Strongly elliptic systems and boundary integral equations.

*(English)*Zbl 0948.35001
Cambridge: Cambridge University Press. xiv, 357 p. (2000).

Classically the solvability of the Dirichlet and Neumann problems were proved by reformulating them as integral equations of Fredholm type on the boundary. Now, having established the solvability properties of the associated partial differential equations by direct methods (Gårdings inequality!), we can reverse the strategy and derive from them the key properties of the boundary integral equations. The contents of this book. One-third of the text is background material needed to present the main topics: Chapter 1, Introduction; Chapter 2, Functional analysis; Chapter 3, Spobolev spaces on Lipschitz domains; Chapter 5, Homogeneous distributions; Appendices, Calderón’s extension theorem, Interpolation for Sobolev spaces, spherical harmonics. In Chapter 4 the author begins the investigations of strongly elliptic systems. He derives the Gårding inequality (the name “Gårding” does not appear in the text, why?) and gets from it by functional analysis the solvability properties of the Dirichlet and Neumann boundary problems. There are also some standard results on the regularity of solutions including the transmission property. Chapter 6 deals first with parametrics and fundamental solutions, then with the main properties of single- and double-layer potentials, including the familar jump relations. Chapter 7 derives the boundary integral equations for the Dirichlet, Neumann and mixed problems, treating interior as well exterior problems. The Fredholm alternative for the various boundary integral equations is established by showing positive-definiteness up to a compact perturbation, a property that is intimately related to the strong ellipticity of the associated partial differential operator. Chapter 8-10 study three examples of elliptic equations: Laplace equation, Helmholtz equation and the equations of linear elasticity.

Reviewer: Josef Wloka (Kiel)

##### MSC:

35-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations |

35J55 | Systems of elliptic equations, boundary value problems (MSC2000) |

47F05 | General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX) |

47G10 | Integral operators |

47N20 | Applications of operator theory to differential and integral equations |

35B65 | Smoothness and regularity of solutions to PDEs |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

74B05 | Classical linear elasticity |