Harnack inequality for degenerate elliptic equations and sum operators

Previous Article
The Liouville theorems for elliptic equations with nonstandard growth
CPAA Home
This Issue
Next Article
Nonlinear elliptic systems involving the fractional Laplacian in the unit ball and on a half space

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

Abstract
Related Papers

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 and 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, 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.

show all references

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.

[1]	Fabio Paronetto. A Harnack type inequality and a maximum principle for an elliptic-parabolic and forward-backward parabolic De Giorgi class. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 853-866. doi: 10.3934/dcdss.2017043
[2]	Daniela De Silva, Ovidiu Savin. A note on higher regularity boundary Harnack inequality. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 6155-6163. doi: 10.3934/dcds.2015.35.6155
[3]	Pablo Raúl Stinga, Chao Zhang. Harnack's inequality for fractional nonlocal equations. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 3153-3170. doi: 10.3934/dcds.2013.33.3153
[4]	Carlo Morosi, Livio Pizzocchero. On the constants in a Kato inequality for the Euler and Navier-Stokes equations. Communications on Pure and Applied Analysis, 2012, 11 (2) : 557-586. doi: 10.3934/cpaa.2012.11.557
[5]	Jinggang Tan, Jingang Xiong. A Harnack inequality for fractional Laplace equations with lower order terms. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 975-983. doi: 10.3934/dcds.2011.31.975
[6]	James Scott, Tadele Mengesha. A fractional Korn-type inequality. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3315-3343. doi: 10.3934/dcds.2019137
[7]	Wen Wang, Dapeng Xie, Hui Zhou. Local Aronson-Bénilan gradient estimates and Harnack inequality for the porous medium equation along Ricci flow. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1957-1974. doi: 10.3934/cpaa.2018093
[8]	Teo Kukuljan. Higher order parabolic boundary Harnack inequality in C¹ and C^{k, α} domains. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2667-2698. doi: 10.3934/dcds.2021207
[9]	Shu-Yu Hsu. Some results for the Perelman LYH-type inequality. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3535-3554. doi: 10.3934/dcds.2014.34.3535
[10]	Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad, Saed F. Mallak, Hussam Alrabaiah. Lyapunov type inequality in the frame of generalized Caputo derivatives. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2335-2355. doi: 10.3934/dcdss.2020212
[11]	Piotr Pokora. The orbifold Langer-Miyaoka-Yau Inequality and Hirzebruch-type inequalities. Electronic Research Announcements, 2017, 24: 21-27. doi: 10.3934/era.2017.24.003
[12]	José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla. Hardy-Sobolev type inequality and supercritical extremal problem. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3345-3364. doi: 10.3934/dcds.2019138
[13]	Jun Zhou, Jun Shen, Weinian Zhang. A powered Gronwall-type inequality and applications to stochastic differential equations. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7207-7234. doi: 10.3934/dcds.2016114
[14]	Leszek Gasiński. Optimal control problem of Bolza-type for evolution hemivariational inequality. Conference Publications, 2003, 2003 (Special) : 320-326. doi: 10.3934/proc.2003.2003.320
[15]	Bum Ja Jin, Kyungkeun Kang. Caccioppoli type inequality for non-Newtonian Stokes system and a local energy inequality of non-Newtonian Navier-Stokes equations without pressure. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4815-4834. doi: 10.3934/dcds.2017207
[16]	Jorge A. Becerril, Javier F. Rosenblueth. Necessity for isoperimetric inequality constraints. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1129-1158. doi: 10.3934/dcds.2017047
[17]	Gisella Croce, Bernard Dacorogna. On a generalized Wirtinger inequality. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1329-1341. doi: 10.3934/dcds.2003.9.1329
[18]	Tomasz Cieślak. Trudinger-Moser type inequality for radially symmetric functions in a ring and applications to Keller-Segel in a ring. Discrete and Continuous Dynamical Systems - B, 2013, 18 (10) : 2505-2512. doi: 10.3934/dcdsb.2013.18.2505
[19]	Vincenzo Ferone, Carlo Nitsch, Cristina Trombetti. On a conjectured reverse Faber-Krahn inequality for a Steklov--type Laplacian eigenvalue. Communications on Pure and Applied Analysis, 2015, 14 (1) : 63-82. doi: 10.3934/cpaa.2015.14.63
[20]	Anouar Bahrouni. Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity. Communications on Pure and Applied Analysis, 2017, 16 (1) : 243-252. doi: 10.3934/cpaa.2017011

2020 Impact Factor: 1.916

Tools

Metrics

Other articles
by authors

on AIMS
Giuseppe Di Fazio Maria Stella Fanciullo Pietro Zamboni
on Google Scholar
Giuseppe Di Fazio Maria Stella Fanciullo Pietro Zamboni

[Back to Top]