Let be a sub-laplacian on a stratified Lie group . In this paper we study the Dirichlet problem for with -boundary data, on domains which are contractible with respect to the natural dilations of . One of the main difficulties we face is the presence of non-regular boundary points for the usual Dirichlet problem for . A potential theory approach is followed. The main results are applied to study a suitable notion of Hardy spaces.
@article{ASNSP_2006_5_5_4_579_0, author = {Bonfiglioli, Andrea and Lanconelli, Ermanno}, title = {Dirichlet problem with $L^p$-boundary data in contractible domains of {Carnot} groups}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {579--610}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 5}, number = {4}, year = {2006}, zbl = {1170.35429}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2006_5_5_4_579_0/} }
TY - JOUR AU - Bonfiglioli, Andrea AU - Lanconelli, Ermanno TI - Dirichlet problem with $L^p$-boundary data in contractible domains of Carnot groups JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2006 DA - 2006/// SP - 579 EP - 610 VL - Ser. 5, 5 IS - 4 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2006_5_5_4_579_0/ UR - https://zbmath.org/?q=an%3A1170.35429 LA - en ID - ASNSP_2006_5_5_4_579_0 ER -
Bonfiglioli, Andrea; Lanconelli, Ermanno. Dirichlet problem with $L^p$-boundary data in contractible domains of Carnot groups. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 4, pp. 579-610. http://www.numdam.org/item/ASNSP_2006_5_5_4_579_0/
[1] “Classical Potential Theory", Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2001. | MR 1801253 | Zbl 0972.31001
and ,[2] “Harmonic Function Theory", Graduate Texts in Mathematics, Vol. 137, Springer-Verlag, New York, 1992. | MR 1184139 | Zbl 0765.31001
, and ,[3] The Poisson kernel for certain degenerate elliptic operators, J. Funct. Anal. 56 (1984), 171-209. | MR 738578 | Zbl 0556.35036
, and ,[4] A Poisson-Jensen type representation formula for subharmonic functions on stratified Lie groups, Potential Anal. 22 (2005), 151-169. | MR 2137059 | Zbl 1069.31001
and ,[5] The theory of energy for sub-Laplacians with an application to quasi-continuity Manuscripta Math. 118 (2005), 283-309. | MR 2183041 | Zbl 1131.35006
and ,[6] Liouville-type theorems for real sub-Laplacians, Manuscripta Math. 105 (2001), 111-124. | MR 1885817 | Zbl 1016.35014
and ,[7] Subharmonic functions on Carnot groups, Math. Ann. 325 (2003), 97-122. | MR 1957266 | Zbl 1017.31003
and ,[8] Uniform Gaussian estimates of the fundamental solutions for heat operators on Carnot groups, Adv. Differential Equations 7 (2002), 1153-1192. | MR 1919700 | Zbl 1036.35061
, and ,[9] A note on lifting of Carnot groups, Rev. Mat. Iberoamericana 21 (2005), to appear. | MR 2232674 | Zbl 1100.35029
and ,[10] Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier 19 (1969), 277-304. | Numdam | MR 262881 | Zbl 0176.09703
,[11] Boundary behavior of nonnegative solutions of subelliptic equations in NTA domains for Carnot-Carathéodory metrics, J. Fourier Anal. Appl. 4 (1998), 403-432. | MR 1658616 | Zbl 0926.35043
and ,[12] A version of a theorem of Dahlberg for the subelliptic Dirichlet problem, Math. Res. Lett. 5 (1998), 541-549. | MR 1653336 | Zbl 0934.22017
, and ,[13] Properties of harmonic measures in the Dirichlet problem for nilpotent Lie groups of Heisenberg type, Amer. J. Math. 124 (2002), 273-306. | MR 1890994 | Zbl 0998.22001
, and ,[14] “The Dirichlet Problem with -Boundary Data for Elliptic Linear Equations", Lecture Notes in Mathematics, Vol. 1482, Springer-Verlag, Berlin, 1991. | MR 1165533 | Zbl 0734.35024
,[15] On the geometry and dynamics of crystalline continua, Ann. Inst. H. Poincaré Phys. Theor. 69 (1998), 335-358. | Numdam | MR 1648987 | Zbl 0916.73013
,[16] Nuovo tipo di condizione al contorno e nuovo metodo di trattazione per il problema generalizzato di Dirichlet Rend. Circ. Mat. Palermo 61 (1937), 177-221. | JFM 64.1163.01
,[17] Equazione di Poisson e problema generalizzato di Dirichlet, Atti Acc. Italia, Rend. Cl. Sci. Fis. Mat. Nat. 1 (1940), 322-329. | JFM 66.0445.01 | MR 1888
