Neuman and Dirichlet heat kernels in inner uniform domains

Gyrya, Pavel ; Saloff-Coste, Laurent

Astérisque, no. 336 (2011) , 152 p.

MR Zbl | 2 citations dans Numdam

@book{AST_2011__336__R1_0,
     author = {Gyrya, Pavel and Saloff-Coste, Laurent},
     title = {Neuman and {Dirichlet} heat kernels in inner uniform domains},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {336},
     year = {2011},
     mrnumber = {2807275},
     zbl = {1222.58001},
     language = {en},
     url = {http://www.numdam.org/item/AST_2011__336__R1_0/}
}

TY  - BOOK
AU  - Gyrya, Pavel
AU  - Saloff-Coste, Laurent
TI  - Neuman and Dirichlet heat kernels in inner uniform domains
T3  - Astérisque
PY  - 2011
IS  - 336
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/AST_2011__336__R1_0/
LA  - en
ID  - AST_2011__336__R1_0
ER  -

%0 Book
%A Gyrya, Pavel
%A Saloff-Coste, Laurent
%T Neuman and Dirichlet heat kernels in inner uniform domains
%S Astérisque
%D 2011
%N 336
%I Société mathématique de France
%U http://www.numdam.org/item/AST_2011__336__R1_0/
%G en
%F AST_2011__336__R1_0

Gyrya, Pavel; Saloff-Coste, Laurent. Neuman and Dirichlet heat kernels in inner uniform domains. Astérisque, no. 336 (2011), 152 p. http://numdam.org/item/AST_2011__336__R1_0/

Bibliographie
Cité par

[1] H. Aikawa - "Boundary Harnack principle and Martin boundary for a uniform domain", J. Math. Soc. Japan 53 (2001), p. 119-145. | MR | Zbl | DOI

[2] H. Aikawa, T. Lundh & T. Mizutani - "Martin boundary of a fractal domain", Potential Anal. 18 (2003), p. 311-357. | MR | Zbl | DOI

[3] A. Ancona - "Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien", Ann. Inst. Fourier (Grenoble) 28 (1978), p. 169-213. | MR | Zbl | EuDML | Numdam | DOI

[4] A. Ancona, "Sur la théorie du potentiel dans les domaines de John", Publ. Mat. 51 (2007), p. 345-396. | MR | Zbl | EuDML | DOI

[5] D. G. Aronson - "Bounds for the fundamental solution of a parabolic equation", Bull. Amer. Math. Soc. 73 (1967), p. 890-896. | MR | Zbl | DOI

[6] D. G. Aronson, "Non-negative solutions of linear parabolic equations", Ann. Scuola Norm. Sup. Pisa 22 (1968), p. 607-694. | MR | Zbl | EuDML | Numdam

[7] D. G. Aronson, "Addendum: "Non-negative solutions of linear parabolic equations" (Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 607-694)". | MR | Zbl | EuDML | Numdam

D. G. Aronson, "Addendum: "Non-negative solutions of linear parabolic equations Ann. Scuola Norm.Sup. Pisa 25 (1971), p.221-228. | MR | Zbl | EuDML | Numdam

[8] D. G. Aronson & J. Serrin - "Local behavior of solutions of quasilinear parabolic equations", Arch. Rational Mech. Anal. 25 (1967), p. 81-122. | MR | Zbl | DOI

[9] R. Bañuelos, R. D. Deblassie & R. G. Smits - "The first exit time of planar Brownian motion from the interior of a parabola", Ann. Probab. 29 (2001), p. 882-901. | MR | Zbl

[10] R. Bañuelos & R. G. Smits - "Brownian motion in cones", Probab. Theory Related Fields 108 (1997), p. 299-319. | MR | Zbl | DOI

[11] H. Bauer - Harmonische Räume und ihre Potentialtheorie, Lecture Notes in Math., vol. 22, Springer, 1966. | MR | Zbl

[12] A. Bendikov - "Markov processes and partial differential equations on a group: the spatially homogeneous case", Uspekhi Mat. Nauk 42 (1987), p. 41-78. | MR | Zbl

[13] A. Bendikov, Potential theory on infinite-dimensional abelian groups, de Gruyter Studies in Mathematics, vol. 21, Walter de Gruyter & Co., 1995. | MR | Zbl

