Abstract
Given a compact doubling metric measure space X that supports a 2-Poincaré inequality, we construct a Dirichlet form on \(N^{1,2}(X)\) that is comparable to the upper gradient energy form on \(N^{1,2}(X)\). Our approach is based on the approximation of X by a family of graphs that is doubling and supports a 2-Poincaré inequality (see [20]). We construct a bilinear form on \(N^{1,2}(X)\) using the Dirichlet form on the graph. We show that the \(\Gamma \)-limit \(\mathcal {E}\) of this family of bilinear forms (by taking a subsequence) exists and that \(\mathcal {E}\) is a Dirichlet form on X. Properties of \(\mathcal {E}\) are established. Moreover, we prove that \(\mathcal {E}\) has the property of matching boundary values on a domain \(\Omega \subseteq X\). This construction makes it possible to approximate harmonic functions (with respect to the Dirichlet form \(\mathcal {E}\)) on a domain in X with a prescribed Lipschitz boundary data via a numerical scheme dictated by the approximating Dirichlet forms, which are discrete objects.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alonso-Ruiz, P., Baudoin, F.: Dirichlet forms on metric measure spaces as mosco limits of \(l^2\) korevaar-schoen energies, arXiv:2301.08273 (2023)
Alvarado, R., Hajłasz, P., Malý, L.: A simple proof of reflexivity and separability of \(N^{1, p}\) Sobolev spaces. Ann. Fenn. Math. 48(1), 255–275 (2023)
Beurling, A., Deny, J.: Espaces de Dirichlet. I. Le cas élémentaire. Acta Math. 99, 203–224 (1958)
Beurling, A., Deny, J.: Dirichlet spaces. Proc. Nat. Acad. Sci. U.S.A. 45, 208–215 (1959)
Björn, A., Björn, J.: Nonlinear potential theory on metric spaces, EMS Tracts in Mathematics, vol. 17. European Mathematical Society (EMS), Zürich (2011)
Björn, A., Björn, J., Shanmugalingam, N.: The Dirichlet problem for \(p\)-harmonic functions on metric spaces. J. Reine Angew. Math. 556, 173–203 (2003)
Bonk, M., Saksman, E.: Sobolev spaces and hyperbolic fillings. J. Reine Angew. Math. 737, 161–187 (2018)
Bourdon, M., Pajot, H.: Cohomologie \(l_p\) et espaces de Besov. J. Reine Angew. Math. 558, 85–108 (2003)
Braides, A.: A handbook of \(\Gamma \)-convergence, Handbook of Differential Equations: Stationary Partial Differential Equations, vol. 3, pp. 101–213. North-Holland, (2006)
Carrasco Piaggio, M.: On the conformal gauge of a compact metric space. Ann. Sci. Éc. Norm. Supér (4) 46(3), 495–548 (2013)
Carron, G., Tewodrose, D.: A rigidity result for metric measure spaces with Euclidean heat kernel. J. Éc. polytech. Math. 9, 101–154 (2022)
Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9(3), 428–517 (1999)
Christ, M.: A \(T(b)\) theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. 60/61(2), 601–628 (1990)
Dal Maso, Gianni: An introduction to \(\Gamma \)-convergence, Progress in Nonlinear Differential Equations and their Applications, vol. 8. Birkhäuser Boston Inc, Boston, MA (1993)
De Giorgi, E., Spagnolo, S.: Sulla convergenza degli integrali dell’energia per operatori ellittici del secondo ordine. Boll. Un. Mat. Ital. (4) 8, 391–411 (1973)
Durand-Cartagena, E., Shanmugalingam, N.: An elementary proof of Cheeger’s theorem on reflexivity of Newton-Sobolev spaces of functions in metric measure spaces. J. Anal. 21, 73–83 (2013)
Eriksson-Bique, S., Sarsa, S.: Duality for the gradient of a \(p\)-harmonic function and the existence of gradient curves, in preparation (2024)
Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov processes, extended ed., De Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, (2011)
Gigli, N., Mondino, A., Rajala, T.: Euclidean spaces as weak tangents of infinitesimally Hilbertian metric measure spaces with Ricci curvature bounded below. J. Reine Angew. Math. 705, 233–244 (2015). 3377394
James, T.G., Lopez M.: Discrete approximations of metric measure spaces of controlled geometry. J. Math. Anal. Appl. 431(1), 73–98 (2015)
Grothaus, M., Wittmann, S.: Mosco convergence of gradient forms with non-convex interaction potential, arXiV:2105.05140 (2021)
Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Amer. Math. Soc. 145(688), x+101 (2000)
Heinonen, J.: Lectures on analysis on metric spaces. Universitext, Springer-Verlag, New York (2001)
Heinonen, J., Koskela, P., Shanmugalingam, N., Tyson, J.T.: Sobolev spaces on metric measure spaces: An approach based on upper gradients, New Mathematical Monographs, vol. 27. Cambridge University Press, Cambridge (2015)
Holopainen, I., Soardi, P.M.: \(p\)-harmonic functions on graphs and manifolds. Manuscripta Math. 94(1), 95–110 (1997)
Holopainen, I.: A strong Liouville theorem for \(p\)-harmonic functions on graphs. Ann. Acad. Sci. Fenn. Math. 22(1), 205–226 (1997)
Ishiwata, S., Kawabi, H.: A graph discretized approximation of semigroups for diffusion with drift and killing on a complete riemannian manifold. Math, Ann (2024)
Kumagai, T., Sturm, K.-T.: Construction of diffusion processes on fractals, \(d\)-sets, and general metric measure spaces. J. Math. Kyoto Univ. 45(2), 307–327 (2005)
Kuwae, K., Shioya, T.: Convergence of spectral structures: a functional analytic theory and its applications to spectral geometry. Comm. Anal. Geom. 11(4), 599–673 (2003)
Kuwae, K., Shioya, T.: Variational convergence over metric spaces: Trans. Amer. Math. Soc. 360(1), 35–75 (2008)
Mosco, U.: Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123(2), 368–421 (1994)
Shanmugalingam, N.: Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoamericana 16(2), 243–279 (2000). 1809341
Shanmugalingam, N.: Harmonic functions on metric spaces: Illinois. J. Math. 45(3), 1021–1050 (2001)
Sturm, K.-T.: Diffusion processes and heat kernels on metric spaces. Ann. Probab. 26(1), 1–55 (1998)
Acknowledgements
N.S.’s work is partially supported by the NSF (U.S.A.) grant DMS #2054960. This work was begun during the residency of L.L. and N.S. at the Mathematical Sciences Research Institute (MSRI, Berkeley, CA) as members of the program Analysis and Geometry in Random Spaces which is supported by the National Science Foundation (NSF U.S.A.) under Grant No. 1440140, during Spring 2022. They thank MSRI for its kind hospitality. The authors thank Patricia Alonso-Ruiz and Fabrice Baudoin for sharing with us their early manuscript [1], which helped us work out the nature of the tool of Mosco convergence. The authors also thank the anonymous referee for the kind suggestions that helped improve the exposition of the paper
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Butaev, A., Luo, L. & Shanmugalingam, N. Construction of a Dirichlet form on Metric Measure Spaces of Controlled Geometry. Potential Anal (2024). https://doi.org/10.1007/s11118-024-10144-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11118-024-10144-6
Keywords
- Dirichlet form
- \(N^{1,2}(X)\)
- Graph approximation
- \(\Gamma \)-convergence
- Boundary values
- Doubling spaces
- Poincaré inequality
- Lipschitz approximations