Neuman and Dirichlet heat kernels in inner uniform domains
Astérisque, no. 336 (2011) , 152 p.
Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez le site de la revue.
@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},
     zbl = {1222.58001},
     mrnumber = {2807275},
     language = {en},
     url = {http://www.numdam.org/item/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/

[1] H. Aikawa - "Boundary Harnack principle and Martin boundary for a uniform domain", J. Math. Soc. Japan 53 (2001), p. 119-145. | Article | MR 1800526 | Zbl 0976.31002

[2] H. Aikawa, T. Lundh & T. Mizutani - "Martin boundary of a fractal domain", Potential Anal. 18 (2003), p. 311-357. | Article | MR 1953266 | Zbl 1021.31002

[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. | Article | EuDML 74379 | Numdam | MR 513885 | Zbl 0377.31001

[4] A. Ancona, "Sur la théorie du potentiel dans les domaines de John", Publ. Mat. 51 (2007), p. 345-396. | Article | EuDML 41899 | MR 2334795 | Zbl 1134.31009

[5] D. G. Aronson - "Bounds for the fundamental solution of a parabolic equation", Bull. Amer. Math. Soc. 73 (1967), p. 890-896. | Article | MR 217444 | Zbl 0153.42002

[6] D. G. Aronson, "Non-negative solutions of linear parabolic equations", Ann. Scuola Norm. Sup. Pisa 22 (1968), p. 607-694. | EuDML 83474 | Numdam | MR 435594 | Zbl 0182.13802

[7] D. G. Aronson, "Addendum: "Non-negative solutions of linear parabolic equations" (Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 607-694)". | EuDML 83474 | Numdam | MR 435594 | Zbl 0182.13802

D. G. Aronson, "Addendum: "Non-negative solutions of linear parabolic equations Ann. Scuola Norm.Sup. Pisa 25 (1971), p.221-228. | EuDML 83560 | Numdam | MR 435595 | Zbl 0223.35046

[8] D. G. Aronson & J. Serrin - "Local behavior of solutions of quasilinear parabolic equations", Arch. Rational Mech. Anal. 25 (1967), p. 81-122. | Article | MR 244638 | Zbl 0154.12001

[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 1849181 | Zbl 1013.60060

[10] R. Bañuelos & R. G. Smits - "Brownian motion in cones", Probab. Theory Related Fields 108 (1997), p. 299-319. | Article | MR 1465162 | Zbl 0884.60037

[11] H. Bauer - Harmonische Räume und ihre Potentialtheorie, Lecture Notes in Math., vol. 22, Springer, 1966. | MR 210916 | Zbl 0142.38402

[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 928774 | Zbl 0636.60077

[13] A. Bendikov, Potential theory on infinite-dimensional abelian groups, de Gruyter Studies in Mathematics, vol. 21, Walter de Gruyter & Co., 1995. | MR 1445559 | Zbl 0869.31001

[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. | Article | MR 1797661 | Zbl 0969.31008

[15] M. Van Den Berg - "Heat equation on the arithmetic von Koch snowflake", Probab. Theory Related Fields 118 (2000), p. 17-36. | Article | MR 1785451 | Zbl 0963.35072

[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. | Article | MR 1378473 | Zbl 0851.31008

[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. | EuDML 244300 | MR 1340280 | Zbl 0837.31006

[18] M. Biroli & U. Mosco, "Sobolev inequalities on homogeneous spaces", Potential Anal. 4 (1995), p. 311-324. | Article | MR 1354886 | Zbl 0833.46020

[19] J. Bliedtner & W. Hansen - Potential theory, Universitext, Springer, 1986, An analytic and probabilistic approach to balayage. | MR 850715 | Zbl 0706.31001

[20] R. M. Blumenthal & R. K. Getoor - Markov processes and potential theory, Pure and Applied Mathematics, vol. 29, Academic Press, 1968. | MR 264757 | Zbl 0169.49204

[21] M. Brelot - "Le problème de Dirichlet géodésique", C. R. Acad. Sci. Paris 228 (1949), p. 1790-1792. | MR 31139 | Zbl 0035.06602

[22] D. Burago, Y. Burago & S. Ivanov - A course in metric geometry, Graduate Studies in Math., vol. 33, Amer. Math. Soc, 2001. | Article | Zbl 0981.51016

[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. | Article | EuDML 59574 | Zbl 0926.35043

[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 1017.30020

[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. | Article | Zbl 0998.22001

[26] L. Capogna & P. Tang - "Uniform domains and quasiconformal mappings on the Heisenberg group", Manuscripta Math. 86 (1995), p. 267-281. | Article | EuDML 156054 | Zbl 0824.30011

[27] Z. Q. Chen - "On reflected Dirichlet spaces", Probab. Theory Related Fields 94 (1992), p. 135-162. | Article | Zbl 0767.60073

