Harnack inequality for degenerate elliptic equations and sum operators

Giuseppe Di  Fazio; Maria Stella  Fanciullo; Pietro  Zamboni

doi:10.3934/cpaa.2015.14.2363

Article Contents

2015, Volume 14, Issue 6: 2363-2376. Doi: 10.3934/cpaa.2015.14.2363

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Harnack inequality for degenerate elliptic equations and sum operators

1.
Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Italy

Received: January 2015

Revised: July 2015

Published: November 2015

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Abstract

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.

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Mathematics Subject Classification: Primary: 35B45; Secondary: 35B65.

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References

[1]	S. M. Buckley, Inequalities of John-Nirenberg type in doubling spaces, J. Anal. Math., 79 (1999), 215-240.doi: 10.1007/BF02788242.
[2]	F. Chiarenza and M. Frasca, A remark on a paper by C. Fefferman, Proc. Amer. Math. Soc., 108 (1990), 407-409.doi: 10.2307/2048289.
[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, Manuscripta Math., 135 (2011), 361-380.doi: 10.1007/s00229-010-0420-y.
[4]	G. Di Fazio, M. S. Fanciullo and P. Zamboni, Regularity for a class of strongly degenerate quasilinear operators, J. Differential Equations, 255 (2013), 3920-3939.doi: 10.1016/j.jde.2013.07.062.
[5]	G. Di Fazio, M. S. Fanciullo and P. Zamboni, Sum Operators and Fefferman - Phong Inequalities, Geometric Methods in PDE's, Springer INdAM Series, Vol. 13 (2015).
[6]	G. Di Fazio and P. Zamboni, A Fefferman-Poincaré type inequality for Carnot-Carathéodory vector fields, Proc. Amer. Math. Soc., 130 (2002), 2655-2660.doi: 10.1090/S0002-9939-02-06394-3.
[7]	G. Di Fazio and P. Zamboni, Hölder continuity for quasilinear subelliptic equations in Carnot Carathéodory spaces, Math. Nachr., 272 (2004), 3-10.doi: 10.1002/mana.200310185.
[8]	G. Di Fazio and P. Zamboni, Regularity for quasilinear degenerate elliptic equations, Math. Z., 253 (2006), 787-803.doi: 10.1007/s00209-006-0933-y.
[9]	G. Di Fazio and P. Zamboni, Local regularity of solutions to quasilinear subelliptic equations in Carnot Caratheodory spaces, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 9 (2006), 485-504.
[10]	B. Franchi, G. Lu and R. L. Wheeden, Representation formulas and weighted Poincaré type inequalities for Hörmander vector fields, Ann. inst. Fourier tome, 45 (1995), 577-604.
[11]	B. Franchi, G. Lu and R. L. Wheeden, A relationship between Poincaré-type inequalities and representation formulas in spaces of homogeneous type, Int. Math. Res. Not., (1996), 1-14.doi: 10.1155/S1073792896000013.
[12]	B. Franchi, C. Perez and R. L. Wheeden, Self-improving properties of John-Nirenberg and Poincaré inequalities on spaces of homogeneous type, J. of functional Analysis, 153 (1998), 108-146.doi: 10.1006/jfan.1997.3175.
[13]	B. Franchi, C. Perez and R. L. Wheeden, A sum operator with applications to self-improving properties of Poincaré inequalities in metric spaces, The Journal of Fourier Analysis and Applications, 9 (2003), 511-540.doi: 10.1007/s00041-003-0025-x.
[14]	B. Franchi, R. Serapioni and F. Serra Cassano, Approximation and imbedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields, Boll. Un. Mat. Ital. B (7), 11 (1997), 83-117.
[15]	C. E. Gutierrez, Harnack's inequality for degenerate Schrödinger operators, Trans. AMS, 312 (1989), 403-419.doi: 10.2307/2001222.
[16]	P. Hajlasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc., 688, (2000).doi: 10.1090/memo/0688.
[17]	J. Heinonen and P. Koskela, Quasiconformal maps on metric spaces with controlled geometry, Acta Math., 181 (1998), 1-61.doi: 10.1007/BF02392747.
[18]	D. Jerison, The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J., 53 (1986), 503-523.doi: 10.1215/S0012-7094-86-05329-9.
[19]	G. Lu, Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications, Rev. Mat. Iberoamericana, 8(1992), 367-439.doi: 10.4171/RMI/129.
[20]	G. Lu, The sharp Poincaré inequality for free vector fields: an endpoint result, Rev. Mat. Iberoamericana, 10 (1994), 453-466.doi: 10.4171/RMI/158.
[21]	G. Lu, A Fefferman-Phong type inequality for degenerate vector fields and applications, Panamer. Math. J., 6 (1996), 37-57.
[22]	G. Lu and R. L. Wheeden, An optimal representation formula for Carnot- Carathéodory vector fields, Bull. London Math. Soc., 30 (1998), 578-584.doi: 10.1112/S0024609398004895.
[23]	M. A. Ragusa and P. Zamboni, Local regularity of solutions to quasilinear elliptic equations with general structure, Commun. Appl. Anal., 3 (1999), 131-147.
[24]	J. Serrin, Local behavior of solutions of quasilinear equations, Acta Math., 111 (1964), 247-302.
[25]	P. Zamboni, Some function spaces and elliptic partial differential equations, Le Matematiche (Catania), 42 (1987), 171-178.
[26]	P. Zamboni, The Harnack inequality for quasilinear elliptic equations under minimal assumptions, Manuscripta Math., 102 (2000), 311-323.doi: 10.1007/s002290050002.
[27]	P. Zamboni, Unique continuation for non-negative solutions of quasilinear elliptic equations, Bull. Austral. Math. Soc., 64 (2001), 149-156.doi: 10.1017/S0004972700019766.
[28]	P. Zamboni, Hölder continuity for solutions of linear degenerate elliptic equations under minimal assumptions, J. Differential Equations, 182 (2002), 121-140.doi: 10.1006/jdeq.2001.4094.