Green functions for a class of nonlinear degenerate operators with X-ellipticity
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- by Shenzhou Zheng and Zhaosheng Feng PDF
- Trans. Amer. Math. Soc. 364 (2012), 3627-3655 Request permission
Abstract:
A maximum principle and some a priori estimates of a class of degenerate equations with X-ellipticity in the sense of distributions are established. A local comparison of the generalized Green function with its fundamental solutions is obtained. As an application, by means of the power of the Green function as a kernel function of a local integral, we also derive local Hölder continuity for nonlinear degenerate subelliptic equations.References
- Alain Bensoussan and Jens Frehse, Regularity results for nonlinear elliptic systems and applications, Applied Mathematical Sciences, vol. 151, Springer-Verlag, Berlin, 2002. MR 1917320, DOI 10.1007/978-3-662-12905-0
- A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni, Stratified Lie groups and potential theory for their sub-Laplacians, Springer Monographs in Mathematics, Springer, Berlin, 2007. MR 2363343
- André Bellaïche and Jean-Jacques Risler (eds.), Sub-Riemannian geometry, Progress in Mathematics, vol. 144, Birkhäuser Verlag, Basel, 1996. MR 1421821, DOI 10.1007/978-3-0348-9210-0
- Luca Capogna, Donatella Danielli, and Nicola Garofalo, Subelliptic mollifiers and a basic pointwise estimate of Poincaré type, Math. Z. 226 (1997), no. 1, 147–154. MR 1472145, DOI 10.1007/PL00004330
- Luca Capogna and Qing Han, Pointwise Schauder estimates for second order linear equations in Carnot groups, Harmonic analysis at Mount Holyoke (South Hadley, MA, 2001) Contemp. Math., vol. 320, Amer. Math. Soc., Providence, RI, 2003, pp. 45–69. MR 1979931, DOI 10.1090/conm/320/05598
- Luca Capogna, Donatella Danielli, and Nicola Garofalo, An embedding theorem and the Harnack inequality for nonlinear subelliptic equations, Comm. Partial Differential Equations 18 (1993), no. 9-10, 1765–1794. MR 1239930, DOI 10.1080/03605309308820992
- Luca Capogna, Donatella Danielli, and Nicola Garofalo, Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations, Amer. J. Math. 118 (1996), no. 6, 1153–1196. MR 1420920
- S. Chanillo and R. L. Wheeden, Existence and estimates of Green’s function for degenerate elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15 (1988), no. 2, 309–340 (1989). MR 1007400
- Gui-Qiang Chen, Yanyan Li, Xiping Zhu, and Daomin Cao (eds.), Advances in nonlinear partial differential equations and related areas, World Scientific Publishing Co., Inc., River Edge, NJ, 1998. A volume in honour of Professor Xiaqi Ding [Xia Xi Ding]; Papers from the conference held in Beijing, August 9–10, 1997. MR 1690817, DOI 10.1142/9789812815811
- Giovanna Citti, Nicola Garofalo, and Ermanno Lanconelli, Harnack’s inequality for sum of squares of vector fields plus a potential, Amer. J. Math. 115 (1993), no. 3, 699–734. MR 1221840, DOI 10.2307/2375077
- G. Citti and M. Manfredini, Implicit function theorem in Carnot-Carathéodory spaces, Commun. Contemp. Math. 8 (2006), no. 5, 657–680. MR 2263950, DOI 10.1142/S0219199706002234
- G. Di Fazio, A. Domokos, M. S. Fanciullo, and J. J. Manfredi, Subelliptic Cordes estimates in the Grušin plane, Manuscripta Math. 120 (2006), no. 4, 419–433. MR 2245893, DOI 10.1007/s00229-006-0025-7
- Donatella Danielli and Nicola Garofalo, Geometric properties of solutions to subelliptic equations in nilpotent Lie groups, Reaction diffusion systems (Trieste, 1995) Lecture Notes in Pure and Appl. Math., vol. 194, Dekker, New York, 1998, pp. 89–105. MR 1472512