,[18] Neuronal oscillations in the visual cortex: -convergence to the Riemannian Mumford-Shah functional SIAM J. Math. Anal. 35 (2004), 1394-1419. | MR 2083784 | Zbl 1058.49010
, and ,[19] “Potential Theory on Harmonic Spaces", Die Grundlehren der mathematischen Wissenschaften, Band 158, Springer-Verlag, New York-Heidelberg, 1972. | MR 419799 | Zbl 0248.31011
and ,[20] A Poisson kernel on Heisenberg type nilpotent groups, Colloq. Math. 53 (1987), 239-247. | MR 924068 | Zbl 0661.53035
,[21] Subelliptic Estimates and Function Spaces on Nilpotent Groups, Ark. Mat. 13 (1975), 161-207. | MR 494315 | Zbl 0312.35026
,[22] “Hardy spaces on homogeneous groups", Mathematical Notes, Vol. 28, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. | MR 657581 | Zbl 0508.42025
and ,[23] Capacités, mouvement brownien et problème de l'épine de Lebesgue sur les groupes de Lie nilpotents, In: Probability measures on groups, Oberwolfach, 1981, 96-120, Lecture Notes in Math., Vol. 928, Springer, Berlin-New York, 1982. | MR 669065 | Zbl 0483.60072
,[24] Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathéodory spaces, J. Anal. Math. 74 (1998), 67-97. | MR 1631642 | Zbl 0906.46026
- ,[25] Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents Acta Math. 139 (1977) 95-153. | MR 461589 | Zbl 0366.22010
,[26] The Dirichlet problem for sub-Laplacians on nilpotent Lie groups - geometric criteria for regularity Math. Ann. 276 (1987), 537-547. | MR 879533 | Zbl 0601.31007
and ,[27] “Introduction to Potential Theory", Pure and Applied Mathematics, Vol. 22, Wiley-Interscience, John Wiley & Sons, New York-London-Sydney, 1969. | MR 261018 | Zbl 0188.17203
,[28] Les fonctions surharmoniques dans l'axiomatique de M. Brelot associées á un opérateur elliptique dégénéré, Ann. Inst. Fourier (Grenoble) 22 (1972), 131-145. | Numdam | MR 377092 | Zbl 0224.31014
and ,[29] Hypoelliptic second-order differential equations, Acta Math. 121 (1968), 147-171. | MR 222474 | Zbl 0156.10701
,[30] Wiener criterion in potential theory with applications to nilpotent Lie groups Math. Z. 190 (1985), 527-542. | MR 808920 | Zbl 0585.31004
,[31] Examples of irregular domains for some hypoelliptic differential operators Expo. Math. 4 (1986), 189-192. | MR 879912 | Zbl 0597.58036
,[32] Boundary regularity in the Dirichlet problem for on CR manifolds, Comm. Pure Appl. Math. 36 (1983) 143-181. | MR 690655 | Zbl 0544.35069
,[33] “Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems", Conference Board of the Mathematical Sciences, Regional Conference Series in Mathematics, Vol. 83, American Mathematical Society, Providence, RI, 1994. | MR 1282720 | Zbl 0812.35001
,[34] Nonlinear equations on Carnot groups and curvature problems for CR manifolds, Renato Caccioppoli and modern analysis. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 14 (2003), 227-238. | MR 2064269
,[35] On Carnot-Carathéodory metrics, J. Differential Geom. 21 (1985), 35-45. | MR 806700 | Zbl 0554.53023
,[36] Pseudoconvex fully nonlinear partial differential operators: strong comparison theorems, J. Differential Equations 202 (2004), 306-331. | MR 2068443 | Zbl 1161.35414
and ,[37] “A Tour of subRiemannian Geometries, their Geodesics and Applications", Mathematical Surveys and Monographs, Vol. 91, American Mathematical Society, Providence, RI, 2002. | MR 1867362 | Zbl 1044.53022
,[38] Trace theorems for vector fields, Math. Z. 239 (2002), 747-776. | MR 1902060 | Zbl 1030.46041
and ,[39] Wiener criterion for a class of degenerate elliptic operators, J. Differential Equations 66 (1987), 151-164. | MR 871992 | Zbl 0633.35018
and ,[40] MR 1449186 | Zbl 0897.01011
, (1841-1915) and his investigations on the Dirichlet problem, In: “Studies in the history of modern mathematics”, II, Rend. Circ. Mat. Palermo (2) Suppl. No. 44 (1996), 43-55. |[41] Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), 247-320. | MR 436223 | Zbl 0346.35030
and ,[42] “Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals”, Princeton Mathematical Series, Vol. 43, Princeton, NJ: Princeton University Press, 1993. | MR 1232192 | Zbl 0821.42001
,