Strong ellipticity of boundary integral operators. (English) Zbl 0628.35027

Costabel, M.; Wendland, W. L.

J. Reine Angew. Math. 372, 34-63 (1986).

The problem area is that of replacing the boundary value problem \(Pu=f\) in \(\Omega_ 1\), \(R\gamma u=g\) on \(\Gamma\) by a boundary integral equation, where \(\Omega_ 1\) is a bounded domain in \({\mathbb{R}}^ n\) and \(\Gamma\) its boundary. The simplicity of this statement disguises the fact that in the framework of this paper P is a linear differential operator of order 2m which has \(C^{\infty}\) coefficients which can be \(N\times N\) matrices of differential operators which act on the vector valued function u, and the boundary integral is to be interpreted as a pseudo-differential operator.
The paper is in three sections together with an introduction. The first section sets out the assumptions and presents background material. P is an elliptic operator with fundamental solution G, i.e. G is the two sided inverse to P on the space of distributions with compact support and is a pseudo-differential operator of order -2m. There is a representation formula which can be interpreted as a generalization of Green’s third theorem. Included in this section is a generalization of the Plemelj formulae in terms of Calderón projectors.
The following section contains the precise statement of the boundary value problem. The authors set out the assumptions for the subsequent analysis. These include those of the existence of a certain sesquilinear form, the boundedness of a bilinear form for which a Gårding inequality holds. The assumptions are such that P is strongly elliptic.
In the final section the authors set up boundary integral equations in the form of pseudodifferential operator equations. As is well known in the simplest context of Laplace’s equation the types of equation can be found depending on whether single or double layer representations are used. A similar situation arises in the more general situation. The authors choose to consider the equation of the first kind and present results on the relation between solutions of the boundary value problem and the integral equation. The external problem can be included when suitable radiation conditions are imposed.
There are two bilinear forms associated with an elliptic linear boundary value problem in variational form namely the quadratic variational form of the boundary value problem on spaces of functions defined in the domain and also the bilinear form of corresponding boundary integral operators defined on the boundary. The authors examine the question of whether among all possible types of boundary integral equations there will be a special one whose bilinear form coincides with the variational bilinear form of the bilinear problem.
The authors take the reader on a journey across mountain tops, but in the valley one espies the villages of Laplace, Poisson, Helmholtz, Fredholm and Mikhlin which may give sustenance to the traveller.

Reviewer: D.Kershaw

MSC:

35J35	Variational methods for higher-order elliptic equations
35J40	Boundary value problems for higher-order elliptic equations
35S15	Boundary value problems for PDEs with pseudodifferential operators
35A35	Theoretical approximation in context of PDEs
35C15	Integral representations of solutions to PDEs

Keywords:

boundary integral equation; order 2m; fundamental solution; representation formula; Plemelj formulae; Calderón projectors; Gårding inequality; strongly elliptic; single or double layer representations; quadratic variational form; bilinear form