Nonlinear Hypoellipticity of Infinite Type

Funkcialaj Ekvacioj

Volumn 50, Issue 1, 2007, Pages 33-65

  Morimoto, Yoshinori ^a   Xu, Chao Jiang ^b  

^a KYOTO UNIVERSITY   (Japan)

^b UNIVERSITÉ DE ROUEN   (France)

Author keywords

Controllability; Hormander s operators; Maximum principle; Nonlinear hypoellipticity

Indexed keywords

EID: 56349161363     PISSN: 05328721     EISSN: None     Source Type: Journal    
DOI: 10.1619/fesi.50.33     Document Type: Article

References (26)

1. Bony, J.-M., Principe du maximum, inégalité de Harnack et unicité du probleme de Cauchy pour les opérateurs elliptiques dégénérées, Ann. Inst. Fourier, 19 (1969), 227–304.
2. Bony, J.-M., Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéáires, Ann. Sci. École Norm. Sup., 14 (1981), 209–246.
3. Brezis, H. and Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437–477.
4. Christ, M., Hypoellipticity: geometrization and speculation, Complex analysis and geometry (Paris, 1997), Progr. Math., 188, Birkhauser, Basel, (2000), 91–109.
5. Christ, M., Hypoellipticity in the infinitely degenerate regime, Complex analysis and geometry (Columbus, OH, 1999), Ohio State Univ. Math. Res. Inst. Publ., 9 de Gruyter, Berlin (2001), 59–84.
6. Derridj, M., Un probleme aux limites pour une classe d’opérateurs du second ordre hypoelliptiques, Ann. Inst. Fourier, 21 (1971), 99–148.
7. Derridj, M., Sur un théoréme de traces, Ann. Inst. Fourier, 22 (1972), 73–83.
8. Fefferman, C. L. and Phong, D. H., Subelliptic eigenvalue problems, Proceedings of the conference on harmonic analysis in honor of Antoni Zygmund, Wadsworth Math. Series, (1981), 590–606.
9. Jerison, D., The Dirichlet problem for the Kohn-Laplacian on the Heisenberg group, Parts I and II, J. Funct. Anal., 43 (1981), 97–142.
10. Jerison, D. and Sanchez-Calle, A., Subelliptic second order differential operators, Complex analysis, III (College Park, Md., 1985-86), Lecture Notes in Math., 1277, Springer, Berlin, (1987), 46–77.
11. Kohn, J. J., Hypoellipticity at points of infinite type, Analysis, geometry, number theory: the mathematics of Leon Ehrenpreis (Philadelphia, 1998), Contemp. Math., 251 (2000), 393–398.
12. Kohn, J. J., Superlogarithmic estimates on pseudoconvex domains and CR manifolds, Ann. of Math., 156 (2002), 213–248.
13. Kumano-go, H., Pseudodifferential operators, MIT Press, 1982.
14. Kusuoka, S. and Stroock, D., Applications of the Malliavin calculus, Part II, J. Fac. Sci. Univ. Tokyo Sect. IA, Math., 32 (1985), 1–76.
15. Morimoto, Y., Hypoellipticity for infinitely degenerate elliptic operators, Osaka J. Math., 24 (1987), 13–35.
16. Morimoto, Y., A criterion for hypoellipticity of second order differential operators, Osaka J. Math., 24 (1987), 651–675.
17. Morimoto, Y. and Morioka, T., The positivity of Schrödinger operators and the hypoellipticity of second order degenerate elliptic operators, Bull. Sci. Math., 121 (1997), 507–547.
18. Morimoto, Y. and Morioka, T., Hypoellipticity for elliptic operators with infinite degeneracy, ‘Partial Differential Equations and Their Applications’ (Chen Hua and L. Rodino, eds.), World Sci. Publishing, River Edge, NJ, (1999), 240–259.
19. Morimoto, Y. and Xu, C.-J., Logarithmic Sobolev inequality and semi-linear Dirichlet problems for infinitely degenerate elliptic operators, Astérisque, 284 (2003), 245–264.
20. Xu, C.-J., The Dirichlet problems for a class of semilinear sub-elliptic equations, Nonlinear Anal., 37 (1999), 1039–1049.
21. Xu, C.-J., Nonlinear microlocal analysis, General theory of partial differential equations and microlocal analysis (Trieste, 1995), Pitman Res. Notes Math. Ser., 349, Longman, Harlow, (1996), 155–182.
22. Xu, C.-J., Regularity problem for quasi-linear second order subelliptic equations, Comm. Pure Appl. Math., 45 (1992), 77–96.
23. Xu, C.-J., Semilinear subelliptic equations and Sobolev inequality for vector fields satisfying Hormander’s condition, Chinese J. Contemp. Math., 15 (1994), 183–193.
24. Xu, C.-J., Subelliptic variational problems, Bull. Soc. Math. France, 118 (1990), 147–169.
25. Xu, C.-J., Operateurs sous-elliptiques et régularité des solutions d’équations aux dérivées partielles non lineaires du second ordre en deux variables, Comm. Partial Differential Equations, 11 (1986), 1575–1603.
26. Xu, C.-J. and Zuily, C., Smoothness up to the boundary for solutions of the nonlinear and nonelliptic Dirichlet problem, Trans. Amer. Math. Soc., 308 (1988), 243–257.