Periodic solutions of Lagrangian systems with Lipschitz obstacle
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.
scientific article; zbMATH DE number 1735272