Ellipticity of boundary integral operators
Contents
2 Boundary integral operator related to the single layer potential
3 Natural density and capacity
4 Hypersingular boundary integral operator
5 Steklov-Poincaré operator
1 Ellipticity and solution uniqueness
Lax-Milgram Lemma is an existence and uniqueness theorem for the solution of an operator equation as well as its equivalent variational formulation, whose key feature is governed by the operator \(A: X \rightarrow X'\) or the corresponding bilinear form \(a(\cdot ,\cdot )\).
The operator equation is \begin{equation} Au = f \end{equation} and its variational formulation is \begin{equation} a(u,v) = \left \langle Au,v \right \rangle = \left \langle f,v \right \rangle \quad \forall v\in X. \end{equation} The boundedness and \(X\)-ellipticity of \(A\) ensures the existence and uniqueness of the solution \(u\). N.B. Ellipticity of an operator is defined with respect to with its domain.
Analogy
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For a matrix equation \(Ax=b\), the ellipticity condition is equivalent to that the minimum eigenvalue of \(A\) is larger than 0 (see here). Therefore, the matrix is invertible and the solution exists and is unique.
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For the simple division in elementary arithmetic, the ellipticity is equivalent to that the denominator is not zero.
2 Boundary integral operator related to the single layer potential
When the spatial dimension \(d=3\), \(V: H^{-1/2}(\Gamma ) \rightarrow H^{1/2}(\Gamma )\) is \(H^{-1/2}(\Gamma )\)-elliptic, i.e. it is elliptic on its whole domain. However, when \(d=2\), \(V\) is only \(H_{\ast }^{-1/2}(\Gamma )\)-elliptic, i.e. it is elliptic on a subspace of its domain, where \begin{equation} H_{\ast }^{-1/2}(\Gamma ) := \left \{ v\in H^{-1/2}(\Gamma ): \left \langle v,1 \right \rangle _{\Gamma }=0 \right \}. \end{equation} The functions in this space are orthogonal to constant functions, or we say the kernel of \(V\) when \(d=2\) is \(\mathrm {span}\{u_0\}\) where \(u_0 \equiv 1\).
3 Natural density and capacity
By introducing a natural density \(w_{\mathrm {eq}}\) and capacity \(\lambda \), it can be proved that when \(\mathrm {diam}(\Omega )< 1\) and \(d=2\), \(V\) is \(H^{-1/2}(\Gamma )\)-elliptic (Steinbach, Theorem 6.23 on P143).
The variational equation for \(w_{\mathrm {eq}}\) is \begin{equation} \label {eq:weq-var-eq} \begin{aligned} \left \langle V w_{\mathrm {eq}},\tau \right \rangle _{\Gamma } - \lambda \left \langle 1,\tau \right \rangle _{\Gamma } &= 0 \\ \left \langle w_{\mathrm {eq}}, 1 \right \rangle _{\Gamma } &= 1 \end{aligned} \quad \forall \tau \in H^{-1/2}(\Gamma ), \end{equation} where \(\lambda \) is the Lagrange multiplier to restrict the test function \(\tau \) within the subspace \(H_{\ast }^{-1/2}(\Gamma )\). This variational equation is equivalent to the following operator equation: \begin{equation} \label {eq:weq-op-eq} \begin{aligned} (V w_{\mathrm {eq}})(x) &= \lambda \\ \left \langle w_{\mathrm {eq}}, 1 \right \rangle _{\Gamma } &= 1 \end{aligned} \quad x\in \Gamma . \end{equation} From the variational equation, we have \(\langle Vw_{\mathrm {eq}},w_{\mathrm {eq}} \rangle _{\Gamma }=\lambda \). We can also derive a normalized version of \(w_{\mathrm {eq}}\): \begin{equation} \tilde {w}_{\mathrm {eq}} := \frac {w_{\mathrm {eq}}}{\lambda }, \end{equation} which satisfies \begin{equation} (V\tilde {w}_{\mathrm {eq}})(x) = 1, \; \langle \tilde {w}_{\mathrm {eq}},1 \rangle _{\Gamma }=\frac {1}{\lambda }. \end{equation}
4 Hypersingular boundary integral operator
Unlike the single layer potential boundary integral operator \(V\), the hypersingular boundary integral operator \(D: H^{1/2}(\Gamma ) \rightarrow H^{-1/2}(\Gamma )\) has a non-trivial kernel \(\mathrm {ker}(D)=\mathrm {span} \{ u_0 \}\), so the ellipticity is not available on the whole space \(H^{1/2}(\Gamma )\) but is only valid in some subspace, which is the orthogonal space of \(\mathrm {ker}(D)\). The concept of orthogonality requires an inner product structure assigned to the space. Because \(H^{1/2}(\Gamma )\) is a Hilbert space, the inner product is available. Different definitions of the inner product for \(H^{1/2}(\Gamma )\) lead to different definitions of orthogonality. Hence, different orthogonal subspaces with respect to \(\mathrm {ker}(D)\) will be obtained. Here we have three choices.