- G. Di Fazio, D. K. Palagachev, and M. A. Ragusa, On Morrey’s regularity of strong solutions to elliptic boundary value problems, C. R. Acad. Bulgare Sci. 50 (1997), no. 11-12, 17–20. MR 1709471
- Giuseppe Di Fazio, Dian K. Palagachev, and Maria Alessandra Ragusa, Global Morrey regularity of strong solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients, J. Funct. Anal. 166 (1999), no. 2, 179–196. MR 1707751, DOI 10.1006/jfan.1999.3425
- Giuseppe Di Fazio and Pietro Zamboni, Unique continuation of non negative solutions to quasilinear subelliptic equations in Carnot Caratheodory spaces, Comm. Appl. Nonlinear Anal. 10 (2003), no. 2, 97–105. Nonlinear elliptic and parabolic equations and systems (Pisa, 2002). MR 1992310
- Giuseppe Di Fazio and Pietro Zamboni, Hölder continuity for quasilinear subelliptic equations in Carnot Carathéodory spaces, Math. Nachr. 272 (2004), 3–10. MR 2079757, DOI 10.1002/mana.200310185
- Fausto Ferrari and Bruno Franchi, A local doubling formula for the harmonic measure associated with subelliptic operators and applications, Comm. Partial Differential Equations 28 (2003), no. 1-2, 1–60. MR 1974448, DOI 10.1081/PDE-120019372
- Fausto Ferrari, Harnack inequality for two-weight subelliptic $p$-Laplace operators, Math. Nachr. 279 (2006), no. 8, 815–830. MR 2228656, DOI 10.1002/mana.200410396
- Bruno Franchi, Cristian E. Gutiérrez, and Richard L. Wheeden, Weighted Sobolev-Poincaré inequalities for Grushin type operators, Comm. Partial Differential Equations 19 (1994), no. 3-4, 523–604. MR 1265808, DOI 10.1080/03605309408821025
- Bruno Franchi and Ermanno Lanconelli, Une métrique associée à une classe d’opérateurs elliptiques dégénérés, Rend. Sem. Mat. Univ. Politec. Torino Special Issue (1983), 105–114 (1984) (French). Conference on linear partial and pseudodifferential operators (Torino, 1982). MR 745979
- Bruno Franchi and Ermanno Lanconelli, An embedding theorem for Sobolev spaces related to nonsmooth vector fields and Harnack inequality, Comm. Partial Differential Equations 9 (1984), no. 13, 1237–1264. MR 764663, DOI 10.1080/03605308408820362
- G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), no. 2, 161–207. MR 494315, DOI 10.1007/BF02386204
- C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 590–606. MR 730094
- Mariano Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton University Press, Princeton, NJ, 1983. MR 717034
- Cristian E. Gutiérrez and Ermanno Lanconelli, Maximum principle, nonhomogeneous Harnack inequality, and Liouville theorems for $X$-elliptic operators, Comm. Partial Differential Equations 28 (2003), no. 11-12, 1833–1862. MR 2015404, DOI 10.1081/PDE-120025487
- Nicola Garofalo and Duy-Minh Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math. 49 (1996), no. 10, 1081–1144. MR 1404326, DOI 10.1002/(SICI)1097-0312(199610)49:10<1081::AID-CPA3>3.0.CO;2-A
- Nicola Garofalo and Duy-Minh Nhieu, Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathéodory spaces, J. Anal. Math. 74 (1998), 67–97. MR 1631642, DOI 10.1007/BF02819446
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
- Michael Grüter and Kjell-Ove Widman, The Green function for uniformly elliptic equations, Manuscripta Math. 37 (1982), no. 3, 303–342. MR 657523, DOI 10.1007/BF01166225
- Piotr Hajłasz and Pekka Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688, x+101. MR 1683160, DOI 10.1090/memo/0688
- Juha Heinonen, Tero Kilpeläinen, and Olli Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1207810
- Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 222474, DOI 10.1007/BF02392081
- David Jerison and John M. Lee, Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc. 1 (1988), no. 1, 1–13. MR 924699, DOI 10.1090/S0894-0347-1988-0924699-9
