Appl Math Optim 39:309–326 (1999)
1999 Springer-Verlag New York Inc.
On the Uniqueness Theorem for Generalized Solutions of
Initial-Boundary Problems for the Marguerre–Vlasov Vibrations
of Shallow Shells with Clamped Boundary Conditions
V. I. Sedenko
Department of Mathematics, National Academy of Economy,
Bolshaya Sadovaya 69, Rostov-on-Don 344007, Russia
Communicated by I. Lasiecka
Abstract. The uniqueness theorem for generalized solutions of initial-boundary
problems for the Marguerre–Vlasov vibrations of shallow shells with clamped
boundary conditions is proved. A unique method developed by the author, based
upon a nonstandard treatment of smoothing operators, is applied instead of using
an enclosure theorem at the critical values of indices.
Key Words. Generalized solutions, Uniqueness, Estimates, Enclosure theorem,
Smoothing operators, Integral inequalities.
AMS Classiﬁcation. 35L70, 73C50.
In 1957 Vorovich  proved the existence of generalized solutions of initial-boundary
problems for the Marguerre–Vlasov nonlinear equations of shallow shells. An analogous
result was obtained by Morozov  for the oscillations of plates. Recently, the author
has proved a uniqueness theorem for generalized solutions of initial-boundary problems
for the Marguerre–Vlasov-type equations, with the assumption that the inertia of the
longitudinal displacements on the middle surface is small (see  and ). This result
was obtained with the use of a special technique, proposed by the author, that is based on
This research was supported by the Russian Foundation for Fundamental Investigations (RFFI) under