A variational principle for a nonlinear differential equation of second order. (Q1415384)

The paper deals with the evolution equation of second order \[ u''+ \partial \phi (u) \ni f,\,\,u(0)=u_0,\,\,u'(0)=u_1, \tag{*} \] where \(\phi :H \to [0,1]\) is a convex lower semi-continuous positive functional with int(dom \(\phi) \neq \emptyset\), \(H\) is a Hilbert space and \(f\) is a bounded vector measure. The existence of generalized solutions with discontinuous velocities is proved. This is done in two ways. The first one uses the sequence of regularized problems \[ u''_{\lambda}+ \partial \phi_{\lambda} (u) = f_{\lambda},\,\,u(0)=u_0,\,\,u'(0)=u_1, \] and applies then compactness arguments derived from a priori estimates. The second one consists in using epi-convergence techniques. Further, it is shown that there exists a variational principle verified by the solution of problem (*). More precisely, it is proved that all solutions of (*) minimize the same functional on a subspace of continuous functions, and that any minimizer of this functional is a solution of problem (*). The continuous depedence on initial data is also studied. Finally, it is proved that every solution of problem (*) can be obtained as limits of smooth approximate solutions. The above problem is stimulated by the paper of \textit{M. Schatzman} [Nonlinear Anal., Theory, Methods Appl. 2, 355--373 (1978; Zbl 0382.34003)] (concerning energy preserving solutions) and the technique developed by \textit{J. J. Moreau} [Unilateral problems in structural analysis, Proc. 2nd Meet., Ravello/Italy 1983, CISM Courses Lect. 288, 173--221 (1985; Zbl 0619.73115)] and \textit{J. J. Moreau} (ed.), \textit{P. D. Panagiotopoulos} (ed.) and \textit{G. Strang} (ed.) [Topics in nonsmooth mechanics. Basel etc.: Birkhäuser Verlag (1988; Zbl 0646.00014)] and \textit{J. J. Morau} [Nonsmooth mechanics and applications, CISM Courses Lect. 302, 1--82 (1988; Zbl 0703.73070)].