Investigation of solvability of variational problems in the nonlinear theory of thin shells. (Q1599448)

This is an attempt to establish the solvability of a general boundary value problem of equilibrium of a geometrically and physically nonlinear non-shallow shell (Koiter's version) of nonzero Gaussian curvature with clamped edge. The problem is considered as the problem of minimum of total energy of the shell in some Hilbert spaces of three-component vector functions taking the components of displacement vector as unknowns. The author applies the techniques developed by \textit{I. I. Vorovich} [Functional analysis in mechanics. Springer Monographs in Mathematics. New York, NY: Springer. xi (2003; Zbl 1014.46002)] for nonlinear shallow shells. Particular problems for this type of shells were studied by \textit{I. I. Vorovich} and \textit{Sh. M. Shlafman} [J. Appl. Math. Mech. 38, 894--897 (1974); translation from Prikl. Mat. Mekh. 38, 944--946 (1974; Zbl 0311.73051)] and by \textit{I. I. Vorovich, L. P. Lebedev} and \textit{Sh. M. Shlafman} [J. Appl. Math. Mech. 38, 310--319 (1974); translation from Prikl. Math. Mekh. 38, 339--348 (1974; Zbl 0304.73068)]. Unfortunately, the attempt is not successful. The main difficulty of the proof is to show that the total energy functional is growing. Repeating the ideas of paper by \textit{I. I. Vorovich} and \textit{L. P. Lebedev} [J. Appl. Math. Mech. 36, 652--665 (1972); translation from Prikl. Mat. Mekh. 36, 691--704 (1972; Zbl 0282.73056)], the author considers the energy functional on the ellipsoid. The ellipsoid is obtained by the deformation of the components of displacement \((u_1^*,u_2^*,u_3^*)\) of unit sphere in energy space (the author uses other functions but his transformations are equivalent to what is described) by formulas \(u_1=R u_1^*, u_2=R u_2^*, u_3=R^2 u_3^*\). Then the author states that the structure of the total energy functional with respect to \(R\) is a polynomial of 4th order (page 72). This is correct for the shallow shell theory but it is incorrect for the theory considered by the author. In Shlafman's papers and his Ph.D. thesis, to get this structure of the functional with respect to \(R\), instead of the tangential components of the displacement vector, the angles of rotations of the normal to the mid-surface of the shell were introduced as new unknowns. It seems that this approach could be used for the problem under consideration since the author assumes the Gaussian curvature to be non-vanishing. There are other troubles for the paper. In particular, the reviewer mentions an unclear system of notations and a wrong statement on page 70 that operator \(\operatorname{grad} U(\gamma)\) is monotone.