Elliptic boundary value problems in domains with point singularities. (English) Zbl 0947.35004

Kozlov, V. A.; Maz’ya, V. G.; Rossmann, J.

Mathematical Surveys and Monographs. 52. Providence, RI: American Mathematical Society (AMS). ix, 414 p. (1997).

The book gives a self contained, detailed and systematic treatment of elliptic boundary value problems on domains with smooth boundary as well as on domains with singular points. The authors concentrate on three fundamental features: ellipticity condition of the boundary value problem, Fredholm property of the corresponding operator and validity of a priori estimates for solutions in Sobolev spaces of both positive and negative order.
The first part of the book (Chapters 1-4) deals with smoothly bounded domains; the second part (Chapters 5-8) with domains with conical points and the third part with domains whose boundary admits interior or exterior cuspidal points.
The authors start the first, preparatory part with a thorough study of a boundary value problem for operators of the order \(2m\) with constant coefficients on a half line, passing then to periodic problems in more variables and, finally, to a boundary value problem on a half space. These results are then extended to smoothly bounded domains and operators with variable coefficients. The Green function for a boundary value problem is studied and used for the representation of solutions. While in the first three chapters the boundary operators are supposed to have order less then \(2m\), in Chapter 4 this condition is released and boundary operators without any restriction on their order are treated. The equivalence of ellipticity and Fredholm properties is carefully investigated through the whole book.
The second part starts with a study of elliptic boundary value problems in an infinite cylinder \(C = \{ (x,t)\); \(x \in \Omega, t\in \mathbb{R}\}\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^n.\) If the coefficients of the problem do not depend on \(t\), necessary and sufficient conditions for unique sovability of the boundary value problem are given. If the coefficients depend on \(t\) and stabilize in infinity, analogous conditions guarantee Fredholm properties. Moreover, the asymptotic behaviour of solutions in infinity is given.
Chapter 6 deals with domains with conical points. Solvability is studied here in weighted spaces \(V^l_{2,\beta}\) with homogeneus norms (index \(\beta\) characterizes the growth of the solution). Under some additional conditions on the right hand side, it is proved that any solution is a sum of finitely many singular functions and a regular remainder. As the class \(V^l_{2,\beta}\) does not include standard Sobolev spaces without weight, in Chapter 7 analogies of previous results are given in a new class \(W^l_{2,\beta}\), which for \(\beta = 0\) coincides with the Sobolev space \(W^l_2.\) Chapter 8 extends obtained results to boundary value problems for elliptic systems. Special care is given to problems in variational form, Fredholm property and asymptotics of variational solutions near singular points of the boundary.
In Chapter 9 there are treated boundary value problems on domains which can be transformed by a suitable deformation onto a halfcylinder (including cusps or points in infinity).
The book is very well written with a clear and concise style of exposition and elegant proofs. This fact together with the deep and profound content and the excellent choice of material make it a distinguished and valuable reading for researchers and graduate students in the field of partial differential equations.

Reviewer: J.Stará (Praha)

MSC:

35-02	Research exposition (monographs, survey articles) pertaining to partial differential equations
35J40	Boundary value problems for higher-order elliptic equations
35B40	Asymptotic behavior of solutions to PDEs
35D05	Existence of generalized solutions of PDE (MSC2000)
35D10	Regularity of generalized solutions of PDE (MSC2000)
35R05	PDEs with low regular coefficients and/or low regular data

Keywords:

elliptic equations and systems; boundary value problems; point singularity of the boundary; solvability; ellipticity; a priori estimates