In a cylinder we study the boundary behavior of nonnegative solutions of second order parabolic equations of the form
where is a system of vector fields in satisfying Hörmander’s rank condition (1.2), and is a non-tangentially accessible domain with respect to the Carnot-Carathéodory distance induced by . Concerning the matrix-valued function , we assume that it is real, symmetric and uniformly positive definite. Furthermore, we suppose that its entries are Hölder continuous with respect to the parabolic distance associated with . Our main results are: 1) a backward Harnack inequality for nonnegative solutions vanishing on the lateral boundary (Theorem 1.1); 2) the Hölder continuity up to the boundary of the quotient of two nonnegative solutions which vanish continuously on a portion of the lateral boundary (Theorem 1.2); 3) the doubling property for the parabolic measure associated with the operator (Theorem 1.3). These results generalize to the subelliptic setting of the present paper, those in Lipschitz cylinders by Fabes, Safonov and Yuan in [20,39]. With one proviso: in those papers the authors assume that the coefficients be only bounded and measurable, whereas we assume Hölder continuity with respect to the intrinsic parabolic distance.
@article{ASNSP_2012_5_11_2_437_0, author = {Frentz, Marie and Garofalo, Nicola and G\"otmark, Elin and Munive, Isidro and Nystr\"om, Kaj}, title = {Non-divergence form parabolic equations associated with non-commuting vector fields: boundary behavior of nonnegative solutions}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {437--474}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 11}, number = {2}, year = {2012}, mrnumber = {3011998}, zbl = {1258.31005}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2012_5_11_2_437_0/} }
TY - JOUR AU - Frentz, Marie AU - Garofalo, Nicola AU - Götmark, Elin AU - Munive, Isidro AU - Nyström, Kaj TI - Non-divergence form parabolic equations associated with non-commuting vector fields: boundary behavior of nonnegative solutions JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2012 SP - 437 EP - 474 VL - 11 IS - 2 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2012_5_11_2_437_0/ LA - en ID - ASNSP_2012_5_11_2_437_0 ER -
%0 Journal Article %A Frentz, Marie %A Garofalo, Nicola %A Götmark, Elin %A Munive, Isidro %A Nyström, Kaj %T Non-divergence form parabolic equations associated with non-commuting vector fields: boundary behavior of nonnegative solutions %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2012 %P 437-474 %V 11 %N 2 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2012_5_11_2_437_0/ %G en %F ASNSP_2012_5_11_2_437_0
Frentz, Marie; Garofalo, Nicola; Götmark, Elin; Munive, Isidro; Nyström, Kaj. Non-divergence form parabolic equations associated with non-commuting vector fields: boundary behavior of nonnegative solutions. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 2, pp. 437-474. http://archive.numdam.org/item/ASNSP_2012_5_11_2_437_0/
[1] P. Bauman, Positive solutions of elliptic equations in nondivergence form and their adjoints, Ark. Mat. 22 (1984), 153–173. | MR | Zbl
[2] A. Bonfiglioli and F. Uguzzoni, Maximum principle and propagation for intrinsicly regular solutions of differential inequalities structured on vector fields, J. Math. Anal. Appl. (2) 322 (2006), 886–900. | MR | Zbl
[3] J. M. Bony, Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les operateurs elliptique degeneres, Ann. Inst. Fourier (Grenoble) 119 (1969), 277–304. | EuDML | Numdam | MR | Zbl
[4] M. Bramanti, L. Brandolini, E. Lanconelli and F. Uguzzoni, Heat kernels for non-divergence operators of Hörmander type, C. R. Math. Acad. Sci. Paris 343 (2006), 463–466. | MR | Zbl
[5] M. Bramanti, L. Brandolini, E. Lanconelli and F. Uguzzoni, “Non-Divergence Equations Structured on Hörmander Vector Fields: heat Kernels and Harnack Inequalities”, Mem. Amer. Math. Soc., Vol. 240, 2010. | MR | Zbl
[6] L. Caffarelli, E. Fabes, S. Mortola and S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana Univ. Math. J. 30 (1981), 621–640. | MR | Zbl
[7] 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. | EuDML | MR | Zbl
[8] L. Capogna, N. Garofalo and D. M. Nhieu, A subelliptic version of a theorem of Dahlberg for the subelliptic Dirichlet problem, Math. Res. Lett. 5 (1998), 541–549. | MR | Zbl
[9] L. Capogna, N. Garofalo and D. M. Nhieu, Examples of uniform and NTA domains in Carnot groups, In: “Proceedings on Analysis and Geometry” (Russian) (Novosibirsk Akademgorodok, 1999), Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 2000, 103–121. | MR | Zbl
[10] 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