[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 0859.46021

[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. | Article | Zbl 1053.60084

[30] K. L. Chung & Z. X. Zhao - From Brownian motion to Schrddinger's equation, Grund. Math. Wiss., vol. 312, Springer, 1995. | Zbl 1195.60002

[31] C. Constantinescu & A. Cornea - Potential theory on harmonic spaces, Springer, 1972. | Article | Zbl 0248.31011

[32] E. B. Davies - "Two-dimensional Riemannian manifolds with fractal boundaries", J. London Math. Soc. 49 (1994), p. 343-356. | Article | Zbl 0824.58043

[33] E. B. Davies, "Non-Gaussian aspects of heat kernel behaviour", J. London Math. Soc. 55 (1997), p. 105-125. | Article | Zbl 0879.35064

[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. | Article | EuDML 58147 | Zbl 0797.58087

[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. | Article | Zbl 0568.47034

[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 0084.31901

[37] J. L. Doob - Classical potential theory and its probabilistic counterpart, Classics in Mathematics, Springer, 2001, Reprint of the 1984 edition. | Zbl 0990.31001

[38] E. B. Dynkin - Markov processes. Vols. I, II, Grund. Math. Wiss., vol. 121, 122, Academic Press Inc., 1965. | Zbl 0132.37901

[39] J. Eells & B. Fuglede - Harmonic maps between Riemannian polyhedra, Cambridge Tracts in Mathematics, vol. 142, Cambridge Univ. Press, 2001. | Zbl 0979.31001

[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 0625.35006

[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. | EuDML 59543 | Zbl 0939.35082

[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. | Article | Zbl 0976.35031

[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. | Article | Zbl 0652.35052

[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. | Article | Zbl 1009.35013

[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. | Article | Zbl 1146.60064

[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 0838.31001

[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 0537.46001

[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 50119 | Zbl 1015.30009

[51] A. Grigor'Yan - "Stochastically complete manifolds and summable harmonic functions", Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), p. 1102-1108, 1120. | Zbl 0677.60086

[52] A. Grigor'Yan, "The heat equation on noncompact Riemannian manifolds", Mat. Sb. 182 (1991), p. 55-87. | EuDML 72232 | Zbl 0743.58031

[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. | Article | Zbl 0927.58019

[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. | Article | Zbl 1037.58018

[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 0988.28002

[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. | Article | Zbl 1048.46033

[57] P. Hajłasz & P. Koskela - "Sobolev met Poincaré", Mem. Amer. Math. Soc. 145 (2000). | Zbl 0954.46022

[58] W. Hebisch & L. Saloff-Coste - "On the relation between elliptic and parabolic Harnack inequalities", Ann. Inst. Fourier Grenoble) 51 (2001), p. 1437-1481. | Article | EuDML 115954 | Numdam | Zbl 0988.58007

[59] J. Heinonen - Lectures on analysis on metric spaces, Universitext, Springer, 2001. | Article | Zbl 0985.46008

[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 0661.47042

[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. | Article | EuDML 74931 | Numdam | Zbl 0734.35040

[62] Y. Heurteaux, "Mesure harmonique et équation de la chaleur", Ark. Mat. 34 (1996), p. 119-139. | Article | Zbl 0916.31001

[63] F. Hirsch - "Intrinsic metrics and Lipschitz functions", J. Evol. Equ. 3 (2003), p. 11-25, Dedicated to Philippe Bénilan. | Article | Zbl 1031.31006

[64] L. Hörmander - "Hypoelliptic second order differential equations", Acta Math. 119 (1967), p. 147-171. | Article | Zbl 0156.10701

[65] D. S. Jerison & C. E. Kenig - "Boundary behavior of harmonic functions in nontangentially accessible domains", Adv. in Math. 46 (1982), p. 80-147. | Article | Zbl 0514.31003

[66] P. W. Jones - "Extension theorems for BMO", Indiana Univ. Math. J. 29 (1980), p. 41-66. | Article | Zbl 0432.42017

[67] M. Kassmann - "Harnack inequalities: an introduction", Bound. Value Probl. (2007), Art. ID 81415, 21. | EuDML 54459 | Zbl 1144.35002

[68] K. Kuwae - "Reflected Dirichlet forms and the uniqueness of Silverstein's extension", Potential Anal. 16 (2002), p. 221-247. | Article | Zbl 0998.31006

[69] M. L. Lapidus & M. M. H. Pang - "Eigenfunctions of the Koch snowflake domain", Comm. Math. Phys. 172 (1995), p. 359-376. | Article | Zbl 0857.35093

[70] O. Martio & J. Sarvas - "Injectivity theorems in plane and space", Ann. Acad. Sci. Fenn. Ser. A I Math. 4 (1979), p. 383-401. | Article | Zbl 0406.30013

