Periodic solutions of Lagrangian systems with Lipschitz obstacle
type: Item
instance of: scholarly article
title: Periodic solutions of Lagrangian systems with Lipschitz obstacle
zbMATH Open document ID: 1024.70008
DOI: 10.1016/S0362-546X(00)00237-6
author: Marco Marzocchi
author: Laura Morbini
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-11-24T00:00:00Z
review text: Let \(M\) be closure of an open subset of \(\mathbb{R}^n\) with locally Lipschitz boundary, and let \(L=L(s,q,v): \mathbb{R}\times \mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}\) be a function of class \(C^1\) satisfying \(|D_qL |<C (1+|v |^2)\), \(|D_vL |<1+ |v|\) and \((D_vL(.,.,v) -D_vL(.,.w)) (v-w)>C |v-w|^2\), where \(C>0\). This ensures the existence of the integral \[ f_{a,b}(\gamma)= \int^t_a L\bigl(s,\gamma(s), \gamma'(s) \bigr)ds,\;\gamma\in W^{1,2} (a,b,M). \] Applying subtle devices of nonsmooth analysis, two main results are as follows. Every stationary curve \(\gamma\) is of the class \(W^1_{\text{loc}}\) with \(D_vL(s,\gamma (s),\gamma'(s)) \in BV_{\text{loc}}\) and satisfies the generalized Lagrange system \[ d\bigl(D_v L(s,\gamma, \gamma') \bigr)/ds- D_qL(s,\gamma, \gamma')=\nu d\mu, \] where \(\mu\) is a positive Borel measure and \(\nu\) a bounded Borel function such that \(\nu(s) \in N_{\gamma(s)}\) a.e. (\(N_x\) denotes the Clarke normal cone at \(x\in M)\). If \(M\) is compact, 1-connected and noncontractible in itself and \(L(s+1,.,.)= L(s,. ,.)\), then there exists a sequence of 1-periodic stationary curves \(\mu_n\) such that \(\lim f_{0,1} (\gamma_n)= \infty\). The article is rather brief and technical, however, it provides a remarkable insight into the mechanisms of nonsmooth analysis even for nonspecialists.
reviewed by: Jan Chrastina
Mathematics Subject Classification ID: 70H03
Mathematics Subject Classification ID: 70K42
zbMATH DE Number: 1735272
zbMATH Keywords: Lipschitz obstacle
zbMATH Keywords: nonsmooth critical points
zbMATH Keywords: periodic stationary curves
zbMATH Keywords: generalized Lagrange system
zbMATH Keywords: Clarke normal cone
MaRDI profile type: Publication
cites work: Q4947578
cites work: Periodic solutions of Lagrangian systems on a compact manifold
cites work: Subdifferential Calculus and Nonsmooth Critical Point Theory
cites work: Periodic solutions of Lagrangian systems on manifolds with boundary
cites work: Periodic solutions of quadratic lagrangian systems on $p$-convex sets
cites work: Q4860444
cites work: Variational methods for non-differentiable functionals and their applications to partial differential equations
cites work: Optimization and nonsmooth analysis
cites work: Q3917070
cites work: A critical point theory for nonsmooth functionals
cites work: Multiple solutions of hemivariational inequalities with area-type term
cites work: Closed geodesics with Lipschitz obstacle
cites work: Category of loop spaces of open subsets in euclidean space
cites work: Metric critical point theory. I: Morse regularity and homotopic stability of a minimum
cites work: Q4358798
cites work: Mountain pass theorems and global homeomorphism theorems
cites work: Q3678189
description: scientific article; zbMATH DE number 1735272