Uniqueness in the elastic bounce problem. II (Q1176495)

In a recent paper [the author, J. Differ. Equations 56, 206-215 (1985; Zbl 0542.73004)] uniqueness for the elastic bounce problem has been studied in a very general framework. More precisely given \(T>0\) and \(f\in C^ 3 (\mathbb{R}^ n; \mathbb{R})\), a pair \((x,U)\in \text{Lip} (0,T; \mathbb{R}^ n)\times L^ 1(0,T; C^ 2 (\mathbb{R}^ n))\) is said to be a solution to the elastic bounce problem iff (i) \(f(x(t)) \geq 0\) in \([0,T]\); (ii) there exists a bounded measure \(\mu\geq 0\) on \([0,T]\) such that \(x(t)\) is an extremal for the functional \(F(y)= \int_ 0^ T \{{1\over 2} |\dot y|^ 2+ U(t, y(t))\}dt+ \int_ 0^ T f(y(t)) d\mu\) and \(\text{spt } \mu\subseteq \{t\in [0,T]\): \(f(x(t))=0\}\); (iii) the function \({\mathcal E}: t\to | \dot x(t)|^ 2\) is continuous on \([0,T]\). The aim of this paper is to show that if \(u\) is assumed to be real analytic and \(U\equiv 0\), then the solution to the elastic bounce problem is unique for every choice of the Cauchy data.