Resolvent Flows for Convex Functionals and p-Harmonic Maps

Abstract

Abstract:We prove the unique existence of the (non-linear) resolvent associated to a coercive proper lower semicontinuous function satisfying a weak notion of p-uniform λ-convexity on a complete metric space, and establish the existence of the minimizer of such functions as the large time limit of the resolvents, which generalizing pioneering work by Jost for convex functionals on complete CAT( )-spaces. The results can be applied to L p -Wasserstein space over complete p-uniformly convex spaces. As an application, we solve an initial boundary value problem for p-harmonic maps into CAT( )-spaces in terms of Cheeger type p-Sobolev spaces.

Multi-step iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces

“…Kuwae [24] defined another version of p-resolvent operator which is more general than (4) in p-uniformly convex metric spaces as follows:…”

Convex functions and $p$-barycenter on CAT(1)-spaces of small radii

“…Definition 11 (p-uniformly convex space, [NS], [Ku3,Ku2]). A geodesic space, i.e., an ∞-geodesic space, (X, d) is called a p-uniformly convex space for p ≥ 2 if there exists a constant c p > 0 for which…”

“…Ohta [Oh] proved Inequality (10) with p = 2 and the sharp constant k 2 = 2r/ tan r. We refer to Naor-Silberman [NS] and Kuwae [Ku2,Ku3] for puniformly convex spaces. It is not possible to improve the power max{p, 2} to p in Inequality (10), e.g.…”

“…It is not possible to improve the power max{p, 2} to p in Inequality (10), e.g. [NS], [Ku3]. Inequality (10) might be a candidate for a definition of p-uniformly convex spaces when p < 2, but it forces the space to have finite diameter.…”

“…Yang [Ya], is a generalization of Fermat(-Torricelli) points of plane triangles and Steiner points in Sakai [Sa]. For example, p-barycenter appears in the works of Afsari [Af], Naor-Silberman [NS] and Kuwae [Ku2,Ku3].…”