for degenerate elliptic equations and sum operators. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2363-2376. doi: 10.3934/cpaa.2015.14.2363
References:
[1]

S. M. Buckley, Inequalities of John-Nirenberg type in doubling spaces,, \emph{J. Anal. Math.}, 79 (1999), 215.  doi: 10.1007/BF02788242.  Google Scholar

[2]

F. Chiarenza and M. Frasca, A remark on a paper by C. Fefferman,, \emph{Proc. Amer. Math. Soc.}, 108 (1990), 407.  doi: 10.2307/2048289.  Google Scholar

[3]

G. Di Fazio, M. S. Fanciullo and P. Zamboni, Harnack inequality for a class of strongly degenerate elliptic operators formed by Hörmander vector fields,, \emph{Manuscripta Math.}, 135 (2011), 361.  doi: 10.1007/s00229-010-0420-y.  Google Scholar

[4]

G. Di Fazio, M. S. Fanciullo and P. Zamboni, Regularity for a class of strongly degenerate quasilinear operators,, \emph{J. Differential Equations}, 255 (2013), 3920.  doi: 10.1016/j.jde.2013.07.062.  Google Scholar

[5]

G. Di Fazio, M. S. Fanciullo and P. Zamboni, Sum Operators and Fefferman - Phong Inequalities,, Geometric Methods in PDE's, (2015).   Google Scholar

[6]

G. Di Fazio and P. Zamboni, A Fefferman-Poincaré type inequality for Carnot-Carathéodory vector fields,, \emph{Proc. Amer. Math. Soc.}, 130 (2002), 2655.  doi: 10.1090/S0002-9939-02-06394-3.  Google Scholar

[7]

G. Di Fazio and P. Zamboni, Hölder continuity for quasilinear subelliptic equations in Carnot Carathéodory spaces,, \emph{Math. Nachr.}, 272 (2004), 3.  doi: 10.1002/mana.200310185.  Google Scholar

[8]

G. Di Fazio and P. Zamboni, Regularity for quasilinear degenerate elliptic equations,, \emph{Math. Z.}, 253 (2006), 787.  doi: 10.1007/s00209-006-0933-y.  Google Scholar

[9]

G. Di Fazio and P. Zamboni, Local regularity of solutions to quasilinear subelliptic equations in Carnot Caratheodory spaces,, \emph{Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8)}, 9 (2006), 485.   Google Scholar

[10]

B. Franchi, G. Lu and R. L. Wheeden, Representation formulas and weighted Poincaré type inequalities for Hörmander vector fields,, \emph{Ann. inst. Fourier tome}, 45 (1995), 577.   Google Scholar

[11]

B. Franchi, G. Lu and R. L. Wheeden, A relationship between Poincaré-type inequalities and representation formulas in spaces of homogeneous type,, \emph{Int. Math. Res. Not.}, (1996), 1.  doi: 10.1155/S1073792896000013.  Google Scholar

[12]

B. Franchi, C. Perez and R. L. Wheeden, Self-improving properties of John-Nirenberg and Poincaré inequalities on spaces of homogeneous type,, \emph{J. of functional Analysis}, 153 (1998), 108.  doi: 10.1006/jfan.1997.3175.  Google Scholar

[13]

B. Franchi, C. Perez and R. L. Wheeden, A sum operator with applications to self-improving properties of Poincaré inequalities in metric spaces,, \emph{The Journal of Fourier Analysis and Applications}, 9 (2003), 511.  doi: 10.1007/s00041-003-0025-x.  Google Scholar

[14]

B. Franchi, R. Serapioni and F. Serra Cassano, Approximation and imbedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields,, \emph{Boll. Un. Mat. Ital. B (7)}, 11 (1997), 83.   Google Scholar

[15]

C. E. Gutierrez, Harnack's inequality for degenerate Schrödinger operators,, \emph{Trans. AMS}, 312 (1989), 403.  doi: 10.2307/2001222.  Google Scholar

[16]

P. Hajlasz and P. Koskela, Sobolev met Poincaré,, Mem. Amer. Math. Soc., 688 (2000).  doi: 10.1090/memo/0688.  Google Scholar

[17]

J. Heinonen and P. Koskela, Quasiconformal maps on metric spaces with controlled geometry,, \emph{Acta Math.}, 181 (1998), 1.  doi: 10.1007/BF02392747.  Google Scholar

[18]

