Variational convergence over metric spaces

Abstract

Abstract. We introduce a natural definition of L p -convergence of maps, p ≥ 1, in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a Gromov-Hausdorff convergent sequence. With the L p -convergence, we establish a theory of variational convergences. We prove that the Poincaré inequality with some additional condition implies the asymptotic compactness. The asymptotic compactness is equivalent to the Gromov-Hausdorff compactness of the energy-sublevel sets. Supposing that the targets are CAT(0)-spaces, we study convergence of resolvents. As applications, we investigate the approximating energy functional over a measured metric space and convergence of energy functionals with a lower bound of Ricci curvature.

Resolvent Flows for Convex Functionals and p-Harmonic Maps

“…For any τ > and u ∈ H, there exists a unique minimizer, say J E τ (u) ∈ H, of v → E(v) + pτ p− d p H (u, v). This de nes a map J E τ : H → H, called the resolvent of E (see Propositions 3.26 below and [10,17,22] for the case p = ). The minimum E τ (u) := min v∈H (E(v) + pτ p− d p H (u, v)) is called Moreau-Yosida approximation, Hamilton-Jacobi semigroup or Hopf-Lax formula.…”

Ricci curvature and Lp -convergence

“…In Section 4 we will apply several results given in Section 3 to prove theorems introduced in this section. Moreover we will show a Rellich type compactness result for Sobolev functions with respect to the Gromov-Hausdorff topology which is a generalization of a KuwaeShioya's result about L 2 -energy functionals given in [39,41] to L p -case. See Theorem 4.9 and Remark 4.11.…”

“…On the other hand, we recall the definition of L p -convergence of functions with respect to the Gromov-Hausdorff topology given by Kuwae-Shioya in [39,41]. Note that without loss of generality we can assume that every (1) We say that f i L p -converges to f 1 on…”

“…Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 8/6/15 6:57 AM Note that in [41] Kuwae-Shioya gave the definitions above for the case of r D s D 0 (i.e., each T i is a function) and showed several important properties. A difficulty to give the definitions above for tensor fields is that we cannot consider the difference 'T i T 1 ' canonically because it would be hard to compare between T r s M i and T r s M 1 .…”

“…More precisely in Section 3:1 we will give the definitions of (W) and of (S) for the case of functions by a somewhat different way from Kuwae-Shioya given in [41] and their fundamental properties. We will also show that this formulation is equivalent to that by KuwaeShioya.…”