The Hoff Equation as a Model of Elastic Shell

The solvability of the Hoff equation modelling the process of I-beam buckling under a constant load and high temperatures is under investigation. This equation is a part of large class of Sobolev type semilinear (we can select the linear and non-linear parts from the operator acting on the original function) equation. G.A. Sviridyuk and his followers in their works research the solvability of the abstract Sobolev type equations in Banach spaces using the phase space method. We consider the Hoff equations on the smooth compact oriented Riemannian manifold without boundary. In this case we understand manifold as the two-side elastic shell. We can reduce this problem to the Cauchy problem for the abstract Sobolev type equation and apply the general theory for it. We reduce it basing on the Sviridyuk theory of relatively p-bounded operators and the Hodge - Kodaira theory of the decomposition of spaces of the differential forms in a direct sum of the subspaces. As a result we obtain a theorem of the simplicity phase space of the Hoff equation in case of contact or not the parameter characterizing the load in the spectrum of the Laplace - Beltrami operator.

Keywords

Sobolev type equation, phase space, Riemannian manifolds, differential k-forms

References

1. Sviridyuk G.A., Kazak V.O. The Phase Space of an Initial-Boundary Value Problem for the Hoff Equation. Mathematical Notes, 2002, vol. 71, no. 1-2, pp. 262-266.
2. Sviridyuk G.A., Shemetova V.V. Hoff Equations on Graphs. Differential Equations, 2006, vol. 42, no. 1, pp. 139-145.
3. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht; Boston; Koln; Tokyo, VSP, 2003. 268 p.
4. K. Moren Metody gil'bertova prostranstva [Hilbert space methods]. Moskow, Mir, 1965. 570 p.