Elliptic boundary value problems on corner domains. Smoothness and asymptotics of solutions.

*(English)*Zbl 0668.35001
Lecture Notes in Mathematics, 1341. Berlin etc.: Springer-Verlag. viii, 259 p. DM.42.50 (1988).

The book under review is a rather complete (and thorough) treatment of elliptic boundary value problems in domains with nonsmooth boundaries. It is divided into four parts.

Part 1 contains a review of previous results and a comparison to some of the author’s original work which is incorporated in the book. The highlight of this part is a very careful definition of the author’s “corner domains” (by recursion on space dimension) which is both precise and general enough so as to include domains with “holes” (single points removed), “slits”, “cracks”, “edges” and “conical points”.

Part 2 is devoted to Dirichlet’s problem both for elliptic equations and strongly elliptic systems. Thanks to the author’s careful definition of corner domains the “splitting” of the resulting operators L by introducing “spherical coordinates” can be described with unusual ease and clarity. Furthermore the author’s idea of replacing “injectivity” by “injectivity modulo polynomials” for the “split operators” is introduced and yields improvements over the classical results of V. A. Kondrat’ev. Namely in great generality the properties of “L is (semi-) Fredholm”, “global regularity holds for L”,... are tied to “injectivity modulo polynomials” properties for the “split operators”. Also “subtracting off singularities” can be described in great generality. In the second half of Part 2 the general machinery is applied to special situations namely i) domain in \({\mathbb{R}}^ 2\); ii) edges modelled over smooth cones; iii) edges in \({\mathbb{R}}^ 3\) meet in a conical point (polyhedra); iv) the Laplacian in various geometries.

Part 3 is devoted to non-Dirichlet-problems in variational setting. It starts with a regularity theory in fractional order Sobolev spaces for smooth operators in smooth domains based on a-priori estimates. This theory is used in treating non-Dirichlet-operators in “polyhedral domains”. (This means that by modelling the corner singularities by recursion on the dimension one ends up with cones with “plane faces”.) Revising the definition of “injectivity modulo polynomials” for the split operators results similar to those of Part 2 are obtained.

Part 4 is in the form of an appendix devoted to four topics: A) Fractional order (weighted) Sobolev spaces: Comparison of the various possible definitions of this space (in particular by using the Mellin transform); B) An abstract lemma connecting the indices of two Fredholm operators differing by “subtracted singularities”. C) An analytic perturbation result for the range of a semi-Fredholm operator. D) An algebraic lemma on polynomials vanishing on the boundary of a cone.

Part 1 contains a review of previous results and a comparison to some of the author’s original work which is incorporated in the book. The highlight of this part is a very careful definition of the author’s “corner domains” (by recursion on space dimension) which is both precise and general enough so as to include domains with “holes” (single points removed), “slits”, “cracks”, “edges” and “conical points”.

Part 2 is devoted to Dirichlet’s problem both for elliptic equations and strongly elliptic systems. Thanks to the author’s careful definition of corner domains the “splitting” of the resulting operators L by introducing “spherical coordinates” can be described with unusual ease and clarity. Furthermore the author’s idea of replacing “injectivity” by “injectivity modulo polynomials” for the “split operators” is introduced and yields improvements over the classical results of V. A. Kondrat’ev. Namely in great generality the properties of “L is (semi-) Fredholm”, “global regularity holds for L”,... are tied to “injectivity modulo polynomials” properties for the “split operators”. Also “subtracting off singularities” can be described in great generality. In the second half of Part 2 the general machinery is applied to special situations namely i) domain in \({\mathbb{R}}^ 2\); ii) edges modelled over smooth cones; iii) edges in \({\mathbb{R}}^ 3\) meet in a conical point (polyhedra); iv) the Laplacian in various geometries.

Part 3 is devoted to non-Dirichlet-problems in variational setting. It starts with a regularity theory in fractional order Sobolev spaces for smooth operators in smooth domains based on a-priori estimates. This theory is used in treating non-Dirichlet-operators in “polyhedral domains”. (This means that by modelling the corner singularities by recursion on the dimension one ends up with cones with “plane faces”.) Revising the definition of “injectivity modulo polynomials” for the split operators results similar to those of Part 2 are obtained.

Part 4 is in the form of an appendix devoted to four topics: A) Fractional order (weighted) Sobolev spaces: Comparison of the various possible definitions of this space (in particular by using the Mellin transform); B) An abstract lemma connecting the indices of two Fredholm operators differing by “subtracted singularities”. C) An analytic perturbation result for the range of a semi-Fredholm operator. D) An algebraic lemma on polynomials vanishing on the boundary of a cone.

Reviewer: N.Week

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35J25 | Boundary value problems for second-order elliptic equations |

35B45 | A priori estimates in context of PDEs |

35J67 | Boundary values of solutions to elliptic equations and elliptic systems |

35B40 | Asymptotic behavior of solutions to PDEs |

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

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