[14] A. Bendikov & L. Saloff-Coste - "On- and off-diagonal heat kernel behaviors on certain infinite dimensional local Dirichlet spaces", Amer. J. Math. 122 (2000), p. 1205-1263. | MR | Zbl | DOI

[15] M. Van Den Berg - "Heat equation on the arithmetic von Koch snowflake", Probab. Theory Related Fields 118 (2000), p. 17-36. | MR | Zbl | DOI

[16] M. Biroli & U. Mosco - "A Saint-Venant type principle for Dirichlet forms on discontinuous media", Ann. Mat. Pura Appl. 169 (1995), p. 125-181. | MR | Zbl | DOI

[17] M. Biroli & U. Mosco, "Sobolev and isoperimetric inequalities for Dirichlet forms on homogeneous spaces", Atti Accad. Naz. Lincei CL Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 6 (1995), p. 37-44. | MR | Zbl | EuDML

[18] M. Biroli & U. Mosco, "Sobolev inequalities on homogeneous spaces", Potential Anal. 4 (1995), p. 311-324. | MR | Zbl | DOI

[19] J. Bliedtner & W. Hansen - Potential theory, Universitext, Springer, 1986, An analytic and probabilistic approach to balayage. | MR | Zbl

[20] R. M. Blumenthal & R. K. Getoor - Markov processes and potential theory, Pure and Applied Mathematics, vol. 29, Academic Press, 1968. | MR | Zbl

[21] M. Brelot - "Le problème de Dirichlet géodésique", C. R. Acad. Sci. Paris 228 (1949), p. 1790-1792. | MR | Zbl

[22] D. Burago, Y. Burago & S. Ivanov - A course in metric geometry, Graduate Studies in Math., vol. 33, Amer. Math. Soc, 2001. | Zbl | DOI

[23] L. Capogna & N. Garofalo - "Boundary behavior of nonnegative solutions of subelliptic equations in NTA domains for Carnot-Carathéodory metrics", J. Fourier Anal. Appl. 4 (1998), p. 403-432. | Zbl | EuDML | DOI

[24] L. Capogna, N. Garofalo & 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, p. 103-121. | Zbl

[25] L. Capogna, N. Garofalo & D.-M. Nhieu, "Properties of harmonic measures in the Dirichlet problem for nilpotent Lie groups of Heisenberg type", Amer. J. Math. 124 (2002), p. 273-306. | Zbl | DOI

[26] L. Capogna & P. Tang - "Uniform domains and quasiconformal mappings on the Heisenberg group", Manuscripta Math. 86 (1995), p. 267-281. | Zbl | EuDML | DOI

[27] Z. Q. Chen - "On reflected Dirichlet spaces", Probab. Theory Related Fields 94 (1992), p. 135-162. | Zbl | DOI

[28] Z. Q. Chen, "Reflecting Brownian motions and a deletion result for Sobolev spaces of order (1,2)", Potential Anal. 5 (1996), p. 383-401. | Zbl

[29] Z. Q. Chen, P. J. Fitzsimmons, M. Takeda, J. Ying & T.-S. Zhang - "Absolute continuity of symmetric Markov processes", Ann. Probab. 32 (2004), p. 2067-2098. | Zbl | DOI

[30] K. L. Chung & Z. X. Zhao - From Brownian motion to Schrddinger's equation, Grund. Math. Wiss., vol. 312, Springer, 1995. | Zbl

[31] C. Constantinescu & A. Cornea - Potential theory on harmonic spaces, Springer, 1972. | Zbl | DOI

[32] E. B. Davies - "Two-dimensional Riemannian manifolds with fractal boundaries", J. London Math. Soc. 49 (1994), p. 343-356. | Zbl | DOI

[33] E. B. Davies, "Non-Gaussian aspects of heat kernel behaviour", J. London Math. Soc. 55 (1997), p. 105-125. | Zbl | DOI

[34] E. B. Davies & M. Lianantonakis - "Heat kernel and Hardy estimates for locally Euclidean manifolds with fractal boundaries", Geom. Fund. Anal. 3 (1993), p. 527-563. | Zbl | EuDML | DOI