- David Jerison and John M. Lee, Intrinsic CR normal coordinates and the CR Yamabe problem, J. Differential Geom. 29 (1989), no. 2, 303–343. MR 982177
- Alessia Elisabetta Kogoj and Ermanno Lanconelli, Liouville theorem for $X$-elliptic operators, Nonlinear Anal. 70 (2009), no. 8, 2974–2985. MR 2509383, DOI 10.1016/j.na.2008.12.029
- Tero Kilpeläinen and Jan Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994), no. 1, 137–161. MR 1264000, DOI 10.1007/BF02392793
- Alexander Koshelev, Regularity problem for quasilinear elliptic and parabolic systems, Lecture Notes in Mathematics, vol. 1614, Springer-Verlag, Berlin, 1995. MR 1442954, DOI 10.1007/BFb0094482
- Yan Yan Li, Interior gradient estimates for solutions of certain fully nonlinear elliptic equations, J. Differential Equations 90 (1991), no. 1, 172–185. MR 1094454, DOI 10.1016/0022-0396(91)90166-7
- Ermanno Lanconelli and Alessia Elisabetta Kogoj, $X$-elliptic operators and $X$-control distances, Ricerche Mat. 49 (2000), no. suppl., 223–243. Contributions in honor of the memory of Ennio De Giorgi (Italian). MR 1826225
- P. Lindqvist and O. Martio, Two theorems of N. Wiener for solutions of quasilinear elliptic equations, Acta Math. 155 (1985), no. 3-4, 153–171. MR 806413, DOI 10.1007/BF02392541
- W. Littman, G. Stampacchia, and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 43–77. MR 161019
- Giovanni Mazzoni, Green function for $X$-elliptic operators, Manuscripta Math. 115 (2004), no. 2, 207–238. MR 2098471, DOI 10.1007/s00229-004-0494-5
- Juan J. Manfredi and Giuseppe Mingione, Regularity results for quasilinear elliptic equations in the Heisenberg group, Math. Ann. 339 (2007), no. 3, 485–544. MR 2336058, DOI 10.1007/s00208-007-0121-3
- Paolo Negrini and Vittorio Scornazzani, Wiener criterion for a class of degenerate elliptic operators, J. Differential Equations 66 (1987), no. 2, 151–164. MR 871992, DOI 10.1016/0022-0396(87)90029-5
- Linda Preiss Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), no. 3-4, 247–320. MR 436223, DOI 10.1007/BF02392419
- Neil S. Trudinger and Xu-Jia Wang, On the weak continuity of elliptic operators and applications to potential theory, Amer. J. Math. 124 (2002), no. 2, 369–410. MR 1890997
- Chao-Jiang Xu and Claude Zuily, Higher interior regularity for quasilinear subelliptic systems, Calc. Var. Partial Differential Equations 5 (1997), no. 4, 323–343. MR 1450714, DOI 10.1007/s005260050069
- S. Zheng and Z. Feng, Regularity for quasi-linear elliptic systems with discontinuous coefficients, Dyn. Partial Differ. Equ. 5 (2008), no. 1, 87–99. MR 2397307, DOI 10.4310/DPDE.2008.v5.n1.a4
- Shenzhou Zheng, Xueliang Zheng, and Zhaosheng Feng, Regularity for a class of degenerate elliptic equations with discontinuous coefficients under natural growth, J. Math. Anal. Appl. 346 (2008), no. 2, 359–373. MR 2431532, DOI 10.1016/j.jmaa.2008.05.059
Additional Information
- Shenzhou Zheng
- Affiliation: Department of Mathematics, Beijing Jiaotong University, Beijing 100044, People’s Republic of China
- MR Author ID: 605970
- Email: shzhzheng@bjtu.edu.cn
- Zhaosheng Feng
- Affiliation: Department of Mathematics, University of Texas-Pan American, Edinburg, Texas 78539
- Email: zsfeng@utpa.edu
- Received by editor(s): July 27, 2010
- Published electronically: March 7, 2012
- Additional Notes: This work was supported by NSF (China) Grant No.11071012 and UTPA Faculty Research Council Grant 145MATH04
The second author was the corresponding author - © Copyright 2012 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 364 (2012), 3627-3655
- MSC (2010): Primary 35J70, 35H20, 35D10
- DOI: https://doi.org/10.1090/S0002-9947-2012-05617-0
- MathSciNet review: 2901227