On the uniqueness theorem for generalized solutions of initial-boundary problems for the Maguerre-Vlasov vibrations of shallow shells with clamped boundary conditions
The author studies the problem of uniqueness of (weak) solutions of a nonlinear system of evolution equations describing the deformations of elastic shallow shells in the variables \(u\), \(v\) and \(w\). The first two describe the longitudinal point displacements on the middle surface of the shell, and \(w\) describes the tranverse displacements. Such a system is known in the literature as the complete Marguerre-Vlasov nonlinear equations of shallow shells. Global existence of (weak) solutions is already known. Uniqueness is much harder. Essentially because of the nonlinearity it is a critical case taking into account available embedding techniques. In this paper the author uses a new technique (due to him) to show uniqueness. The method relies on the energy inequality in negative spaces and on estimates of the type \[ \max_{x\in \Omega}\bigl |(T_Nh)(x) \bigr |\leq C(\log N)^{1/2}\|h\|_{H^1} \] and \(\|(I-T_N)h\|_{L^2} \leq N^{-1} \|h\|_{H^2}\) for a suitable sequence of operators \(T_N\), \(N=2, 3, \dots,\) where \(B\) is a positive constant (independent of \(N)\). This paper is a good contribution to the subject.
scientific article; zbMATH DE number 1289880