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

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 in R 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 unusual—and thereby more interesting to the analyst—than 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.

Содержание

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.

Об авторе

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.