Boundary value problems of finite elasticity. Local theorems on existence, uniqueness, and analytic dependence on data. (English) Zbl 0648.73019

Valent, Tullio

Springer Tracts in Natural Philosophy, Vol. 31, New York etc.: Springer- Verlag. XII, 191 p.; DM 124.00 (1988).

The book under revision is a welcome addition to Continuum Mechanics - Partial Differential Equations literature, since it contains a wealth of results not easily found elsewhere. However it should be said, that the book discusses mainly results obtained by the author and it does not try to be complete.
It treats in systematic form some local theorems on existence, uniqueness and analytical dependence on load for problems of finite elasticity of compressible materials. The function spaces used are Sobolev and Schauder, and the main analytical tool is the implicit function theorem. Problems of dead and live loads are discussed.
The book has six chapters, two appendices, and a short bibliography. Chapter 1 gives a quick survey to the general concepts of elasticity and it is useful only if one already knows the subject. Chapter 2 discusses the analytical basis necessary for the rest of the book, and it is very nice. It gives useful results on Sobolev and Schauder spaces. Chapter 3 discusses the linear problem, necessary for the application of the implicit function theorem. The following three chapters discuss special problems in elastostatics, starting with the boundary problem of place, then the general traction problem, that presents some analytical difficulties, and finally the boundary problem of pressure type.
One of the appendices deals with the implicit function theorem and the other with the representation of orthogonal matrices.

Reviewer: R.Sampaio

MSC:

74B20	Nonlinear elasticity
35A07	Local existence and uniqueness theorems (PDE) (MSC2000)
35B30	Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
74-02	Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
34B99	Boundary value problems for ordinary differential equations

Keywords:

Sobolev space; Schauder space; dead loads; compressible materials; implicit function theorem; live loads; elastostatics; boundary problem of place; general traction problem; boundary problem of pressure type