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Parabolic Harnack inequality for divergence form second order differential operators. (English) Zbl 0840.31006
Cet article est une analyse approfondie des liens entre: (1) le principe de Harnack parabolique pour les opérateurs différentiels du deuxième ordre sous forme divergence; et (2) deux propriétés géométriques, à savoir inégalité de Poincaré et propriété de doublement.
Il donne en particulier une preuve complète de l’équivalence entre (1) et (2), annoncée par l’Auteur [Int. Math. Res. Not. 1992, No. 2, 27-38 (1992; Zbl 0769.58054)], sous les hypothèses suivantes: \(M\) étant une variété \({\mathcal C}^\infty\) connexe, \(\lll\) un opérateur différentiel du deuxième ordre à coefficients \({\mathcal C}^\infty\), sans terme d’ordre 0, autoadjoint et positif sur \(L^2 (M, \mu)\), \(\mu\) une mesure \({\mathcal C}^\infty \) positive sur \(M\), on suppose que la “distance” \(\rho\) associée à \(\lll\) est finie sur \(M\), qu’elle est continue et induit la topologie donnée de \(M\), enfin que l’espace métrique \((M, \rho)\) est complet; on note \(B(x,r) = \{y \in M : \rho (x,y) < r\}\). Alors, \(R > 0\) étant fixé, il y a équivalence entre
(1) \(\lll\) satisfait \([PHP(R)]\): Il existe \(C > 0\) tel que, \(\forall x \in M\), \( \forall s \in \mathbb{R}\), \(\forall r \in ]0,R[\), toute solution \(u \geq 0\) de \[ \left( {\partial \over \partial t} + \lll \right) u = 0 \quad \text{sur} \quad Q = ]s,\;s + r^2[ \times B(x,r)\tag \(*\) \] vérifie \(\sup_{Q_-} u \leq C \sup_{Q_+} u\), où \(Q_- = ]s + r^2/6\), \(s + r^2/3 [\times B (x,r/2)\) et \(Q_+ = ]s + 2r^2/3\), \(s + r^2 [\times B (x,r/2)\)
(2) \(\lll\) satisfait \([D(R)]\): Il existe \(D > 0\) tel que, \(\forall x \in M\), \(\forall r \in ]0,R[\), on ait \(\mu [B(x,2r)] \leq D\) \(\mu [B(x,r)]\); \(\lll\) satisfait \([P(R)]\): Il existe \(P > 0\) tel que, \(\forall x \in M\), \(\forall r \in ]0,R[\), \(\forall \psi \in {\mathcal C}^\infty [B(x,2r)]\), on ait \[ \int_B |\psi - \psi_B |^2 d \mu \leq Pr^2 \int_{2B} |\nabla \psi |^2 d \mu \] en notant \(B = B(x,r)\), \(2B = B(x,2r)\), \(\psi_B = \mu\)-Valeur moyenne de \(\psi\) sur \(B\).
On montre assez facilement \(PHP(R) \Rightarrow D(R), P(R)\), ainsi que la continuité hölderienne des solutions de \((*)\) et un encadrement gaussien de la solution fondamentale sur \(M\). La preuve de la réciproque est beaucoup plus difficile et utilise en particulier la technique itérative de Moser.

MSC:
31B35 Connections of harmonic functions with differential equations in higher dimensions
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
26D10 Inequalities involving derivatives and differential and integral operators
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[1] Alexopoulos G.: An application of homogenization theory to harmonic analysis: Harnack inequalities and Riesz transforms on Lie groups of polynomial growth.Canadian J. Math. 44, 1992, 691-727. · Zbl 0792.22005 · doi:10.4153/CJM-1992-042-x
[2] Aronson D. G.: Bounds for the fundamental solution of a parabolic equation.Bull. Amer. Math. Soc. 73, 1967, 890-896. · Zbl 0153.42002 · doi:10.1090/S0002-9904-1967-11830-5
[3] Aronson D. G.: Non-negative solutions of linear parabolic equationsAnn. Scu. Norm. Sup. Pisa. Cl. Sci. 22, 1968, 607-694; Addendum25, 1971, 221-228. · Zbl 0182.13802
[4] Aronson D. G. and Serrin J.: Local behavior of solutions of quasilinear parabolic equations.Arch. Rat. Mech. Anal. 25, 1967, 81-122. · Zbl 0154.12001 · doi:10.1007/BF00281291
[5] Bakry D., Coulhon Th., Ledoux M. Saloff-Coste L.: Sobolev inequalities in disguise. Preprint 1994. · Zbl 0857.26006
[6] Biroli M. and Mosco U.: Formes de Dirichlet et estimations structurelles dans les milieux discontinus.C. R. Acad. Sci. Paris,313, 1991, 593-598. · Zbl 0760.49004
[7] Biroli M. and Mosco U.: A Saint-Venant Principle for Dirichlet forms on discontinuous media.Ann. di Mat. Pura e Appl. · Zbl 0851.31008
[8] Biroli M. Mosco U.: Sobolev and isoperimetric inequalities for Dirichlet forms on homogeneous spaces.Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur., 1994 · Zbl 0837.31006
[9] Bôcher M.: Singular points of functions which satisfy partial differential equations of elliptic type.Bull. Amer. Math. Soc. 9, 1903, 455-465. · JFM 35.0356.03 · doi:10.1090/S0002-9904-1903-01017-9
[10] Bombieri E.:Theory of mininal surfaces and a counter-example to the Berstein conjecture in high dimensions. Mineographed Notes of Lectures held at Courant Institute, New-York University, 1970.
