Notions of Dirichlet problem for functions of least gradient in metric measure spaces
Korte, R., Lahti, P., Li, X., & Shanmugalingam, N. (2019). Notions of Dirichlet problem for functions of least gradient in metric measure spaces. Revista Matematica Iberoamericana, 35(6), 1603-1648. https://doi.org/10.4171/rmi/1095
Published inRevista Matematica Iberoamericana
© 2019 European Mathematical Society
We study two notions of Dirichlet problem associated with BV energy minimizers (also called functions of least gradient) in bounded domains in metric measure spaces whose measure is doubling and supports a (1, 1)-Poincaré inequality. Since one of the two notions is not amenable to the direct method of the calculus of variations, we construct, based on an approach of Juutinen and Mazón-Rossi–De León, solutions by considering the Dirichlet problem for p-harmonic functions, p>1, and letting p→1. Tools developed and used in this paper include the inner perimeter measure of a domain.
PublisherEuropean Mathematical Society Publishing House
Publication in research information system
MetadataShow full item record
Additional information about fundingR. Korte was supported by Academy of Finland grant number 308063, P. Lahti was supported by a grant from the Finnish Cultural Foundation, and X. Li was supported by NNSF of China (No. 11701582). The research of N. Shanmugalingam is partially supported by the grant # DMS–1500440 of NSF (USA).
Showing items with similar title or keywords.
Existence and almost uniqueness for p-harmonic Green functions on bounded domains in metric spaces Björn, Anders; Björn, Jana; Lehrbäck, Juha (Elsevier, 2020)We study (p-harmonic) singular functions, defined by means of upper gradients, in bounded domains in metric measure spaces. It is shown that singular functions exist if and only if the complement of the domain has positive ...
Julin, Vesa; Liimatainen, Tony; Salo, Mikko (International Press, 2017)We show that on any Riemannian manifold with H¨older continuous metric tensor, there exists a p-harmonic coordinate system near any point. When p = n this leads to a useful gauge condition for regularity results in ...
Schultz, Timo (American Mathematical Society (AMS), 2021)In this paper, we prove that a metric measure space which has at least one open set isometric to an interval, and for which the (possibly non-unique) optimal transport map exists from any absolutely continuous measure to ...
Varpanen, Harri (University of Jyväskylä, 2008)
Coulhon, Thierry; Jiang, Renjin; Koskela, Pekka; Sikora, Adam (Elsevier, 2020)Let (X,d,μ) be a doubling metric measure space endowed with a Dirichlet form E deriving from a “carré du champ”. Assume that (X,d,μ,E) supports a scale-invariant L2-Poincaré inequality. In this article, we study the following ...