Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Green functions for a class of nonlinear degenerate operators with X-ellipticity


Authors: Shenzhou Zheng and Zhaosheng Feng
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
Published electronically: March 7, 2012
MathSciNet review: 2901227
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. Bensoussan A. and Frehse J., Regularity results for nonlinear elliptic systems and applications, Applied Math. Sci., Vol. 151, Springer-Verlag, 2002. MR 1917320 (2004a:35001)
  • 2. Bonfigliol A., Lanconelli E. and Uguzzoni F., Stratified Lie groups and potential theory for their sub-Laplacians, Springer Monographs in Mathematics, Springer, Berlin, 2007. MR 2363343 (2009m:22012)
  • 3. Bellaiche A. and Risler J. J., Sub-Riemannian geometry, Birkhäuser Verlag, Boston, 1996. MR 1421821 (97f:53002)
  • 4. Capogna L., Subelliptic mollifiers and a basic pointwise estimate of Poincaré type, Math. Z. 226 (1) (1997) 147-154. MR 1472145 (98i:35025)
  • 5. Capogna L., Pointwise Schauder estimates for second order linear equations in Carnot groups, Harmonic Analysis at Mount Holyoke, 320 (2003) 45-69. MR 1979931 (2004m:35029)
  • 6. Capogna L., Danielli D. and Garofalo N., An embedding theorem and the Harnack inequality for nonlinear subelliptic equations, Comm. Part. Diff. Equs. 18 (1993) 1765-1794. MR 1239930 (94j:35038)
  • 7. Capogna L., Danielli D. and Garofalo N., Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations, Amer. J. Math. 118 (1997) 1153-1196. MR 1420920 (97k:35033)
  • 8. Chanillo S. and Wheeden R. L., Existence and estimates of Green's function for degenerate elliptic equations, Ann. Scuola Norm. Sup. Pisa 15 (1998) 309-340. MR 1007400 (91b:35045)
  • 9. Chen G., Li Y., Zhu X. and Cao D.M., Advances in Nonlinear Partial Differential Equations and Related Areas (Beijing, 1997), World Sci. Publ., River Edge, NJ, 1998. MR 1690817 (99m:35002)
  • 10. Citti G., Garofalo N. and Lanconelli E., Harnack's inequality for sum of squares of vector fields plus a potential, Amer. J. Math. 115 (1993) 699-734. MR 1221840 (94m:35069)
  • 11. Citti G. and Manfredini M., Implicit function theorem in Carnot-Carathéodory spaces, Comm. Contemp. Math. 8 (2006) 657-680. MR 2263950 (2007k:53031)
  • 12. Di Fazio G., Domokos A., Fanciullo M. S. and Manfredi J. J., Subelliptic Cordes estimates in the Grusin plane, Manuscripta Math. 120 (4) (2006) 419-433. MR 2245893 (2008e:35027)
  • 13. Danielli D. and Garofalo N., Geometric properties of solutions to subelliptic equations in nilpotent Lie groups, Reaction Diffusion Systems, 194 (1998) 89-105. MR 1472512 (99a:22015)
  • 14. Di Fazio G., Palagachev D. K. and Ragusa M. A., On Morrey's regularity of strong solutions to elliptic boundary value problems, C. R. Acad. Bulgare Sci. 50 (11-12) (1997) 17-20. MR 1709471
  • 15. Di Fazio G., Palagachev D. K. and Ragusa M. A., Global Morrey regularity of strong solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients, J. Funct. Anal. 166 (2) (1999) 179-196. MR 1707751 (2000d:35037)
  • 16. Di Fazio G. and Zamboni P., Unique continuation of non negative solutions to quasilinear subelliptic equations in Carnot Carathéodory spaces, Comm. Appl. Nonlinear Anal. 10 (2) (2003) 97-105. MR 1992310 (2004g:35048)
  • 17. Di Fazio G. and Zamboni P., Hölder continuity for quasilinear subelliptic equations in Carnot-Carathéodory spaces, Math. Nachr. 272 (2004) 3-10. MR 2079757 (2006a:35039)
  • 18. Ferrari F., A local doubling formula for the harmonic measure associated with subelliptic operators and applications, Comm. Part. Diff. Equs. 28 (1-2) (2003) 1-60. MR 1974448 (2004g:35050)
  • 19. Ferrari F., Harnack inequality for two-weight subelliptic $ p$-Laplace operators, Math. Nach. 279 (8) (2006) 815-830. MR 2228656 (2007b:35061)
  • 20. Franchi B., Gutiérrez C. E. and Wheeden R. L., Weight Sobolev-Poincaré inequalities for Grushin type operators, Comm. Part. Diff. Equs. 19 (3-4) (1994) 523-604. MR 1265808 (96h:26019)
  • 21. Franchi B. and Lanconelli E., Une métrique associée à une classe d'opérateurs elliptiques dégénérés (in French), Conference on linear partial and pseudo-differential operators (Torino, 1982), Rend. Sem. Mat. Univ. Politec. Torino 1983, Special Issue, (1984) 105-114. MR 745979 (86d:35057)
  • 22. Franchi B. and Lanconelli E., An embedding theorem for Sobolev spaces related to non-smooth vector fields and Harnack inequality, Comm. Part. Diff. Equs. 9 (13) (1984) 1237-1264. MR 764663 (86b:46048)
  • 23. Folland G. B., Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (2) (1975) 161-207. MR 0494315 (58:13215)
  • 24. Fefferman C. and Phong D. H., Subelliptic eigenvalue problems, Proceedings of Conference on Harmonic Analysis, in honor of Antoni Zygmund, Vols. I-II, 590-606, Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983. MR 730094 (86c:35112)
