Minimizing movements and evolution problems in Euclidean spaces
type: Item
instance of: scholarly article
title: Minimizing movements and evolution problems in Euclidean spaces
zbMATH Open document ID: 0973.49008
DOI: 10.1007/BF02505987
author: Massimo Gobbino
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2001-11-27T00:00:00Z
review text: The paper provides new mathematical results in the theory of minimizing movements proposed by E. De Giorgi. The feature of the theory is to build approximate solutions of a large class of evolution problems, typically of steepest descent type, by introducing a time step \(\tau\) and recursively minimizing a suitable functional. In the classical case of the steepest descent problem for a function \(f\) on a metric space \((X,d)\), the functional is \[ G(\tau,u,v):=f(u)+{1\over 2\tau}d^2(u,v) \] and, given \(u_\tau(k)\), \(u_\tau(k+1)\) is obtained by minimizing \(u\mapsto G(\tau,u,v)\) with \(v=u_tau(k)\). The minimizing movement associated to \(G\) is the pointwise limit of the family of functions \(u_\tau([t/\tau])\). Simple examples show that this limit does not exist in general, and this leads to the definition of generalized minimizing movement, the pointwise limit of \(u_{\tau_i}([t/\tau_i])\) for some infinitesimal sequence \(\tau_i\). Another phenomenon which can occur is the fact that not all solutions of the steepest descent Cauchy problem can be obtained by the time discretization. The author shows that in Euclidean spaces, under very general circumstances, any solution can be obtained as generalized minimizing movement of a suitable ``small perturbation'' of \(G\). In the general metric setting the author provides a counterexample showing that the map \(t\mapsto f(u(t))\) is not necessarily nondecreasing on generalized minimizing movements. Therefore, unless additional assumptions are made, a generalized minimizing movement is a very weak solution of the steepest descent problem.
reviewed by: Luigi Ambrosio
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 35D99
Mathematics Subject Classification ID: 35K99
zbMATH DE Number: 1472029
zbMATH Keywords: implicit time discretization
zbMATH Keywords: minimizing movements
zbMATH Keywords: steepest descent equation
MaRDI profile type: Publication
cites work: Curvature-Driven Flows: A Variational Approach
cites work: Q4284701
cites work: Q3670036
cites work: Evolution equations with lack of convexity
cites work: Q4944347
cites work: Evolution problems and minimizing movements
cites work: Q4850854
cites work: Q3477560
description: scientific article; zbMATH DE number 1472029