Dirichlet problem with $L^p$-boundary data in contractible domains of Carnot groups

Bonfiglioli, Andrea; Lanconelli, Ermanno

Dirichlet problem with

L^{p}

-boundary data in contractible domains of Carnot groups

Bonfiglioli, Andrea; Lanconelli, Ermanno

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 5 (2006) no. 4, p. 579-610

Abstract

Let $ℒ$ be a sub-laplacian on a stratified Lie group $G$ . In this paper we study the Dirichlet problem for $ℒ$ with $L^{p}$ -boundary data, on domains $Ω$ which are contractible with respect to the natural dilations of $G$ . One of the main difficulties we face is the presence of non-regular boundary points for the usual Dirichlet problem for $ℒ$ . A potential theory approach is followed. The main results are applied to study a suitable notion of Hardy spaces.

Zbl 1170.35429

Classification: 35J70, 35H20, 31B05, 31C15, 43A80

@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},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 5},
     number = {4},
     year = {2006},
     pages = {579-610},
     zbl = {1170.35429},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2006_5_5_4_579_0}
}

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, Serie 5, Volume 5 (2006) no. 4, pp. 579-610. http://www.numdam.org/item/ASNSP_2006_5_5_4_579_0/

References

[1] D. H. Armitage and S. J. Gardiner, “Classical Potential Theory", Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2001. | MR 1801253 | Zbl 0972.31001

[2] S. Axler, P. Bourdon and W. Ramey, “Harmonic Function Theory", Graduate Texts in Mathematics, Vol. 137, Springer-Verlag, New York, 1992. | MR 1184139 | Zbl 0765.31001

[3] G. Ben Arous, S. Kusuoka and D. W. Stroock, The Poisson kernel for certain degenerate elliptic operators, J. Funct. Anal. 56 (1984), 171-209. | MR 738578 | Zbl 0556.35036

[4] A. Bonfiglioli and C. Cinti, A Poisson-Jensen type representation formula for subharmonic functions on stratified Lie groups, Potential Anal. 22 (2005), 151-169. | MR 2137059 | Zbl 1069.31001

[5] A. Bonfiglioli and C. Cinti, The theory of energy for sub-Laplacians with an application to quasi-continuity Manuscripta Math. 118 (2005), 283-309. | MR 2183041 | Zbl 1131.35006

[6] A. Bonfiglioli and E. Lanconelli, Liouville-type theorems for real sub-Laplacians, Manuscripta Math. 105 (2001), 111-124. | MR 1885817 | Zbl 1016.35014

[7] A. Bonfiglioli and E. Lanconelli, Subharmonic functions on Carnot groups, Math. Ann. 325 (2003), 97-122. | MR 1957266 | Zbl 1017.31003

[8] A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, 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

[9] A. Bonfiglioli and F. Uguzzoni, A note on lifting of Carnot groups, Rev. Mat. Iberoamericana 21 (2005), to appear. | MR 2232674 | Zbl 1100.35029

[10] J.-M. Bony, 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] L. Capogna and N. Garofalo, 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

[12] L. Capogna, N. Garofalo and D. M. Nhieu, A version of a theorem of Dahlberg for the subelliptic Dirichlet problem, Math. Res. Lett. 5 (1998), 541-549. | MR 1653336 | Zbl 0934.22017

[13] L. Capogna, N. Garofalo and D. M. Nhieu, 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

[14] J. Chabrowski, “The Dirichlet Problem with $L^{2}$ -Boundary Data for Elliptic Linear Equations", Lecture Notes in Mathematics, Vol. 1482, Springer-Verlag, Berlin, 1991. | MR 1165533 | Zbl 0734.35024

[15] D. Christodoulou, 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] G. Cimmino, 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] G. Cimmino, 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] G. Citti, M. Manfredini and A. Sarti, Neuronal oscillations in the visual cortex: $Γ$ -convergence to the Riemannian Mumford-Shah functional SIAM J. Math. Anal. 35 (2004), 1394-1419. | MR 2083784 | Zbl 1058.49010

[19] C. Constantinescu and A. Cornea, “Potential Theory on Harmonic Spaces", Die Grundlehren der mathematischen Wissenschaften, Band 158, Springer-Verlag, New York-Heidelberg, 1972. | MR 419799 | Zbl 0248.31011

[20] E. Damek, A Poisson kernel on Heisenberg type nilpotent groups, Colloq. Math. 53 (1987), 239-247. | MR 924068 | Zbl 0661.53035

[21] G. B. Folland, Subelliptic Estimates and Function Spaces on Nilpotent Groups, Ark. Mat. 13 (1975), 161-207. | MR 494315 | Zbl 0312.35026

[22] G. B. Folland and E. M. Stein, “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

[23] L. Gallardo, 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] N. Garofalo - D. M. 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 | Zbl 0906.46026

[25] L. Gaveau, 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] W. Hansen and H. Hueber, 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

[27] L. L. Helms, “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] R. M. Hervé and M. Hervé, 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

[29] L. Hörmander, Hypoelliptic second-order differential equations, Acta Math. 121 (1968), 147-171. | MR 222474 | Zbl 0156.10701

[30] H. Hueber, Wiener criterion in potential theory with applications to nilpotent Lie groups Math. Z. 190 (1985), 527-542. | MR 808920 | Zbl 0585.31004

[31] H. Hueber, Examples of irregular domains for some hypoelliptic differential operators Expo. Math. 4 (1986), 189-192. | MR 879912 | Zbl 0597.58036

[32] D. Jerison, Boundary regularity in the Dirichlet problem for $□_{b}$ on CR manifolds, Comm. Pure Appl. Math. 36 (1983) 143-181. | MR 690655 | Zbl 0544.35069

[33] C. E. Kenig, “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] E. Lanconelli, 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 | Zbl pre02217030

[35] J. Mitchell, On Carnot-Carathéodory metrics, J. Differential Geom. 21 (1985), 35-45. | MR 806700 | Zbl 0554.53023

[36] A. Montanari and E. Lanconelli, Pseudoconvex fully nonlinear partial differential operators: strong comparison theorems, J. Differential Equations 202 (2004), 306-331. | MR 2068443 | Zbl 1161.35414

[37] R. Montgomery, “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] R. Monti and D. Morbidelli, Trace theorems for vector fields, Math. Z. 239 (2002), 747-776. | MR 1902060 | Zbl 1030.46041

[39] P. Negrini and V. Scornazzani, Wiener criterion for a class of degenerate elliptic operators, J. Differential Equations 66 (1987), 151-164. | MR 871992 | Zbl 0633.35018

[40] M. Von Rentlen, Friedrich Prym (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. | MR 1449186 | Zbl 0897.01011

[41] L. P. Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), 247-320. | MR 436223 | Zbl 0346.35030

[42] E. M. Stein, “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