  • 25. Giaquinta M., Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, Ann. Math. Stud. Vol. 105, 1983. MR 717034 (86b:49003)
  • 26. Gutiérrez C. E. and Lanconelli E. W., Maximum principle, non-homogeneous Harnack inequality, and Liouville theorems for X-elliptic operators, Comm. Part. Diff. Equs. 28 (11-12) (2003) 1833-1862. MR 2015404 (2004j:35030)
  • 27. Garofalo N. and Nhieu D. M., Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math. 49 (1996) 1081-1144. MR 1404326 (97i:58032)
  • 28. Garofalo N. and Nhieu D. M., Lipschitz continuity, global smooth approximation and extension theorems for Sobolev functions in Carnot-Carathéodory spaces, J. Analyse Math. 74 (1998) 67-97. MR 1631642 (2000i:46025)
  • 29. Gilbarg D. and Trudinger N. S., Elliptic partial differential equations of second order, 2nd ed. Spinger-Verlag, New York, 1983. MR 737190 (86c:35035)
  • 30. Grüter M. and Widman K., The Green function for uniformly elliptic equations, Manuscripta Math. 37 (1982) 303-342. MR 657523 (83h:35033)
  • 31. Hayslaz P. and Koskela P., Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688, x+101 pp. MR 1683160 (2000j:46063)
  • 32. Heinonen J., Kilpeläinen T. and Martio O., Nonlinear potential theory of degenerate elliptic equations, Clarendon Press, 1993. MR 1207810 (94e:31003)
  • 33. Hörmander L., Hypoelliptic second order differential equations, Acta Math. 119 (1967) 147-171. MR 0222474 (36:5526)
  • 34. Jerison D. and Lee J. M., Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc. 1 (1) (1988) 1-13. MR 924699 (89b:53063)
  • 35. Jerison D. and Lee J. M.: Intrinsic CR coordinate and CR Yamabe problem, J. Diff. Geom. 29 (2) (1989) 303-343. MR 982177 (90h:58083)
  • 36. Kogoj A. E. and Lanconelli E., Liouville theorem for $ X$-elliptic operators, Nonlinear Anal. 70 (8) (2009) 2974-2985. MR 2509383 (2010e:35055)
  • 37. Kilpeläinen T. and Malý J., The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994) 137-161. MR 1264000 (95a:35050)
  • 38. Koshelev A. K., Regularity problem for quasilinear elliptic and parabolic systems, Springer-Verlag, Berlin, 1995. MR 1442954 (98j:35004)
  • 39. Li Y., Interior gradient estimates for solutions of certain fully nonlinear elliptic equations, J. Diff. Equs. 90 (1991) 172-185. MR 1094454 (92a:35071)
  • 40. Lanconelli E. and Kogoj A. E., $ X$-elliptic operators and $ X$-control distances, Contributions in honor of the memory of Ennio De Giorgi, Ricerche Mat. (suppl.) 49 (2000) 223-243. MR 1826225 (2002c:35121)
  • 41. Lindqvist P. and Martio O., Two theorems of N. Wiener for quasilinear elliptic equations, Acta Math. 155 (1985) 153-171. MR 806413 (87g:35074)
  • 42. Littman W., Stampacchia G. and Weinberger H., Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Normale Superiore di Pisa (III), 17 (1963) 45-79. MR 0161019 (28:4228)
  • 43. Mazzoni G., Green function for X-elliptic operators, Manuscripta Math. 115 (2004) 207-238. MR 2098471 (2007b:35063)
  • 44. Manfredi J. J. and Mingione G., Regularity results for quasilinear elliptic equations in the Heisenberg group, Math. Ann. 339 (2007) 485-544. MR 2336058 (2008k:35063)
  • 45. Negrini P. and Scornazzani V., Wiener criterion for a class of degenerate elliptic operators, J. Diff. Equs. 66 (2) (1987) 151-164. MR 871992 (88b:35081)
  • 46. Rothschild L. P. and Stein E. M., Hypoelliptic differential operators and nilpotent groups, Acta Math. 138 (1976) 247-320. MR 0436223 (55:9171)
  • 47. Trudinger N. S. and Wang X. J., On the weak continuity of elliptic operators and applications to potential theory, Amer. J. Math. 124 (2002) 369-410. MR 1890997 (2003c:35025)
  • 48. Xu C. J. and Zuily C., Higher interior regularity for quasilinear subelliptic systems, Calc. Var. 5 (1997) 323-343. MR 1450714 (98e:35039)
  • 49. Zheng S.Z. and Feng Z., Regularity for quasi-linear elliptic systems with discontinuous coefficients, Dyn. Partial Differ. Equs. 5 (2008) 87-99. MR 2397307 (2009b:35084)
  • 50. Zheng S.Z., Zheng X. L. and Feng Z., Regularity for a class of degenerate elliptic equations with discontinuous coefficients under natural growth, J. Math. Anal. Appl. 346 (2008) 359-373. MR 2431532 (2010a:35087)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 35J70, 35H20, 35D10

Retrieve articles in all journals with MSC (2010): 35J70, 35H20, 35D10


Additional Information

Shenzhou Zheng
Affiliation: Department of Mathematics, Beijing Jiaotong University, Beijing 100044, People’s Republic of China
Email: shzhzheng@bjtu.edu.cn

Zhaosheng Feng
Affiliation: Department of Mathematics, University of Texas-Pan American, Edinburg, Texas 78539
Email: zsfeng@utpa.edu

DOI: https://doi.org/10.1090/S0002-9947-2012-05617-0
Keywords: Green function, X-ellipticity, control distance, maximum principle, Morrey’s space, hole-filling technique
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
Article copyright: © Copyright 2012 American Mathematical Society

American Mathematical Society