84 research outputs found
CHARACTERIZATIONS OF SETS OF FINITE PERIMETER USING HEAT KERNELS IN METRIC SPACES
Abstract. The overarching goal of this paper is to link the notion of sets of finite perimeter (a concept associated with N 1,1 -spaces) and the theory of heat semigroups (a concept related to N 1,2 -spaces) in the setting of metric measure spaces whose measure is doubling and supports a 1-Poincaré inequality. We prove a characterization of sets of finite perimeter in terms of a short time behavior of the heat semigroup in such metric spaces. We also give a new characterization of BV functions in terms of a near-diagonal energy in this general setting
CHARACTERIZATIONS OF SETS OF FINITE PERIMETER USING HEAT KERNELS IN METRIC SPACES
Abstract. The overarching goal of this paper is to link the notion of sets of finite perimeter (a concept associated with N 1,1 -spaces) and the theory of heat semigroups (a concept related to N 1,2 -spaces) in the setting of metric measure spaces whose measure is doubling and supports a 1-Poincaré inequality. We prove a characterization of sets of finite perimeter in terms of a short time behavior of the heat semigroup in such metric spaces. We also give a new characterization of BV functions in terms of a near-diagonal energy in this general setting
Local behavior of p-harmonic Green's functions in metric spaces
We describe the behavior of p-harmonic Green's functions near a singularity
in metric measure spaces equipped with a doubling measure and supporting a
Poincar\'e inequality
Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below
This paper is devoted to a deeper understanding of the heat flow and to the
refinement of calculus tools on metric measure spaces (X,d,m). Our main results
are:
- A general study of the relations between the Hopf-Lax semigroup and
Hamilton-Jacobi equation in metric spaces (X,d).
- The equivalence of the heat flow in L^2(X,m) generated by a suitable
Dirichlet energy and the Wasserstein gradient flow of the relative entropy
functional in the space of probability measures P(X).
- The proof of density in energy of Lipschitz functions in the Sobolev space
W^{1,2}(X,d,m).
- A fine and very general analysis of the differentiability properties of a
large class of Kantorovich potentials, in connection with the optimal transport
problem.
Our results apply in particular to spaces satisfying Ricci curvature bounds
in the sense of Lott & Villani [30] and Sturm [39,40], and require neither the
doubling property nor the validity of the local Poincar\'e inequality.Comment: Minor typos corrected and many small improvements added. Lemma 2.4,
Lemma 2.10, Prop. 5.7, Rem. 5.8, Thm. 6.3 added. Rem. 4.7, Prop. 4.8, Prop.
4.15 and Thm 4.16 augmented/reenforced. Proof of Thm. 4.16 and Lemma 9.6
simplified. Thm. 8.6 corrected. A simpler axiomatization of weak gradients,
still equivalent to all other ones, has been propose
Stability and Continuity of Functions of Least Gradient
In this note we prove that on metric measure spaces, functions of least gradient, as well as local
minimizers of the area functional (after modification on a set of measure zero) are continuous everywhere
outside their jump sets. As a tool, we develop some stability properties of sequences of least gradient functions.
We also apply these tools to prove a maximum principle for functions of least gradient that arise as
solutions to a Dirichlet problem
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