D. Jerison, The Poincaré inequality for vector fields satisfying Hörmander's condition,, \emph{Duke Math. J.}, 53 (1986), 503.  doi: 10.1215/S0012-7094-86-05329-9.  Google Scholar

[19]

G. Lu, Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications,, \emph{Rev. Mat. Iberoamericana}, 8 (1992), 367.  doi: 10.4171/RMI/129.  Google Scholar

[20]

G. Lu, The sharp Poincaré inequality for free vector fields: an endpoint result,, \emph{Rev. Mat. Iberoamericana}, 10 (1994), 453.  doi: 10.4171/RMI/158.  Google Scholar

[21]

G. Lu, A Fefferman-Phong type inequality for degenerate vector fields and applications,, \emph{Panamer. Math. J.}, 6 (1996), 37.   Google Scholar

[22]

G. Lu and R. L. Wheeden, An optimal representation formula for Carnot- Carathéodory vector fields,, \emph{Bull. London Math. Soc.}, 30 (1998), 578.  doi: 10.1112/S0024609398004895.  Google Scholar

[23]

M. A. Ragusa and P. Zamboni, Local regularity of solutions to quasilinear elliptic equations with general structure,, \emph{Commun. Appl. Anal.}, 3 (1999), 131.   Google Scholar

[24]

J. Serrin, Local behavior of solutions of quasilinear equations,, \emph{Acta Math.}, 111 (1964), 247.   Google Scholar

[25]

P. Zamboni, Some function spaces and elliptic partial differential equations,, \emph{Le Matematiche (Catania)}, 42 (1987), 171.   Google Scholar

[26]

P. Zamboni, The Harnack inequality for quasilinear elliptic equations under minimal assumptions,, \emph{Manuscripta Math.}, 102 (2000), 311.  doi: 10.1007/s002290050002.  Google Scholar

[27]

P. Zamboni, Unique continuation for non-negative solutions of quasilinear elliptic equations,, \emph{Bull. Austral. Math. Soc.}, 64 (2001), 149.  doi: 10.1017/S0004972700019766.  Google Scholar

[28]

P. Zamboni, Hölder continuity for solutions of linear degenerate elliptic equations under minimal assumptions,, \emph{J. Differential Equations}, 182 (2002), 121.  doi: 10.1006/jdeq.2001.4094.  Google Scholar

show all references

References:
[1]

S. M. Buckley, Inequalities of John-Nirenberg type in doubling spaces,, \emph{J. Anal. Math.}, 79 (1999), 215.  doi: 10.1007/BF02788242.  Google Scholar

[2]

F. Chiarenza and M. Frasca, A remark on a paper by C. Fefferman,, \emph{Proc. Amer. Math. Soc.}, 108 (1990), 407.  doi: 10.2307/2048289.  Google Scholar

[3]

G. Di Fazio, M. S. Fanciullo and P. Zamboni, Harnack inequality for a class of strongly degenerate elliptic operators formed by Hörmander vector fields,, \emph{Manuscripta Math.}, 135 (2011), 361.  doi: 10.1007/s00229-010-0420-y.  Google Scholar

[4]

G. Di Fazio, M. S. Fanciullo and P. Zamboni, Regularity for a class of strongly degenerate quasilinear operators,, \emph{J. Differential Equations}, 255 (2013), 3920.  doi: 10.1016/j.jde.2013.07.062.  Google Scholar

[5]

G. Di Fazio, M. S. Fanciullo and P. Zamboni, Sum Operators and Fefferman - Phong Inequalities,, Geometric Methods in PDE's, (2015).   Google Scholar

[6]

G. Di Fazio and P. Zamboni, A Fefferman-Poincaré type inequality for Carnot-Carathéodory vector fields,, \emph{Proc. Amer. Math. Soc.}, 130 (2002), 2655.  doi: 10.1090/S0002-9939-02-06394-3.  Google Scholar

[7]

G. Di Fazio and P. Zamboni, Hölder continuity for quasilinear subelliptic equations in Carnot Carathéodory spaces,, \emph{Math. Nachr.}, 272 (2004), 3.  doi: 10.1002/mana.200310185.  Google Scholar

[8]

G. Di Fazio and P. Zamboni, Regularity for quasilinear degenerate elliptic equations,, \emph{Math. Z.}, 253 (2006), 787.  doi: 10.1007/s00209-006-0933-y.  Google Scholar

[9]

