This book presents rigorous treatment of boundary value problems in nonlinear theory of shallow shells. The consideration of the problems is carried out using methods of nonlinear functional analysis.

,h In the XIX century, mathematical physics continued to be the main source of new partial differential equations and ofproblems involving them. The study ofLaplace's equation and ofthe wave equation had assumed a more systematic nature. In the beginning of the century, Fourier added the heat equation to the aforementioned two. Marvellous progress in obtaining precise solution repre- sentation formulas is connected with Poisson, who obtained formulas for the solution of the Dirichlet problem in a disc, for the solution of the Cauchy problems for the heat equation, and for the three-dimensional wave equation. The physical setting ofthe problem led to the gradual replacement ofthe search for a general solution by the study of boundary value problems, which arose naturallyfrom the physics ofthe problem. Among these, theCauchy problem was of utmost importance. Only in the context of first order equations, the original quest for general integralsjustified itself. Here again the first steps are connected with the names of D'Alembert and Euler; the theory was being intensively 1h developed all through the XIX century, and was brought to an astounding completeness through the efforts ofHamilton, Jacobi, Frobenius, and E. Cartan. In terms of concrete equations, the studies in general rarely concerned equa- tions of higher than second order, and at most in three variables. Classification 'h ofsecond orderequations was undertaken in the second halfofthe XIX century (by Du Bois-Raymond). An increase in the number of variables was not sanc- tioned by applications, and led to the little understood ultra-hyperbolic case.

This book presents rigorous treatment of boundary value problems in nonlinear theory of shallow shells. The consideration of the problems is carried out using methods of nonlinear functional analysis.

Two general questions regarding partial differential equations are explored in detail in this volume of the Encyclopaedia. The first is the Cauchy problem, and its attendant question of well-posedness (or correctness). The authors address this question in the context of PDEs with constant coefficients and more general convolution equations in the first two chapters. The third chapter extends a number of these results to equations with variable coefficients. The second topic is the qualitative theory of second order linear PDEs, in particular, elliptic and parabolic equations. Thus, the second part of the book is primarily a look at the behavior of solutions of these equations. There are versions of the maximum principle, the Phragmen-Lindel]f theorem and Harnack's inequality discussed for both elliptic and parabolic equations. The book is intended for readers who are already familiar with the basic material in the theory of partial differential equations.