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Minimizing movements. (Italian. English summary) Zbl 0957.49029
The notion of minimizing movement has been introduced by De Giorgi and it unifies under a natural framework many problems and techniques in the calculus of variations, differential equations and geometric measure theory. Minimizing movements cover, in particular, as typical cases, the steepest descent method, the approximation of solutions to the heat equation, the mean curvature flow and they are closely related to some penalization methods and $\Gamma$-convergence theory. Roughly speaking, the minimizing movement is defined as the set of pointwise limits (as $\lambda \to +\infty$) of sequences of functions $\{ w^{\lambda} \}_{\lambda}$ which are obtained in the following two steps: for each $\lambda \in (1, +\infty)$, we define reccurently a sequence of minimizers $\{ w(\lambda,k) \}_{k \in \Bbb Z}$ in some topological space $S$, for an appropriately defined sequence of functionals ${\cal F}: (1,+\infty) \times \Bbb Z \times S^{2} \to \overline{\Bbb R}$; using these minimizers we define a sequence of functions $ \{ w^{\lambda} \}_{\lambda}$ on $\Bbb R$ by formula $ \Bbb R \ni s \mapsto w^{\lambda}(s) = w(\lambda, [\lambda s]) \in S$ ($w^{\lambda}$ are piecewise constant on intervals of length $1/\lambda$). The paper deals with problems which can be treated by exploiting the notion of minimizing movements: the Cauchy problem for the gradient inclusion ${\dot{u}}(t) \in - {\overline {\partial}} f(u(t))$ in a Hilbert space and the problem of evolution of surfaces by mean curvature. Some exercises on minimizing movements and their solutions are also provided.

49Q20Variational problems in a geometric measure-theoretic setting
49J45Optimal control problems involving semicontinuity and convergence; relaxation
35K90Abstract parabolic equations
47J20Inequalities involving nonlinear operators