[35] E. B. Davies & B. Simon - "Ultracontractivity and the heat kernel for Schrodinger operators and Dirichlet Laplacians", J. Fund. Anal. 59 (1984), p. 335-395. | Zbl | DOI

[36] E. De Giorgi - "Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari", Mem. Accad. Sci. Torino. CI. Sci. Fis. Mat. Nat. 3 (1957), p. 25-43. | Zbl

[37] J. L. Doob - Classical potential theory and its probabilistic counterpart, Classics in Mathematics, Springer, 2001, Reprint of the 1984 edition. | Zbl

[38] E. B. Dynkin - Markov processes. Vols. I, II, Grund. Math. Wiss., vol. 121, 122, Academic Press Inc., 1965. | Zbl

[39] J. Eells & B. Fuglede - Harmonic maps between Riemannian polyhedra, Cambridge Tracts in Mathematics, vol. 142, Cambridge Univ. Press, 2001. | Zbl

[40] E. B. Fabes, N. Garofalo & S. Salsa - "A backward Harnack inequality and Fatou theorem for nonnegative solutions of parabolic equations", Illinois J. Math. 30 (1986), p. 536-565. | Zbl

[41] E. B. Fabes & M. V. Safonov - "Behavior near the boundary of positive solutions of second order parabolic equations", in Proceedings of the conference dedicated to Professor Miguel de Guzmán (El Escorial, 1996), vol. 3, 1997, p. 871-882. | Zbl | EuDML

[42] E. B. Fabes, M. V. Safonov & Y. Yuan - "Behavior near the boundary of positive solutions of second order parabolic equations. II", Trans. Amer. Math. Soc. 351 (1999), p. 4947-4961. | Zbl | DOI

[43] E. B. Fabes & D. W. Stroock - "A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash", Arch. Rational Mech. Anal. 96 (1986), p. 327-338. | Zbl | DOI

[44] E. Ferretti & M. V. Safonov - "Growth theorems and Harnack inequality for second order parabolic equations", in Harmonic analysis and boundary value problems (Fayetteville, AR, 2000), Contemp. Math., vol. 277, Amer. Math. Soc., 2001, p. 87-112. | Zbl | DOI

[45] P. J. Fitzsimmons - "The Dirichlet form of a gradient-type drift transformation of a symmetric diffusion", Acta Math. Sin. (Engl. Ser.) 24 (2008), p. 1057-1066. | Zbl | DOI

[46] J. Fleckinger, M. Levitin & D. Vassiliev - "Heat content of the triadic von Koch snowflake", Internat. J. Appl. Sci. Comput. 2 (1995), p. 289-305.

[47] M. Fukushima, Y. Ōshima & M. Takeda - Dirichlet forms and symmetric Markov processes, de Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., 1994. | Zbl

[48] K. Goebel & S. Reich - Uniform convexity, hyperbolic geometry, and non-expansive mappings, Monographs and Textbooks in Pure and Applied Mathematics, vol. 83, Marcel Dekker Inc., 1984. | Zbl

[49] A. V. Greshnov - "On sphere geometry in the Carnot-Caratheodory metric in Carnot groups", in Algebra, geometry, analysis and mathematical physics (Russian) (Novosibirsk, 1996), Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 1997, p. 170-173, 191.

[50] A. V. Greshnov, "On uniform and NTA-domains on Carnot groups", Sibirsk. Mat. Zh. 42 (2001), p. 1018-1035, ii. | EuDML | Zbl

[51] A. Grigor'Yan - "Stochastically complete manifolds and summable harmonic functions", Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), p. 1102-1108, 1120. | Zbl

[52] A. Grigor'Yan, "The heat equation on noncompact Riemannian manifolds", Mat. Sb. 182 (1991), p. 55-87. | Zbl | EuDML

[53] A. Grigor'Yan, "Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds", Bull. Amer. Math. Soc. (N.S.) 36 (1999), p. 135-249. | Zbl | DOI

[54] A. Grigor'Yan & L. Saloff-Coste - "Dirichlet heat kernel in the exterior of a compact set", Comm. Pure Appl. Math. 55 (2002), p. 93-133. | Zbl | DOI

[55] P. Hajłasz - "Sobolev mappings, co-area formula and related topics", in Proceedings on Analysis and Geometry (Novosibirsk, 2000), Sobolev Institut Press, 2000, p. 227-254. | Zbl