[11] Bombieri E. and Giusti E.: Harnack’s inequality for elliptic differential equations on mininal surfaces.Invent. Math. 15, 1972, 24-46. · Zbl 0227.35021 · doi:10.1007/BF01418640
[12] Bony J-M.: Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés.Ann. Inst. Fourier,19, 1969, 277-304. · Zbl 0176.09703
[13] Brelot M.:Elements de la théorie classique du potentiel. Centre de documentation universitaire, Paris, 1965.
[14] Buser P.: A note on the isoperimetric constant.Ann. Sci. Ecole Norm. Sup. 15, 1982, 213-230. · Zbl 0501.53030
[15] Cao H-D. and Yau S-T.: Gradients estimates, Harnack inequalities and estimates for heat kernels of sum of squares of vector fields.Math. Zeit. 211, 1992, 485-504. · Zbl 0808.58037 · doi:10.1007/BF02571441
[16] Chanillo S. and Wheeden R.: Harnack’s inequality and mean-value inequalities for solutions of degenerate elliptic equations.Comm. Part. Diff. Equ. 11, 1986, 1111-1134. · Zbl 0634.35035 · doi:10.1080/03605308608820458
[17] Charienza F. and Serapioni R.: A Harnack inequality for degenerate parabolic equations.Comm. Part. Diff. Equ. 9, 1984, 719-749. · Zbl 0546.35035 · doi:10.1080/03605308408820346
[18] Cheeger J., Gromov M. and Taylor M.: Finite propagation speed, kernel estimates for functions of the Laplace operator and the geometry of complete Riemannian manifolds.J. Diff. Geo. 17, 1982, 15-23. · Zbl 0493.53035
[19] Cheng S., Li P. and Yau S-T.: On the upper estimate of the heat kernel of a complete Riemannian manifold.Amer. J. Math. 103, 1981, 1021-1036. · Zbl 0484.53035 · doi:10.2307/2374257
[20] Cheng S. and Yau S-T.: Differential equations on Riemannian manifolds and their geometric applications.Comm. Pure. Appl. Math. 28, 1975, 333-354. · Zbl 0312.53031 · doi:10.1002/cpa.3160280303
[21] Coulhon Th. and Saloff-Coste L. Variétés riemanniennes isométriques à l’infini. 1994.
[22] Davies E.B.: Explicit constants for Gaussian upper bounds on heat kernels.Amer. J. Math. 109, 319-333. · Zbl 0659.35009
[23] Davies E.B.:Heat kernels and spectral theory. Cambridge University Press, 1989. · Zbl 0699.35006
[24] Doob J.L.:Classical potential theory and its probabilistic counterpart. New-York, Springer-Verlag, 1984. · Zbl 0549.31001
[25] Fabes E.: Gaussian upper bounds on fundamental solutions of parabolic equations: the method of Nash. InDirichlet forms, Lect. Not. Math. 1563, Springer-Verlag, 1993, 1-20. · Zbl 0818.35036
[26] Fabes E. and Stroock D.: A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash.Arch. Rat. Mech. Anal. 96, 1986, 327-338. · Zbl 0652.35052 · doi:10.1007/BF00251802
[27] Fefferman C. and Phong D.H: Subelliptic eigenvalue problems. InProceedings of the conference in harmonic analysis in honor of Antoni Zygmund, Wadsworth Math. Ser., Wadsworth, Belmont, California, 1981, 590-606.
