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, Série 5, Tome 11 (2012) no. 2, pp. 437-474.

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

 $Hu=\sum _{i,j=1}^{m}{a}_{ij}\left(x,t\right){X}_{i}{X}_{j}u-{\partial }_{t}u=0,\phantom{\rule{4pt}{0ex}}\left(x,t\right)\in {ℝ}_{+}^{n+1},$

where $X=\left\{{X}_{1},...,{X}_{m}\right\}$ 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.

Publié le :
Classification : 31C05,  35C15,  65N99
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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},
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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, Série 5, Tome 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 765409 | Zbl 0557.35033

[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 2250624 | Zbl 1106.35149

[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 73982 | Numdam | MR 262881 | Zbl 0176.09703

[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 2267187 | Zbl 1109.35008

[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 2604962 | Zbl 1218.35001

[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 620271 | Zbl 0512.35038

[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 59574 | MR 1658616 | Zbl 0926.35043

[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 1653336 | Zbl 0934.22017

[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 1847513 | Zbl 1017.30020

[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 1890994 | Zbl 0998.22001

[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 2500489 | Zbl 1182.31012

[12] W. L. Chow, Über systeme von linearen partiellen differentialgleichungen erster ordnug, Math. Ann. 117 (1939), 98–105. | EuDML 160043 | JFM 65.0398.01 | MR 1880

[13] G. Citti, Wiener estimates at boundary points for Hörmander’s operators, Boll. Un. Mat. Ital. B (7) 2 (1988), 667–681. | MR 963325 | Zbl 0663.31006

[14] D. Danielli, Regularity at the boundary for solutions of nonlinear subelliptic equations, Indiana Univ. Math. J. 44 (1995), 269–286. | MR 1336442 | Zbl 0828.35022

[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 39377 | MR 1028741 | Zbl 0703.35058

[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 857210 | Zbl 0625.35006

[17] E. Fabes and C. Kenig, Examples of singular parabolic measures and singular transition probability densities, Duke Math. J. 48 (1981), 845–856. | MR 782580 | Zbl 0482.35021

[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 730094 | Zbl 0503.35071

[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 59543 | MR 1600211 | Zbl 0939.35082

[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 1665328 | Zbl 0976.35031

[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 855753 | Zbl 0652.35052

[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 779547 | Zbl 0574.35039

[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 1404326 | Zbl 0880.35032

[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 1828387 | Zbl 1149.35048

[25] H. Hörmander, Hypoelliptic second-order differential equations, Acta Math. 119 (1967), 147–171. | MR 222474 | Zbl 0156.10701

[26] D. Jerison and C. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. Math. 46 (1982), 80–147. | MR 676988 | Zbl 0514.31003

[27] C. Kenig and J. Pipher, The Dirichlet problem for elliptic operators with drift term, Publ. Mat. 45 (2001), 199–217. | EuDML 41425 | MR 1829584 | Zbl 1113.35314

[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 420016 | Zbl 0362.35038

[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 563790 | Zbl 0439.35023

[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 2595724 | Zbl 1191.35104

[31] R. Monti and D. Morbidelli, Non-tangentially accessible domains for vector fields, Indiana Univ. Math. J. 54 (2005), 473–498. | MR 2136818 | Zbl 1143.57013

[32] R. Monti and D. Morbidelli, Regular domains in homogeneous groups, Trans. Amer. Math. Soc. 357 (2005), 2975–3011. | MR 2135732 | Zbl 1067.43003

[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 2988206 | Zbl 1258.35109

[34] K. Nyström, The Dirichlet problem for second order parabolic operators, Indiana Univ. Math. J. 46 (1997), 183–245. | MR 1462802 | Zbl 0878.35010

[35] P. Negrini and V. Scornazzani, Wiener criterion for a class of degenerate elliptic operators, J. Differential Equations 166 (1987), 151–167. | MR 871992 | Zbl 0633.35018

[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 793239 | Zbl 0578.32044

[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 640782 | Zbl 0477.35049

[39] M. Safonov and Y. Yuan, Doubling properties for second order parabolic equations, Ann. of Math. 150 (1999), 313–327. | EuDML 129683 | MR 1715327 | Zbl 1157.35391

[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 2384649 | Zbl 1165.35007