[56] P. Hajłasz, "Sobolev spaces on metric-measure spaces", in Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), Contemp. Math., vol. 338, Amer. Math. Soc., 2003, p. 173-218. | Zbl | DOI

[57] P. Hajłasz & P. Koskela - "Sobolev met Poincaré", Mem. Amer. Math. Soc. 145 (2000). | Zbl

[58] W. Hebisch & L. Saloff-Coste - "On the relation between elliptic and parabolic Harnack inequalities", Ann. Inst. Fourier Grenoble) 51 (2001), p. 1437-1481. | Zbl | EuDML | Numdam | DOI

[59] J. Heinonen - Lectures on analysis on metric spaces, Universitext, Springer, 2001. | Zbl | DOI

[60] Y. Heurteaux - "Inégalités de Harnack à la frontière pour des opérateurs paraboliques", C. R. Acad. Sci. Paris Sér. I Math. 308 (1989), p. 401-404. | Zbl

[61] Y. Heurteaux, "Solutions positives et mesure harmonique pour des opérateurs paraboliques dans des ouverts "lipschitziens"", Ann. Inst. Fourier (Grenoble) 41 (1991), p. 601-649. | Zbl | EuDML | Numdam | DOI

[62] Y. Heurteaux, "Mesure harmonique et équation de la chaleur", Ark. Mat. 34 (1996), p. 119-139. | Zbl | DOI

[63] F. Hirsch - "Intrinsic metrics and Lipschitz functions", J. Evol. Equ. 3 (2003), p. 11-25, Dedicated to Philippe Bénilan. | Zbl | DOI

[64] L. Hörmander - "Hypoelliptic second order differential equations", Acta Math. 119 (1967), p. 147-171. | Zbl | DOI

[65] D. S. Jerison & C. E. Kenig - "Boundary behavior of harmonic functions in nontangentially accessible domains", Adv. in Math. 46 (1982), p. 80-147. | Zbl | DOI

[66] P. W. Jones - "Extension theorems for BMO", Indiana Univ. Math. J. 29 (1980), p. 41-66. | Zbl | DOI

[67] M. Kassmann - "Harnack inequalities: an introduction", Bound. Value Probl. (2007), Art. ID 81415, 21. | Zbl | EuDML

[68] K. Kuwae - "Reflected Dirichlet forms and the uniqueness of Silverstein's extension", Potential Anal. 16 (2002), p. 221-247. | Zbl | DOI

[69] M. L. Lapidus & M. M. H. Pang - "Eigenfunctions of the Koch snowflake domain", Comm. Math. Phys. 172 (1995), p. 359-376. | Zbl | DOI

[70] O. Martio & J. Sarvas - "Injectivity theorems in plane and space", Ann. Acad. Sci. Fenn. Ser. A I Math. 4 (1979), p. 383-401. | Zbl | DOI

[71] P. Mattila - Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Math., vol. 44, Cambridge Univ. Press., 1995. | Zbl

[72] R. Montgomery - A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, vol. 91, Amer. Math. Soc., 2002. | MR | Zbl

[73] U. Mosco - "Composite media and asymptotic Dirichlet forms", J. Funct. Anal. 123 (1994), p. 368-421. | MR | Zbl | DOI

[74] J. Moser - "On Harnack's theorem for elliptic differential equations", Comm. Pure Appl. Math. 14 (1961), p. 577-591. | MR | Zbl | DOI

[75] J. Moser, "A Harnack inequality for parabolic differential equations", Comm. Pure Appl. Math. 17 (1964), p. 101-134. | MR | Zbl | DOI

[76] J. Moser, "Correction to: "A Harnack inequality for parabolic differential equations"", Comm. Pure Appl. Math. 20 (1967), p. 231-236. | MR | Zbl | DOI

[77] J. Nash - "Continuity of solutions of parabolic and elliptic equations", Amer. J. Math. 80 (1958), p. 931-954. | MR | Zbl | DOI

[78] M. Pivarskl & L. Saloff-Coste - "Small time heat kernel behavior on riemannian complexes", New York J. Math. 14 (2008), p. 459-494. | MR | Zbl | EuDML

