Application of Abstract Differential Equations to Some Mechanical Problems, Softcover reprint of the original 1st ed. 2003
Mathematics and Its Applications Series, Vol. 558

PREFACE The theory of differential-operator equations has been described in various monographs, but the initial physical problem which leads to these equations is often hidden. When the physical problem is studied, the mathematical proofs are either not given or are quickly explained. In this book, we give a systematic treatment of the partial differential equations which arise in elastostatic problems. In particular, we study problems which are obtained from asymptotic expansion with two scales. Here the methods of operator pencils and differential-operator equations are used. This book is intended for scientists and graduate students in Functional Analy­ sis, Differential Equations, Equations of Mathematical Physics, and related topics. It would undoubtedly be very useful for mechanics and theoretical physicists. We would like to thank Professors S. Yakubov and S. Kamin for helpfull dis­ cussions of some parts of the book. The work on the book was also partially supported by the European Community Program RTN-HPRN-CT-2002-00274. xiii INTRODUCTION In first two sections of the introduction, a classical mathematical problem will be exposed: the Laplace problem. The domain of definition will be, on the first time, an infinite strip and on the second time, a sector. To solve this problem, a well known separation of variables method will be used. In this way, the structure of the solution can be explicitly found. For more details about the separation of variables method exposed in this part, the reader can refer to, for example, the book by D. Leguillon and E. Sanchez-Palencia [LS].

1. General notions, definitions, and results.- 1. Introduction.- 2. General notions from functional analysis.- 2.4 Dual spaces.- 3. Vector-valued functions in Banach spaces.- 4. Semigroup of linear bounded operators in a Banach space.- 5. Differential-operator equations and fold completeness.- 6. Isomorphism and coerciveness.- 7. Interpolation of spaces.- 8. Useful theorems.- 2. Thermal conduction in a half-strip and a sector.- 1. Asymptotic expansion for the thermal conduction in a plate.- 2. Completeness of a system of root functions for the thermal conduction in a half-strip and a sector with smooth coefficients.- 3. Completeness of a system of root functions for the thermal conduction in a half-strip with piecewise smooth coefficients.- 3. Elasticity problems in a half-strip.- 1. Asymptotic expansion for the elasticity in a plate.- 2. Completeness of a system of root functions for elasticity problems in a half-strip.- 3. Thermoelasticity systems in bounded domains with non-smooth boundaries.- 4. Completeness of elementary solutions of problems for second and fourth orders elliptic equations in semi-infinite tube domains.- 1. Abstract results for second order elliptic equations.- 2. Boundary value problems for second order elliptic equations.- 3. Boundary value problems for fourth order elliptic equations.- 5. Basis property of elementary solutions for second order elliptic equations in semi-infinite tube domains.- 1. Abstract results for second order elliptic equations with a selfadjoint operator coefficient.- 4. Boundary value problems for second order elliptic equations.- Problems.- References.- List of notations.- Author index.