Uniqueness, Generation of Contraction Semigroups, Error Estimates

Abstract

In all this section we consider the “quadratic” approximation scheme (2.0.3b), (2.0.4) for 2-curves of maximal slope and we identify the “weak” topology σ with the “strong” one induced by the distance d as in Remark 2.1.1: thus we are assuming that

(4.0.1)

but we are not imposing any compactness assumptions on the sublevels of φ. Existence, uniqueness and semigroup properties for minimizing movement u ∈ MM(Φ; u₀) (and not simply the generalized ones, recall Definition 2.0.6) are well known in the case of lower semicontinuous convex functionals in Hilbert spaces [38]. In this framework the resolvent operator in J _τ [·] (3.1.2) is single valued and non expansive, i.e.

(4.0.2)

this property is a key ingredient, as in the celebrated Crandall-Ligget generation Theorem [58], to prove the uniform convergence of the exponential formula (cf. (2.0.9))

$$ u\left( t \right) = \mathop {lim}\limits_{n \to \infty } \left( {J_{{t \mathord{\left/ {\vphantom {t n}} \right. \kern-\nulldelimiterspace} n}} } \right)^n \left[ {u_0 } \right],d\left( {u\left( t \right),\left( {J_{{t \mathord{\left/ {\vphantom {t n}} \right. \kern-\nulldelimiterspace} n}} } \right)^n \left[ {u_0 } \right]} \right) \leqslant \frac{{2\left| {\partial \varphi } \right|\left( {u_0 } \right)t}} {{\sqrt n }}, $$

(4.0.3)

and therefore to define a contraction semigroup on $\overline {D\left( \varphi \right)}$. Being generated by a convex functional, this semigroup exhibits a nice regularizing effect [37], since u(t) ∈ D(|∂φ|) whenever t > 0 even if the starting vale u₀ simply belongs to $\overline {D\left( \varphi \right)}$.