Minimizing movements and evolution problems in Euclidean spaces
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.
scientific article; zbMATH DE number 1472029