A variational principle for a nonlinear equation of second order in time (Q2711903)

The author considers the following second-order evolution problem NEWLINE\[NEWLINEu''(t) + \partial \varphi (u(t)) \ni f(t), \text{ a.e. on } [0,T], \qquad u(0)=u_0 ,\quad u'(0)=u_1,NEWLINE\]NEWLINE where \(\varphi :\mathbb{R}^d \to [0,+\infty]\) is convex, lower semicontinuous with intdom \((\varphi)\neq \emptyset\) and \(f\) is a bounded vector measure. He shows that there is a variational principle that underlies the present problem in the sense that every solution with velocity with bounded variation on \((0,T)\) obeys to this variational principle. To do this he first proves the existence of such solutions conserving energy. Some results on approximation and stability are derived.