## 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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