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Uniqueness, Generation of Contraction Semigroups, Error Estimates

Part of the Lectures in Mathematics ETH Zürich book series (LM)

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 but we are not imposing any compactness assumptions on the sublevels of φ. Existence, uniqueness and semigroup properties for minimizing movement uMM(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. 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 \( \overline {D\left( \varphi \right)} \).

Keywords

Variational Inequality Lower Semicontinuity Posteriori Error Estimate Discrete Solution Resolvent Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag AG 2008

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