[71] P. Mattila - Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Math., vol. 44, Cambridge Univ. Press., 1995. | Zbl 0819.28004

[72] R. Montgomery - A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, vol. 91, Amer. Math. Soc., 2002. | MR 1867362 | Zbl 1044.53022

[73] U. Mosco - "Composite media and asymptotic Dirichlet forms", J. Funct. Anal. 123 (1994), p. 368-421. | Article | MR 1283033 | Zbl 0808.46042

[74] J. Moser - "On Harnack's theorem for elliptic differential equations", Comm. Pure Appl. Math. 14 (1961), p. 577-591. | Article | MR 159138 | Zbl 0111.09302

[75] J. Moser, "A Harnack inequality for parabolic differential equations", Comm. Pure Appl. Math. 17 (1964), p. 101-134. | Article | MR 159139 | Zbl 0149.06902

[76] J. Moser, "Correction to: "A Harnack inequality for parabolic differential equations"", Comm. Pure Appl. Math. 20 (1967), p. 231-236. | Article | MR 203268 | Zbl 0149.07001

[77] J. Nash - "Continuity of solutions of parabolic and elliptic equations", Amer. J. Math. 80 (1958), p. 931-954. | Article | MR 100158 | Zbl 0096.06902

[78] M. Pivarskl & L. Saloff-Coste - "Small time heat kernel behavior on riemannian complexes", New York J. Math. 14 (2008), p. 459-494. | EuDML 117119 | MR 2443983 | Zbl 1191.26005

[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 1648049 | Zbl 0910.35025

[80] M. V. Safonov & Y. Yuan - "Doubling properties for second order parabolic equations", Ann. of Math. 150 (1999), p. 313-327. | Article | EuDML 129683 | MR 1715327 | Zbl 1157.35391

[81] L. Saloff-Coste - "A note on Poincaré, Sobolev, and Harnack inequalities", Int. Math. Res. Not. 1992 (1992), p. 27-38. | Article | MR 1150597 | Zbl 0769.58054

[82] L. Saloff-Coste, "Parabolic Harnack inequality for divergence-form second-order differential operators", Potential Anal. 4 (1995), p. 429-467. | Article | MR 1354894 | Zbl 0840.31006

[83] L. Saloff-Coste, Aspects of Sobolev-type inequalities, London Mathematical Society Lecture Note Series, vol. 289, Cambridge Univ. Press, 2002. | MR 1872526 | Zbl 0991.35002

[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. | Article | MR 2195413 | Zbl 1082.31006

[85] S. Semmes - Some novel types of fractal geometry, Oxford Mathematical Monographs, The Clarendon Press Oxford Univ. Press, 2001. | MR 1815356 | Zbl 0970.28001

[86] M. L. Silverstein - Symmetric Markov processes, Springer, 1974, Lecture Notes in Mathematics, Vol. 426. | MR 386032 | Zbl 0296.60038

[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. | Article | MR 2109269 | Zbl 1064.35066

[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. | EuDML 153666 | MR 1301456 | Zbl 0806.53041

[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 1355744 | Zbl 0854.35015

[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. | Article | MR 1360279 | Zbl 0834.58039

[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. | Article | MR 1360279 | Zbl 0834.58039

[92] K.-T. Sturm, "Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality", J. Math. Pures Appl. 75 (1996), p. 273-297. | MR 1387522 | Zbl 0854.35016

[93] J. Väisälä - "Relatively and inner uniform domains", Conform. Geom. Dyn. 2 (1998), p. 56-88. | Article | MR 1637079 | Zbl 0902.30017

[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 892879 | Zbl 0614.22002

[95] N. T. Varopoulos, "Potential theory in conical domains", Math. Proc. Cambridge Philos. Soc. 125 (1999), p. 335-384. | Article | MR 1643806 | Zbl 0918.31008

[96] N. T. Varopoulos, "Potential theory in conical domains. II", Math. Proc. Cambridge Philos . Soc. 129 (2000), p. 301-319. | Article | MR 1765917 | Zbl 0980.31007

[97] N. T. Varopoulos, "Potential theory in conical domains. III", Math. Proc. Cambridge Philos. Soc. 131 (2001), p. 327-361. | Article | MR 1857124 | Zbl 1023.31007

[98] N. T. Varopoulos, "Potential theory in Lipschitz domains", Canad. J. Math. 53 (2001), p. 1057-1120. | Article | MR 1859766 | Zbl 0983.60072

[99] N. T. Varopoulos, "Gaussian estimates in Lipschitz domains", Canad. J. Math. 55 (2003), p. 401-431. | Article | MR 1969798 | Zbl 1042.58013

[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 1218884 | Zbl 0813.22003

[101] N. Weaver - "Lipschitz algebras and derivations. II. Exterior differentiation", J. Funct Anal. 178 (2000), p. 64-112. | Article | MR 1800791 | Zbl 0979.46035