[28] Fefferman C. and Sanchez-Calle A.: Fundamental solutions for second order subelliptic operators.Ann. Math. 124, 1986, 247-272. · Zbl 0613.35002 · doi:10.2307/1971278
[29] Fernandes J.: Mean value and Harnack inequalities for certain class of degenerate parabolic equations.Rev. mat. Iberoamericana,7 1991, 247-286. · Zbl 0761.35050
[30] Franchi B.: Weighted Sobolev-Poincaré inequalities and pointwise estimates for a class of degenerate elliptic equations.Trans. Amer. Math. Soc. 327, 1991, 125-158. · Zbl 0751.46023 · doi:10.2307/2001837
[31] Franchi B. and Lanconelli E.: An embedding theorem for Sobolev spaces related to non-smooth vector fields and Harnack inequality.Commm. P.D.E. 9, 1984, 1237-1264. · Zbl 0589.46023 · doi:10.1080/03605308408820362
[32] Franchi B. and Serapioni R.: Pointwise estimates for a class of strongly degenerate elliptic operators: a geometric approach.Ann. Scul. Norm. Sup. Pisa,14, 1987, 527-568. · Zbl 0685.35046
[33] Gilbarg D. and Trudinger N.:Elliptic partial differential equations of second order. Sec. Ed. Berlin, Heidelberg, Springer-Verlag 1983. · Zbl 0562.35001
[34] de Giorgi E.: Sulla differentiabilita el’analiticita delle estremali degli integrali multipli regolariMem. Accad. Sci. Torino, Cl. Sci. Fis. Mat. Nat., Ser 3,3, 1957, 25-43.
[35] Grigory’an A.: The heat equation on noncompact Riemannian manifold.Math. USSR Sbornik,72, 1992, 47-76. · Zbl 0776.58035 · doi:10.1070/SM1992v072n01ABEH001410
[36] Guivarc’h Y.: Croissance polynômiale et période des functions harmoniques.Bull. Soc. Math. France,101, 1973, 149-152.
[37] Gutiérrez E. and Wheeden R.: Mean value and Harnack inequalties for degenerate parabolic equations.Coll. Math. 60, Volume dédié a M. Anton Zygmund, 1990, 157-194. · Zbl 0785.35057
[38] Hadamard J.: Extension à l’équation de la chaleur d’un théorème de A. Harnack.Rend. Circ. Mat. Palermo, Ser. 2,3, 1954, 337-346. · Zbl 0058.32201 · doi:10.1007/BF02849264
[39] Harnack A.:Die Grundlagen der Theorie des logarthmischen Potentials und der eindeutigen Potentialfunktion. Leipzig, Teubner, 1887. · JFM 19.1026.05
[40] Hörmander L.: Hypoelliptic second order differential equations.Acta Math. 119, 1967, 147-171. · Zbl 0156.10701 · doi:10.1007/BF02392081
[41] Jerison D.: The Poincaré inequality for vector fields satisfying the Hörmander’s condition.Duke Math. J. 53, 1986, 503-523. · Zbl 0614.35066 · doi:10.1215/S0012-7094-86-05329-9
[42] Jerison D. and Sanchez-Calle A.: Subelliptic second order differential operators. InComplexe analysis III, Procedings, Lect. Not. Math. 1277, Springer-Verlag, 1986, 47-77.
[43] Krylov N. and Safonov M.: A certain property of solutions of parabolic equations with mesurable coefficients.Izv. Akad. Nauk. SSSR,44, 1980, 81-98.
[44] Kusuoka S. and Stroock D.: Application of Malliavin calculus, part 3.J. Fac. Sci. Univ. Tokyo, Série IA, Math. 34, 1987, 391-442. · Zbl 0633.60078
[45] Kusuoka S. and Stroock D.: Long time estimates for the heat kernel associated with uniformly subelliptic symmetric second order operator.Ann. Math. 127, 1988, 165-189. 391-442. · Zbl 0699.35025 · doi:10.2307/1971418
[46] Li P. and Yau S-T.: On the parabolic kernel of the Schrodinger operator.Acta Math. 156, 1986, 153-201. · Zbl 0611.58045 · doi:10.1007/BF02399203
[47] Lu G.: Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition and applications.Rev. Mat. Iberoamericana,8, 1992, 367-439. · Zbl 0804.35015
[48] Maheux P. and Saloff-Coste L.: Analyse sur les boules d’un opérateur sous-elliptique. Preprint 1994. · Zbl 0836.35106
[49] Moser J.: On Harnack’s Theorem for elliptic differential equations.Comm. Pure Appl. Math. 14, 1961, 577-591. · Zbl 0111.09302 · doi:10.1002/cpa.3160140329
[50] Moser J.: A Harnack inequality for parabolic differential equations.Comm. Pure Appl. Math. 16, 1964, 101-134. Correction in20, 1967, 231-236. · Zbl 0149.06902 · doi:10.1002/cpa.3160170106
[51] Moser J.: On a pointwise estimate for parabolic differential equations.Comm. Pure Appl. Math. 24, 1971, 727-440. · Zbl 0227.35016 · doi:10.1002/cpa.3160240507
[52] Nagel A., Stein E. and Wainger S.: Balls and metrics defined by vector fields.Acta Math. 155, 1985, 103-147. · Zbl 0578.32044 · doi:10.1007/BF02392539
[53] Nash J.: Continuity of solutions of parabolic and elliptic equations.Amer. J. Math. 80, 1958, 931-953. · Zbl 0096.06902 · doi:10.2307/2372841
[54] Oleinik O. and Radkevic E.:Second order equations with nonnegative characteristic form. American Math. Soc., Providence, 1973.
