Variational and Potential Methods for a Class of Linear Hyperbolic Evolutionary Processes

The book presents variational methods combined with boundary integral equation techniques in application to a model of dynamic bending of plates with transverse shear deformation. The emphasis is on the rigorous mathematical investigation of the model, which covers a complete study of the well-posedness of a number of initial-boundary value problems, their reduction to time-dependent boundary integral equations by means of suitable potential representations, and the solution of the latter in Sobolev spaces.

The analysis, performed in spaces of distributions, is applicable to a wide variety of data with less smoothness than that required in the corresponding classical problems, and is very useful for constructing error estimates in numerical computations. The presentation is detailed and clear, yet reasonably concise. This illustrative model was chosen because of its practical importance and some unusual mathematical features, but the solution technique developed in the book can easily be adapted to many other hyperbolic systems of partial differential equations arising in continuum mechanics.

Auteur

Igor Chudinovich is Professor of Mathematics at the University of Guanajuato, Mexico, and Christian Constanda is Oliphant Professor of Mathematical Sciences at the University of Tulsa, USA.

Résumé
Variational and boundary integral equation techniques are two of the most useful methods for solving time-dependent problems described by systems of equations of the form 2 ? u = Au, 2 ?t 2 where u = u(x,t) is a vector-valued function, x is a point in a domain inR or 3 R,and A is a linear elliptic di?erential operator. To facilitate a better und- standing of these two types of methods, below we propose to illustrate their mechanisms in action on a speci?c mathematical model rather than in a more impersonal abstract setting. For this purpose, we have chosen the hyperbolic system of partial di?erential equations governing the nonstationary bending of elastic plates with transverse shear deformation. The reason for our choice is twofold. On the one hand, in a certain sense this is a hybrid system, c- sistingofthreeequationsforthreeunknownfunctionsinonlytwoindependent variables, which makes it more unusualand thereby more interesting to the analystthan other systems arising in solid mechanics. On the other hand, this particular plate model has received very little attention compared to the so-called classical one, based on Kirchho?'s simplifying hypotheses, although, as acknowledged by practitioners, it represents a substantial re?nement of the latter and therefore needs a rigorous discussion of the existence, uniqueness, and continuous dependence of its solution on the data before any construction of numerical approximation algorithms can be contemplated.

Contenu
Formulation of the Problems and Their Nonstationary Boundary Integral Equations.- Problems with Dirichlet Boundary Conditions.- Problems with Neumann Boundary Conditions.- Boundary Integral Equations for Problems with Dirichlet and Neumann Boundary Conditions.- Transmission Problems and Multiply Connected Plates.- Plate Weakened by a Crack.- Initial-Boundary Value Problems with Other Types of Boundary Conditions.- Boundary Integral Equations for Plates on a Generalized Elastic Foundation.- Problems with Nonhomogeneous Equations and Nonhomogeneous Initial Conditions.