Singularities in boundary value problems. (English) Zbl 0766.35001

Grisvard, P.

Recherches en Mathématiques Appliquées. 22. Paris: Masson. Berlin: Springer-Verlag. xiv, 198 p. (1992).

This book considers second order linear elliptic problems which occur most often in applications, namely various boundary value problems for the Laplace operator, for elasticity and Stokes systems. Evolution problems for the heat and wave equations are also considered. The book is meant to be introductory and self-contained. It begins with some introduction on Sobolev spaces on polygonal domains, as well as trace theorems and Green formula. Then boundary problems for the Laplace operator in \(\mathbb{R}^ 2\) and \(\mathbb{R}^ 3\) are considered. Next the biharmonic problem in \(\mathbb{R}^ 2\) and Kondrat’ev’s method is discussed. Applications are made to elasticity systems in \(\mathbb{R}^ 2\) and \(\mathbb{R}^ 3\), as well as to the heat and wave equations in \(\mathbb{R}^ 2\).

Reviewer: S.Tersian (Russe)

MSC:

35-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
35J25	Boundary value problems for second-order elliptic equations
35L20	Initial-boundary value problems for second-order hyperbolic equations
35K20	Initial-boundary value problems for second-order parabolic equations

Keywords:

heat equation; wave equation; second order linear elliptic problems; Laplace operator; biharmonic problem; Kondrat’ev’s method; elasticity