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New problems on minimizing movements. (English) Zbl 0851.35052

Lions, Jacques-Louis (ed.) et al., Boundary value problems for partial differential equations and applications. Dedicated to Enrico Magenes on the occasion of his 70th birthday. Paris: Masson. Res. Notes Appl. Math. 29, 81-98 (1993).
This article discusses a relatively new concept of a minimizing movement for a function \(F\) defined on \((1, \infty)\times \mathbb{Z}\times S\times S\) with values in \(\mathbb{R}\cup \{- \infty, \infty\}\), where \(S\) is some set. A minimizing movement is a function \(u: \mathbb{R}\to S\) such that there is a function \(w: (1, \infty)\times S\) such that \(\lim_{\lambda\to \infty} w(\lambda, \lambda t)= u(t)\) for any \(t\in \mathbb{R}\) and such that \[ F(\lambda, k, w(\lambda, k+ 1), w(\lambda, k))= \min_{s\in S} F(\lambda, k, s, w(\lambda, k)) \] for any \(\lambda\in (1, \infty)\) and \(k\in \mathbb{Z}\). A number of examples and conjectures are given to show that minimizing movements occur naturally in studying the calculus of variations, differential equations and partial differential equations, and geometric measure theory. In particular, when \(S= \mathbb{R}^n\), the minimizing movement is often the solution to an ordinary differential equation and when \(S\) is a function space, the minimizing movement is (connected to) the solution of a partial differential equation. Obstacle problems and other more complicated structures are introduced by suitable choices of the function \(F\).
For the entire collection see [Zbl 0782.00097].

MSC:

35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
35A15 Variational methods applied to PDEs
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