Abstract
We establish the existence and uniqueness of variational solution to the nonlinear Neumann boundary problem for the pth-sub-Laplacian associated to a system of Hörmander vector fields.
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Baouendi, M.S.: Sur une classe d’opérateurs elliptiques dégénérés. Bull. Soc. Math. Fr. 95, 45–87 (1967)
Bony, J.M.: Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptique dégénérés. Ann. Inst. Fourier Grenoble 119(1), 277–304 (1969)
Capogna, L., Garofalo, N.: Boundary behavior of non-negative solutions of subelliptic equations in NTA domains for Carnot–Carathéodory metrics. J. Fourier Anal. Appl. 4–5(4), 403–432 (1998)
Capogna, L., Garofalo, N., Nhieu, D.M.: Properties of harmonic measures in the Dirichlet problem for nilpotent Lie groups of Heisenberg type. Am. J. Math. 124(2), 273–306 (2002)
Capogna, L., Garofalo, N., Nhieu, D.M.: Mutual absolute continuity of harmonic and surface measures for Hörmander type operators, perspectives in partial differential equations, harmonic analysis and applications. Proc. Sympos. Pure Math. 79, 49–100 (2008)
Capogna, L., Tang, P.: Uniform domains and quasiconformal mappings on the Heisenberg group. Manuscripta Math. 86(3), 267–281 (1995)
Citti, G.: Wiener estimates at boundary points for Hörmander’s operators. Boll. Un. Mat. Ital. 2–B, 667–681 (1988)
Chow, W.L.: Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung. Math. Ann. 117, 98–105 (1939)
Damek, E.: Harmonic functions on semidirect extensions of type H nilpotent groups. Trans. Am. Math. Soc. 290, 375–384 (1985)
Damek, E.: A Poisson kernel on Heisenberg type nilpotent groups. Colloq. Math. 53, 239–247 (1987)
Danielli, D.: Regularity at the boundary for solutions of nonlinear subelliptic equations. Indiana Univ. Math. J. 44, 269–286 (1995)
Danielli, D., Garofalo, N., Munive, I., Nhieu, D.M.: The Neumann problem on the Heisenberg group, regularity and boundary behaviour of solutions, in preparation
Danielli, D., Garofalo, N., Nhieu, D.M.: Non-doubling Ahlfors measures, perimeter measures, and the characterization of the trace spaces of Sobolev functions in Carnot–Carathéodory spaces. Mem. Am. Math. Soc.182, (2006), no. 857
Derridj, M.: Un problème aux limites pour une classe d’opérateurs hypeolliptiques du second ordre. Ann. Inst. Fourier (Grenoble) 21(4), 99–148 (1971)
Garofalo, N., Nguyen, C.P.: Boundary behavior of p-harmonic functions in the Heisenberg group. Math. Ann. 351, 587–632 (2011)
Garofalo, N., Vassilev, D.: Regularity near the characteristic set in the non-linear Dirichlet problem and conformal geometry of sub-Laplacians on Carnot groups. Math. Ann. 318, 453–516 (2000)
Garofalo, N., Nhieu, D.M.: Isoperimetric and Sobolev inequalities for Carnot–Carathéodory spaces and the existence of minimal surfaces. Commun. Pure Appl. Math. 49, 1081–1144 (1996)
Gaveau, B.: Principe de moindre action, propagation de la chaleur et estim’ees sous elliptiques sur certain groupes nilpotents. Acta Math. 139, 95–153 (1977)
Giusti, E.: Direct Methods in the Calculus of Variations. World Scientific Publishing Co., Inc., River Edge (2003)
Hansen, W., Hueber, H.: The Dirichlet problem for sub-Laplacians on nilpotent Lie groups—geometric criteria for regularity. Math. Ann. 276, 537–547 (1987)
Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)
Jerison, D.S.: The Dirichlet problem for the Kohn Laplacian on the Heisenberg group. I. J. Funct. Anal. 43(1), 97–142 (1981)
Jerison, D.S.: The Dirichlet problem for the Kohn Laplacian on the Heisenberg group. II. J. Funct. Anal. 43(2), 224–257 (1981)
Jerison, D., Kenig, C.: Boundary behavior of harmonic functions in non-tangentially accessible domains. Adv. Math. 46, 80–147 (1982)
Jones, P.: Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147(1–2), 71–88 (1981)
Kohn, J.J., Nirenberg, L.: Non-coercive boundary value problems. Commun. Pure Appl. Math. 18, 443–492 (1965)
Nagel, A., Stein, E.M., Wainger, S.: Balls and metrics defined by vector fields I: basic properties. Acta Math. 155, 103–147 (1985)
Nhieu, D.M.: The Neumann problem for sub-Laplacians on Carnot groups and the extension theorem for Sobolev spaces. Ann. Mat. Pura Appl. (4) 180 (2001), no. 1, 1–25
Oleinik, O.A.: On the smoothness of solutions of degenerate elliptic and parabolic equations. Dokl. Akad. Nauk SSSR 163, 577–580 (1965). [Russian]
Oleinik, O.A.: On linear second order equations with non-negative characteristic form. Mat. Sb. 69, 111–140 (1966). [Russian]
Rashevsky, P.K.: Any two points of a totally nonholonomic space may be connected by an admissible line. Uch. Zap. Ped. Inst. im. Liebknechta, Ser. Phys. Math. 2, 83–94 (1938). [Russian]
Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series, vol. 28. Princeton University Press, Princeton (1970)
Uguzzoni, F., Lanconelli, E.: On the Poisson kernel for the Kohn Laplacian. Rend. Mat. Appl. (7) 17(4), 659–677 (1997)
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Supported by the Ministry of Science and Technology, R.O.C., Grant MOST 106-2115-M-008-013.
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Nhieu, DM. Existence and uniqueness of variational solution to the Neumann problem for the pth sub-Laplacian associated to a system of Hörmander vector fields. Nonlinear Differ. Equ. Appl. 25, 44 (2018). https://doi.org/10.1007/s00030-018-0529-3
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DOI: https://doi.org/10.1007/s00030-018-0529-3