Non-divergence form parabolic equations associated with non-commuting vector fields: boundary behavior of nonnegative solutions

Frentz, Marie; Garofalo, Nicola; Götmark, Elin; Munive, Isidro; Nyström, Kaj

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 2, pp. 437-474.

In a cylinder ${\Omega }_T=\Omega \times (0,T)\subset {ℝ}_{+}^{n+1}$ we study the boundary behavior of nonnegative solutions of second order parabolic equations of the form

$$Hu=\sum _{i,j=1}^m{a}_{ij}(x,t)X_i{X}_{j}u-{\partial }_{t}u=0,(x,t)\in {ℝ}_{+}^{n+1},$$

where $X=\{X_1,...,X_m\}$ is a system of $C^{\infty }$ vector fields in ${ℝ}^n$ satisfying Hörmander’s rank condition (1.2), and $\Omega$ is a non-tangentially accessible domain with respect to the Carnot-Carathéodory distance $d$ induced by $X$. Concerning the matrix-valued function $A=\left\{a_{ij}\right\}$, we assume that it is real, symmetric and uniformly positive definite. Furthermore, we suppose that its entries $a_{ij}$ are Hölder continuous with respect to the parabolic distance associated with $d$. 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 $H$ (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 $a_{ij}$ be only bounded and measurable, whereas we assume Hölder continuity with respect to the intrinsic parabolic distance.

Classification : 31C05, 35C15, 65N99

[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