×

A least action principle for steepest descent in a non-convex landscape. (English) Zbl 1084.37060

Conca, Carlos (ed.) et al., Partial differential equations and inverse problems. Proceedings of the Pan-American Advanced Studies Institute on partial differential equations, nonlinear analysis and inverse problems, Santiago, Chile, January 6–18, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3448-7/pbk). Contemporary Mathematics 362, 177-187 (2004).
The authors give a variational proof, which involve a dynamically rescaling space, for the existence and uniqueness of the solution to the problem \[ \dot w(t)+\partial W(w(t))\ni 0\,\,\,\text{a.e. on } [0,T],\text{ in } H,\quad w(0)=w_0. \] Here, \(H\) is a Hilbert space and \(W:H\to \mathbb{R}\cup\{+\infty\}\) is a semiconvex function of the form \(W(u)=\bar W(u)-k| u| ^2/2\), for some \(k\geq 0\), where \(\bar W\) is a strictly convex, proper and lower semicontinuous function.
For the entire collection see [Zbl 1052.35004].

MSC:

37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
47J35 Nonlinear evolution equations
35K55 Nonlinear parabolic equations
49N15 Duality theory (optimization)
PDFBibTeX XMLCite