[79] M. V. Safonov - "Estimates near the boundary for solutions of second order parabolic equations", in Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), vol. I (extra), 1998, p. 637-647. | MR | Zbl

[80] M. V. Safonov & Y. Yuan - "Doubling properties for second order parabolic equations", Ann. of Math. 150 (1999), p. 313-327. | MR | Zbl | EuDML | DOI

[81] L. Saloff-Coste - "A note on Poincaré, Sobolev, and Harnack inequalities", Int. Math. Res. Not. 1992 (1992), p. 27-38. | MR | Zbl | DOI

[82] L. Saloff-Coste, "Parabolic Harnack inequality for divergence-form second-order differential operators", Potential Anal. 4 (1995), p. 429-467. | MR | Zbl | DOI

[83] L. Saloff-Coste, Aspects of Sobolev-type inequalities, London Mathematical Society Lecture Note Series, vol. 289, Cambridge Univ. Press, 2002. | MR | Zbl

[84] L. Saloff-Coste, "Analysis on Riemannian co-compact covers", in Surveys in differential geometry. Vol. IX, Surv. Differ. Geom., IX, Int. Press, Somerville, MA, 2004, p. 351-384. | MR | Zbl | DOI

[85] S. Semmes - Some novel types of fractal geometry, Oxford Mathematical Monographs, The Clarendon Press Oxford Univ. Press, 2001. | MR | Zbl

[86] M. L. Silverstein - Symmetric Markov processes, Springer, 1974, Lecture Notes in Mathematics, Vol. 426. | MR | Zbl

[87] R. Song - "Estimates on the Dirichlet heat kernel of domains above the graphs of bounded $C^{1, 1}$ functions", Glas. Mat. Ser. III 39(59) (2004), p. 273-286. | MR | Zbl | DOI

[88] K.-T. Sturm - "Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and $L^{p}$ -Liouville properties", J. reine angew. Math. 456 (1994), p. 173-196. | MR | Zbl | EuDML

[89] K.-T. Sturm, "Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations", Osaka J. Math. 32 (1995), p. 275-312. | MR | Zbl

[90] K.-T. Sturm, "On the geometry defined by Dirichlet forms", in Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1993), Progr. Probab., vol. 36, Birkhäuser, 1995, p. 231-242. | MR | Zbl | DOI

[91] K.-T. Sturm, "On the geometry defined by Dirichlet forms", in Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1993), Progr. Probab., vol. 36, Birkhäuser, 1995, p. 231-242. | MR | Zbl | DOI

[92] K.-T. Sturm, "Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality", J. Math. Pures Appl. 75 (1996), p. 273-297. | MR | Zbl

[93] J. Väisälä - "Relatively and inner uniform domains", Conform. Geom. Dyn. 2 (1998), p. 56-88. | MR | Zbl | DOI

[94] N. T. Varopoulos - "Fonctions harmoniques sur les groupes de Lie", C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), p. 519-521. | MR | Zbl

[95] N. T. Varopoulos, "Potential theory in conical domains", Math. Proc. Cambridge Philos. Soc. 125 (1999), p. 335-384. | MR | Zbl | DOI

[96] N. T. Varopoulos, "Potential theory in conical domains. II", Math. Proc. Cambridge Philos . Soc. 129 (2000), p. 301-319. | MR | Zbl | DOI

[97] N. T. Varopoulos, "Potential theory in conical domains. III", Math. Proc. Cambridge Philos. Soc. 131 (2001), p. 327-361. | MR | Zbl | DOI

[98] N. T. Varopoulos, "Potential theory in Lipschitz domains", Canad. J. Math. 53 (2001), p. 1057-1120. | MR | Zbl | DOI

[99] N. T. Varopoulos, "Gaussian estimates in Lipschitz domains", Canad. J. Math. 55 (2003), p. 401-431. | MR | Zbl | DOI

[100] N. T. Varopoulos, L. Saloff-Coste & T. Coulhon - Analysis and geometry on groups, Cambridge Tracts in Mathematics, vol. 100, Cambridge Univ. Press, 1992. | MR | Zbl

[101] N. Weaver - "Lipschitz algebras and derivations. II. Exterior differentiation", J. Funct Anal. 178 (2000), p. 64-112. | MR | Zbl | DOI