[55] Pini B.: Sulla soluzione generalizzata di Wiener per il primo problema di valori al contorno nel caso parabolico.Rend. Sem. Mat. Padova,23, 1954, 422-434. · Zbl 0057.32801
[56] Porper F. O. and Eidel’man S.D.: Two-sided estimates of fundamental solutions of second-order parabolic equations, and some applications.Russian Math. Surveys,39, 1984, 119-178. · Zbl 0582.35052 · doi:10.1070/RM1984v039n03ABEH003164
[57] Safonov M.N.: Harnack’s inequality for elliptic equations and Hölder property of their solutions.J. Soviet Math. 21, 1983, 851-863. · Zbl 0511.35029 · doi:10.1007/BF01094448
[58] Saloff-Coste L.: Analyse sur les groupes de Lie à croissance polynômiale.Ark. för Mat. 28, 1990, 315-331. · Zbl 0715.43009 · doi:10.1007/BF02387385
[59] Saloff-Coste L.: Opérateurs unfiormèment elliptiques sur les variétés riemanniennes.C. R. Acad. Sci. Paris, Série I, Math. 312, 1991, 25-30. · Zbl 0714.53031
[60] Saloff-Coste L.: Uniformly elliptic operators on Riemannian manifolds.J. Diff. Geo. 36, 1992, 417-450. · Zbl 0735.58032
[61] Saloff-Coste L.: A note on Poincaré, Sobolev, and Harnack inequality.Duke Math. J., I.M.R.N. 2, 1992, 27-38. · Zbl 0769.58054
[62] Saloff-Coste L. and Stroock D.: Opérateurs uniformèment sous-elliptiques sur les groupes de Lie.J. Funct. Anal. 98, 1991, 97-121. · Zbl 0734.58041 · doi:10.1016/0022-1236(91)90092-J
[63] Sturm K-T.: On the geometry defined by Dirichlet forms. Preprint, 1993.
[64] Sturm K-T.: Analysis on local Dirichlet spaces. I. Recurrence, conservativness andL p -Liouville properties. To appear inJ. Reine Angew. Math. 1994.
[65] Sturm K-T.: Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for fundamental solutions of parabolic equations. To appear inOsaka J. Math. 1994.
[66] Sturm K-T.: Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality. Preprint, 1994. · Zbl 0854.35016
[67] Serrin J.: On the Harnack inequality for linear elliptic differential equations.J. Anal. Math. 4, 1954-1956, 292-308. · Zbl 0070.32302 · doi:10.1007/BF02787725
[68] N. Trudinger: On Harnack type inequalities and their application to quasilinear elliptic equations.Comm. Pure Appl. Math,20, 1967, 721-747. · Zbl 0153.42703 · doi:10.1002/cpa.3160200406
[69] N. Trudinger: Pointwise estimates and quasilinear parabolic equations.Comm. Pure Appl. Math,21, 1968, 205-226. · Zbl 0159.39303 · doi:10.1002/cpa.3160210302
[70] N. Trudinger: Linear elliptic operators with measurable coefficients.Ann. Scuola Norm. sup. Pisa,27, 1973, 265-308. · Zbl 0279.35025
[71] Varopoulos N.: Fonctions harmoniques sur les groupes de Lie.C. R. Acad. Sci. Paris, Série I, Math. 309, 1987, 519-521. · Zbl 0614.22002
[72] Varopoulos N.: Analysis on Lie groups.J. Funct. Anal. 76, 1988, 346-410. · Zbl 0634.22008 · doi:10.1016/0022-1236(88)90041-9
[73] Varopoulos N.: Small time Gaussian estimates of the heat diffusion kernel, Part 1: the semigroup technique.Bull. Sci. Math. 113, 1989, 253-277. · Zbl 0703.58052
[74] Varopoulos N.: Opérateurs sous-elliptique du second ordre.C. R. Acad. Sci. Paris,308, Série I, 1989, 437-440. · Zbl 0699.35054
[75] Varopoulos N., Saloff-Coste L. and Coulhon Th.:Analysis and geometry on groups. Cambridge University Press, 1993. · Zbl 1179.22009
[76] Yau S-T.: Harmonic functions on complete Riemannian manifolds.Comm. Pure. Appl. Math. 28, 1975, 201-228. · Zbl 0297.31005 · doi:10.1002/cpa.3160280203
[77] Yau S-T. Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold.Ann. Sci. Ec. Norm. Sup. Paris,8 1975, 487-507. · Zbl 0325.53039
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