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Boundary integral operators on Lipschitz domains: Elementary results. (English) Zbl 0644.35037

Let \(\Omega\in {\mathbb{R}}^ n\) be a bounded Lipschitz domain with boundary \(\Gamma\) and \(P=-\sum^{n}_{j,k=1}\partial_ ja_{jk}\partial_ k+\sum^{n}_{j=1}b_ j\partial_ j+c,\partial_ j=\partial /\partial x_ j\) a strongly elliptic differential operator with \(C^{\infty}({\mathbb{R}}^ n,C)\)-coefficients \(a_{jk}\), \(b_ j\) and c, which has an inverse G on the space of compactly supported distributions. For a locally integrable function v on \(\Gamma\) one can define the simple resp. double layer potential by \[ K_ 0v(x)=\int_{\Gamma}\{...\}\quad resp.\quad K_ 1v(x)=\int_{\Gamma}\partial_{\nu}\{...\},\quad \{...\}=G(x,y)v(y)ds(y), \] where \(\partial_{\nu}=\sum^{n}_{j,k=1}n_ ja_{jk}\partial_ k\) is the conormal derivative and \(n_ j\) are the components of the outward pointing normal vector. The boundary integral operators are defined by taking the boundary data of \(K_ 0\) and \(K_ 1\) in the following way: \(Av=\gamma_ 0K_ 0v,\quad Bv=\gamma_ 1(K_ 0v|_{\Omega}),\quad Cv=\gamma_ 0(K_ 1v|_{\Omega})\) and \(Dv=-{\tilde \gamma}_ 1(K_ 1v|_{\Omega}),\) where \(\gamma_ 0u=u|_{\Gamma}: H\) \(s_{loc}({\mathbb{R}}^ n) \to H^{s- 1/2}(\Gamma)\) is continuous, \(\gamma_ 1u=\partial_{\nu}u|_{\Gamma}\) and \({\tilde \gamma}_ 1u=\gamma_ 1u-\sum^{n}_{j=1}n_ jb_ ju|_{\Gamma}.\)
The author proves the following facts:
(i) the operators \(K_ 0\), \(K_ 1\), A, B, C, D are continuous on certain spaces, for example \(D: H^{1/2+\sigma}(\Gamma)\to H^{- 1/2+\sigma}(\Gamma)\) for all \(\sigma \in (-1/2,1/2).\)
(ii) there exist compact operators \(T_ A: H^{-1/2}(\Gamma)\to H^{1/2}(\Gamma)\), \(T_ D: H^{1/2}(\Gamma)\to H^{-1/2}(\Gamma)\) and constants \(\lambda_ A,\lambda_ D>0\) such that \(Re<(A+T_ A)v,\bar v)\geq \lambda_ A\| v\|^ 2_{H^{-1/2}(\Gamma)}\) for all \(v\in H^{-}(\Gamma)\), \(Re<(D+T_ D)v,\bar v>\geq \lambda_ D\| v\|^ 2_{H^{1/2}(\Gamma)}\) for all \(v\in H^{1/2}(\Gamma)\). The brackets \(<.,.>\) denote the natural duality between the dual spaces \(H^ s(\Gamma)\) and \(H^{-s}(\Gamma).\)
(iii) if \(\psi \in H^{-1/2}(\Gamma)\) and \(v\in H^{1/2}(\Gamma)\) satisfy \(A\psi \in H^{1/2+\sigma}(\Gamma)\) or \(B\psi \in H^{- 1/2+\sigma}(\Gamma)\) and \(Cv\in H^{1/2+\sigma}(\Gamma)\) or \(Dv\in H^{-1/2+\sigma}(\Gamma)\) for \(\sigma\in [0,1/2]\) then \(\psi \in H^{- 1/2+\sigma}(\Gamma)\) and \(v\in H^{1/2+\sigma}(\Gamma)\), and apriori estimates are given.
(iv) if \(A: H^{-1/2}(\Gamma)\to H^{1/2}(\Gamma)\) is injective then for any \(g\in H^{1/2}(\Gamma)\) there is an \(h_ 0>0\) such that for all \(0<h<h_ 0\) the identity \(<Av_ h,w>\equiv <g,w>\) for all \(w\in S^ h\) has a unique solution \(v_ n\in S^ h\), \(\{S^ h\}_{h>0}\) being a family of subspaces of \(H^{-1/2}(\Gamma)\) with the property that the orthogonal projection operators onto \(S^ h\) tend strongly to the identity in \(H^{-1/2}(\Gamma)\) for \(h\to 0\) and there exists a constant C such that for all \(0<h<h_ 0\) holds \(\| v-v_ h\|_{H^{- 1/2}(\Gamma)}\leq C.\inf_{w\in S^ h}\| v-w\|_{H^{-1/2}(G)},\) where \(v\in H^{-1/2}(\Gamma)\) is the unique solution of \(Av=g.\)
(v) let A be injective as in (iv) and \(g\in H\) 1(\(\Gamma)\), then there is a constant C such that \(\| v-v_ h\|_{H^{1/2}(\Gamma)}\leq Ch^{1/2}\| g\|_{H\quad 1(\Gamma)}\) for all \(0<h<h_ 0\).
Reviewer: F.Rühs

MSC:

35J25 Boundary value problems for second-order elliptic equations
58J40 Pseudodifferential and Fourier integral operators on manifolds
65R20 Numerical methods for integral equations
35C15 Integral representations of solutions to PDEs
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