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Elliptic boundary value problems on Lipschitz domains. (English) Zbl 0624.35029
Beijing lectures in harmonic analysis, Ann. Math. Stud. 112, 131-183 (1986).
[For the entire collection see Zbl 0595.00015.]
This paper contains a thorough description of recent developments on the Dirichlet’s and Neumann’s problems for the Laplace operator in regions with Lipschitz boundaries. The author, a major contributor to these topics, is interested in the employ of layer potentials, whose values on the boundary \(\partial D\) of the domain, e.g. \[ Kf(Q)=(1/\omega_ n)\int_{\partial D} (<Q-P,N_ p>/| P-Q|^ n)f(P)d\sigma (P) \] give rise to the operator \(T:=()I+K.\)
It is the inversion of this T that gives the correct density g for the double-layer potential above, so that \[ u(x)=(1/\omega_ n)\int_{\partial D} g(Q)(<X-Q,N_ Q>/| X-Q|^ n)d\sigma (Q) \] with \(g=T^{-1}(f)\) is the solution of \(\Delta u=0\) in D, \(u| \partial D=f\). The operator K is compact in C(\(\partial D)\) if, e.g., \(\partial D\in C^{1,\alpha}\) and the harmonic function u tends to f non-tangentially. The case of Lipschitz \(\partial D\) is the main concern of this work.
The Sections into which the paper is divided are: Preface, Introduction, § 1. Historical comments and preliminaries: (a) The method of layer potentials for Laplace’s equation on smooth domains, (b) The method of harmonic measure, (c) The method of layer potentials revisited. § 2. Laplace’s equation on Lipschitz domains: (a) The \(L^ 2\) theory, (b) The \(L^ p\) theory. § 3. Systems of equations on Lipschitz domains: (a) The systems of elastostatics, (b) The Stokes system of linear hydrostatics.
Reviewer: J.Bouillet

35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35J25 Boundary value problems for second-order elliptic equations
35C15 Integral representations of solutions to PDEs
35J67 Boundary values of solutions to elliptic equations and elliptic systems
Zbl 0595.00015