G. Di Fazio and P. Zamboni, Local regularity of solutions to quasilinear subelliptic equations in Carnot Caratheodory spaces,, \emph{Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8)}, 9 (2006), 485.   Google Scholar

[10]

B. Franchi, G. Lu and R. L. Wheeden, Representation formulas and weighted Poincaré type inequalities for Hörmander vector fields,, \emph{Ann. inst. Fourier tome}, 45 (1995), 577.   Google Scholar

[11]

B. Franchi, G. Lu and R. L. Wheeden, A relationship between Poincaré-type inequalities and representation formulas in spaces of homogeneous type,, \emph{Int. Math. Res. Not.}, (1996), 1.  doi: 10.1155/S1073792896000013.  Google Scholar

[12]

B. Franchi, C. Perez and R. L. Wheeden, Self-improving properties of John-Nirenberg and Poincaré inequalities on spaces of homogeneous type,, \emph{J. of functional Analysis}, 153 (1998), 108.  doi: 10.1006/jfan.1997.3175.  Google Scholar

[13]

B. Franchi, C. Perez and R. L. Wheeden, A sum operator with applications to self-improving properties of Poincaré inequalities in metric spaces,, \emph{The Journal of Fourier Analysis and Applications}, 9 (2003), 511.  doi: 10.1007/s00041-003-0025-x.  Google Scholar

[14]

B. Franchi, R. Serapioni and F. Serra Cassano, Approximation and imbedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields,, \emph{Boll. Un. Mat. Ital. B (7)}, 11 (1997), 83.   Google Scholar

[15]

C. E. Gutierrez, Harnack's inequality for degenerate Schrödinger operators,, \emph{Trans. AMS}, 312 (1989), 403.  doi: 10.2307/2001222.  Google Scholar

[16]

P. Hajlasz and P. Koskela, Sobolev met Poincaré,, Mem. Amer. Math. Soc., 688 (2000).  doi: 10.1090/memo/0688.  Google Scholar

[17]

J. Heinonen and P. Koskela, Quasiconformal maps on metric spaces with controlled geometry,, \emph{Acta Math.}, 181 (1998), 1.  doi: 10.1007/BF02392747.  Google Scholar

[18]

D. Jerison, The Poincaré inequality for vector fields satisfying Hörmander's condition,, \emph{Duke Math. J.}, 53 (1986), 503.  doi: 10.1215/S0012-7094-86-05329-9.  Google Scholar

[19]

G. Lu, Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications,, \emph{Rev. Mat. Iberoamericana}, 8 (1992), 367.  doi: 10.4171/RMI/129.  Google Scholar

[20]

G. Lu, The sharp Poincaré inequality for free vector fields: an endpoint result,, \emph{Rev. Mat. Iberoamericana}, 10 (1994), 453.  doi: 10.4171/RMI/158.  Google Scholar

[21]

G. Lu, A Fefferman-Phong type inequality for degenerate vector fields and applications,, \emph{Panamer. Math. J.}, 6 (1996), 37.   Google Scholar

[22]

G. Lu and R. L. Wheeden, An optimal representation formula for Carnot- Carathéodory vector fields,, \emph{Bull. London Math. Soc.}, 30 (1998), 578.  doi: 10.1112/S0024609398004895.  Google Scholar

[23]

M. A. Ragusa and P. Zamboni, Local regularity of solutions to quasilinear elliptic equations with general structure,, \emph{Commun. Appl. Anal.}, 3 (1999), 131.   Google Scholar

[24]

J. Serrin, Local behavior of solutions of quasilinear equations,, \emph{Acta Math.}, 111 (1964), 247.   Google Scholar

[25]

P. Zamboni, Some function spaces and elliptic partial differential equations,, \emph{Le Matematiche (Catania)}, 42 (1987), 171.   Google Scholar

[26]

P. Zamboni, The Harnack inequality for quasilinear elliptic equations under minimal assumptions,, \emph{Manuscripta Math.}, 102 (2000), 311.  doi: 10.1007/s002290050002.  Google Scholar

[27]

P. Zamboni, Unique continuation for non-negative solutions of quasilinear elliptic equations,, \emph{Bull. Austral. Math. Soc.}, 64 (2001), 149.  doi: 10.1017/S0004972700019766.  Google Scholar

[28]

P. Zamboni, Hölder continuity for solutions of linear degenerate elliptic equations under minimal assumptions,, \emph{J. Differential Equations}, 182 (2002), 121.  doi: 10.1006/jdeq.2001.4094.  Google Scholar

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