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, Série 5, Tome 5 (2006) no. 4, pp. 579-610.

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 $\Omega$ 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.

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[15] D. Christodoulou, On the geometry and dynamics of crystalline continua, Ann. Inst. H. Poincaré Phys. Theor. 69 (1998), 335-358. | Numdam | MR | Zbl

[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

[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 | MR

[18] G. Citti, M. Manfredini and A. Sarti, Neuronal oscillations in the visual cortex: $\Gamma$-convergence to the Riemannian Mumford-Shah functional SIAM J. Math. Anal. 35 (2004), 1394-1419. | MR | Zbl

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl

[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

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

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

[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 | Zbl

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

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

[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 | Zbl

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

[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 | Zbl