[11] L. Capogna, N. Garofalo and D. M. Nhieu, Mutual absolute continuity of harmonic and surface measure for Hörmander type operators, In: “Perspectives in Partial Differential Equations, Harmonic Analysis and Applications”, Proc. Sympos. Pure Math. Amer. Math. Soc., Vol. 79, Providence, RI, 2008, 49–100. | MR | Zbl
[12] W. L. Chow, Über systeme von linearen partiellen differentialgleichungen erster ordnug, Math. Ann. 117 (1939), 98–105. | EuDML | JFM | MR
[13] G. Citti, Wiener estimates at boundary points for Hörmander’s operators, Boll. Un. Mat. Ital. B (7) 2 (1988), 667–681. | MR | Zbl
[14] D. Danielli, Regularity at the boundary for solutions of nonlinear subelliptic equations, Indiana Univ. Math. J. 44 (1995), 269–286. | MR | Zbl
[15] E. Fabes, N. Garofalo, S. Marin-Malave and S. Salsa, Fatou theorems for some nonlinear elliptic equations, Rev. Mat. Iberoamericana 4 (1988), 227–251. | EuDML | MR | Zbl
[16] E. Fabes, N. Garofalo and S. Salsa, A backward Harnack inequality and Fatou theorem for nonnegative solutions of parabolic equations, Illinois J. Math. 30 (1986), 536–565. | MR | Zbl
[17] E. Fabes and C. Kenig, Examples of singular parabolic measures and singular transition probability densities, Duke Math. J. 48 (1981), 845–856. | MR | Zbl
[18] C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems, Proceedings of the Conference in Harmonic Analysis in Honor of A. Zygmund, Wadsworth Math. Ser., Belmont, CA, (1981), 530–606. | MR | Zbl
[19] E. Fabes and M. Safonov, Behaviour near the boundary of positive solutions of second order parabolic equations, J. Fourier Anal. Appl. 3 (1997), 871–882. | EuDML | MR | Zbl
[20] E. Fabes, M. Safonov and Y. Yuan, Behavior near the boundary of positive solutions of second order parabolic equations.II, Trans. Amer. Math. Soc. 351 (1999), 4947–4961. | MR | Zbl
[21] E. Fabes and D. Stroock, A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash, Arch. Rational Mech. Anal. 96 (1986), 327–338. | MR | Zbl
[22] N. Garofalo, Second order parabolic equations in nonvariational forms: boundary Harnack principle and comparison theorems for nonnegative solutions, Ann. Mat. Pura Appl. 138 (1984), 267–296. | MR | Zbl
[23] N. Garofalo and D. M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math. 49 (1996), 1081–1144. | MR | Zbl
[24] S. Hofmann and J. Lewis, The Dirichlet problem for parabolic operators with singular drift term, Mem. Amer. Math. Soc. 151 (2001), 1–113. | MR | Zbl
[25] H. Hörmander, Hypoelliptic second-order differential equations, Acta Math. 119 (1967), 147–171. | MR | Zbl
[26] D. Jerison and C. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. Math. 46 (1982), 80–147. | MR | Zbl
[27] C. Kenig and J. Pipher, The Dirichlet problem for elliptic operators with drift term, Publ. Mat. 45 (2001), 199–217. | EuDML | MR | Zbl
[28] N. V. Krylov, sequences of convex functions, and estimates of the maximum of the solution of a parabolic equation, Sibirski Math. Zh. 17 (1976), 226–236. | MR | Zbl
[29] N. Krylov and M. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), 161–175. | MR | Zbl
[30] E. Lanconelli and F. Uguzzoni, Potential analysis for a class of diffusion equations: A Gaussian bounds approach, J. Differential Equations 248 (2010), 2329–2367. | MR | Zbl
[31] R. Monti and D. Morbidelli, Non-tangentially accessible domains for vector fields, Indiana Univ. Math. J. 54 (2005), 473–498. | MR | Zbl
[32] R. Monti and D. Morbidelli, Regular domains in homogeneous groups, Trans. Amer. Math. Soc. 357 (2005), 2975–3011. | MR | Zbl
[33] I. Munive, Boundary behavior of nonnegative solutions of the heat equation in sub-Riemannian spaces, Potential Anal., advance online publication doi:10.1007/s11118-011-9258-5. | MR | Zbl
[34] K. Nyström, The Dirichlet problem for second order parabolic operators, Indiana Univ. Math. J. 46 (1997), 183–245. | MR | Zbl
[35] P. Negrini and V. Scornazzani, Wiener criterion for a class of degenerate elliptic operators, J. Differential Equations 166 (1987), 151–167. | MR | Zbl
[36] A. Nagel, E. Stein and S. Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), 103–147. | MR | Zbl
[37] P. K. Rashevsky, Any two points of a totally nonholonomic space may be connected by an admissible line, Uch. Zap. Ped. Inst. im. Liebknechta, Ser. Phys. Math., (Russian) 2 (1938), 83–94.
[38] S. Salsa, Some properties of nonnegative solution to parabolic differential equations, Ann. Mat. Pura Appl. 128 (1981), 193–206. | MR | Zbl
[39] M. Safonov and Y. Yuan, Doubling properties for second order parabolic equations, Ann. of Math. 150 (1999), 313–327. | EuDML | MR | Zbl
[40] F. Uguzzoni, Cone criteria for non-divergence equations modeled on Hörmander vector fields, In: “Subelliptic PDE’s and Applications to Geometry and Finance”, Lect. Notes Semin. Interdiscip. Mat., 6, Semin. Interdiscip. Mat. (S.I.M.), Potenza, 2007, 227–241. | MR | Zbl