Resolvent Flows for Convex Functionals and p-Harmonic Maps

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(0)-spaces. The results can be applied to Lp-Wasserstein space over complete p-uniformly convex spaces. As an application, we solve an initial boundary value problem for p-harmonic maps into CAT(0)-spaces in terms of Cheeger type p-Sobolev spaces.

Keywords

• cat(0)-space
• cat(k)-space
• p-uniformly convex space
• weak convergence
• p-uniformly λ-convex function
• moreau-yosida approximation
• hamilton-jacobi semi-group
• hopf-lax formula
• resolvent
• local slope
• global slope
• stationary point
• cheeger’s energy
• cheeger type sobolev space
• p-harmonic map
• lp- wasserstein space
• generalized geodesics
• primary 53c20; secondary 53c21, 53c23