Osaka Journal of Mathematics

Scale-invariant boundary Harnack principle in inner uniform domains

Janna Lierl and Laurent Saloff-Coste

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Abstract

We prove a scale-invariant boundary Harnack principle in inner uniform domains in the context of non-symmetric local, regular Dirichlet spaces. For inner uniform Euclidean domains, our results apply to divergence form operators that are not necessarily symmetric, and complement earlier results by H. Aikawa and A. Ancona.

Article information

Source
Osaka J. Math., Volume 51, Number 3 (2014), 619-657.

Dates
First available in Project Euclid: 23 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1414090795

Mathematical Reviews number (MathSciNet)
MR3272609

Zentralblatt MATH identifier
1301.31008

Subjects
Primary: 31C256 35K20: Initial-boundary value problems for second-order parabolic equations 58J35: Heat and other parabolic equation methods
Secondary: 60J60: Diffusion processes [See also 58J65] 31C12: Potential theory on Riemannian manifolds [See also 53C20; for Hodge theory, see 58A14] 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60] 50J45

Citation

Lierl, Janna; Saloff-Coste, Laurent. Scale-invariant boundary Harnack principle in inner uniform domains. Osaka J. Math. 51 (2014), no. 3, 619--657. https://projecteuclid.org/euclid.ojm/1414090795


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References

  • H. Aikawa: Boundary Harnack principle and Martin boundary for a uniform domain, J. Math. Soc. Japan 53 (2001), 119–145.
  • H. Aikawa: Potential-theoretic characterizations of nonsmooth domains, Bull. London Math. Soc. 36 (2004), 469–482.
  • H. Aikawa, T. Lundh and T. Mizutani: Martin boundary of a fractal domain, Potential Anal. 18 (2003), 311–357.
  • 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), 169–213.
  • A. Ancona: Erratum: “Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien, Ann. Inst. Fourier (Grenoble) 28 (1978), 169–213”; in Potential Theory, Copenhagen 1979 (Copenhagen, 1979), Lecture Notes in Math. 787, Springer, Berlin, 1980, 28.
  • A. Ancona: Sur la théorie du potentiel dans les domaines de John, Publ. Mat. 51 (2007), 345–396.
  • Z. Balogh and A. Volberg: Boundary Harnack principle for separated semihyperbolic repellers, harmonic measure applications, Rev. Mat. Iberoamericana 12 (1996), 299–336.
  • Z. Balogh and A. Volberg: Geometric localization, uniformly John property and separated semihyperbolic dynamics, Ark. Mat. 34 (1996), 21–49.
  • R.F. Bass and K. Burdzy: A boundary Harnack principle in twisted Hölder domains, Ann. of Math. (2) 134 (1991), 253–276.
  • R.M. Blumenthal and R.K. Getoor: Markov Processes and Potential Theory, Pure and Applied Mathematics 29, Academic Press, New York, 1968.
  • B.E.J. Dahlberg: Estimates of harmonic measure, Arch. Rational Mech. Anal. 65 (1977), 275–288.
  • N. Eldredge and L. Saloff-Coste: Widder's representation theorem for symmetric local Dirichlet spaces, to appear in J. Theoret. Probab., doi:10.1007/s10959-013-0484-1.
  • M. Fukushima, Y. Ōshima and M. Takeda: Dirichlet Forms and Symmetric Markov Processes, De Gruyter Studies in Mathematics 19, de Gruyter, Berlin, 1994.
  • A. Grigor'yan and L. Saloff-Coste: Dirichlet heat kernel in the exterior of a compact set, Comm. Pure Appl. Math. 55 (2002), 93–133.
  • P. Gyrya and L. Saloff-Coste: Neumann and Dirichlet heat kernels in inner uniform domains, Astérisque 336 (2011).
  • W. Hebisch and L. Saloff-Coste: On the relation between elliptic and parabolic Harnack inequalities, Ann. Inst. Fourier (Grenoble) 51 (2001), 1437–1481.
  • L. Hörmander: Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171.
  • D. Jerison: The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J. 53 (1986), 503–523.
  • D. Jerison and A. Sánchez-Calle: Subelliptic, second order differential operators; in Complex Analysis, III (College Park, Md., 1985–86), Lecture Notes in Math. 1277, Springer, Berlin, 1987, 46–77.
  • J.T. Kemper: A boundary Harnack principle for Lipschitz domains and the principle of positive singularities, Comm. Pure Appl. Math. 25 (1972), 247–255.
  • K. Kuwae, Y. Machigashira and T. Shioya: Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces, Math. Z. 238 (2001), 269–316.
  • J. Lierl: Scale-invariant boundary Harnack principle on inner uniform domains in fractal-type spaces, submitted.
  • J. Lierl and L. Saloff-Coste: Parabolic Harnack inequality for time-dependent non-symmetric Dirichlet forms, submitted, http://arxiv.org/abs/1205.6493v3.
  • J. Lierl and L. Saloff-Coste: The Dirichlet heat kernel in inner uniform domains: local results, compact domains and non-symmetric forms, J. Funct. Anal. 266 (2014), 4189–4235.
  • Z.M. Ma and M. Röckner: Introduction to the Theory of (Nonsymmetric) Dirichlet Forms, Universitext, Springer-Verlag, Berlin, 1992.
  • O. Martio and J. Sarvas: Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn. Ser. A I Math. 4 (1979), 383–401. \interlinepenalty10000
  • U. Mosco: Composite media and asymptotic Dirichlet forms, J. Funct. Anal. 123 (1994), 368–421.
  • A. Nagel, E.M. Stein and S. Wainger: Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), 103–147.
  • Y. Oshima: Semi-Dirichlet Forms and Markov Processes, De Gruyter Studies in Mathematics 48, de Gruyter, Berlin, 2013.
  • L. Saloff-Coste: Aspects of Sobolev-Type Inequalities, London Mathematical Society Lecture Note Series 289, Cambridge Univ. Press, Cambridge, 2002.
  • K.-T. Sturm: Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and $L^{p}$-Liouville properties, J. Reine Angew. Math. 456 (1994), 173–196.
  • K.-T. Sturm: Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations, Osaka J. Math. 32 (1995), 275–312.
  • K.-T. Sturm: On the geometry defined by Dirichlet forms; in Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1993), Progr. Probab. 36, Birkhäuser, Basel, 1995, 231–242.
  • K.T. Sturm: Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality, J. Math. Pures Appl. (9) 75 (1996), 273–297.
  • J. Väisälä: Relatively and inner uniform domains, Conform. Geom. Dyn. 2 (1998), 56–88 (electronic).
  • J.M.G. Wu: Comparisons of kernel functions, boundary Harnack principle and relative Fatou theorem on Lipschitz domains, Ann. Inst. Fourier (Grenoble) 28 (1978), 147–167, vi.