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The inner product is \begin{equation} ( u,v )_{H_{\ast }^{1/2}(\Gamma )} := \langle u,w_{\mathrm {eq}} \rangle _{\Gamma } \cdot \langle v,w_{\mathrm {eq}} \rangle _{\Gamma } + \lvert u \rvert _{H^{1/2}(\Gamma )}\cdot \lvert v \rvert _{H^{1/2}(\Gamma )}, \end{equation} where \(\lvert \cdot \rvert _{H^{1/2}(\Gamma )}\) is the Sobolev-Slobodeckii semi-norm.
The corresponding norm is \begin{equation} \lVert v \rVert _{H_{\ast }^{1/2}(\Gamma )} := \left \{ \langle v,w_{\mathrm {eq}} \rangle _{\Gamma }^{2} + \lvert v \rvert _{H^{1/2}(\Gamma )}^{2} \right \}^{1/2}. \end{equation} This norm is equivalent to the Sobolev norm for \(H^{1/2}(\Gamma )\).
Define the following space \begin{equation} H_{\ast }^{1/2}(\Gamma ) := \{ v\in H^{1/2}(\Gamma ): \langle v,w_{\mathrm {eq}} \rangle _{\Gamma }=0 \}. \end{equation} It can be proved that using the above definition of inner product, \(H_{\ast }^{1/2}(\Gamma )\) is orthogonal to \(\mathrm {ker}(D)\). Therefore, \(D\) is \(H_{\ast }^{1/2}(\Gamma )\)-elliptic.
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The inner product is \begin{equation} ( u,v )_{H_{\ast \ast }^{1/2}(\Gamma )} := \langle u,1 \rangle _{\Gamma }\cdot \langle v,1 \rangle _{\Gamma } + \lvert u \rvert _{H^{1/2}(\Gamma )}\cdot \lvert v \rvert _{H^{1/2}(\Gamma )}. \end{equation} The corresponding norm is \begin{equation} \lVert v \rVert _{H_{\ast \ast }^{1/2}(\Gamma )} := \left \{ \langle v,1 \rangle _{\Gamma }^2 + \lvert v \rvert _{H^{1/2}(\Gamma )}^2 \right \}^{1/2}. \end{equation} This norm is equivalent to the Sobolev norm for \(H^{1/2}(\Gamma )\).
Define the following space \begin{equation} H_{\ast \ast }^{1/2}(\Gamma ) := \{ v\in H^{1/2}(\Gamma ): \langle v,1 \rangle _{\Gamma }=0 \}. \end{equation} This space is orthogonal to \(\mathrm {ker}(D)\) and \(D\) is also \(H_{\ast \ast }^{1/2}(\Gamma )\)-elliptic.
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Let \(\Gamma _0\) be an open set contained in \(\Gamma \) and we consider functions on \(\Gamma _0\) instead of on the whole boundary, i.e. the function space is \(\tilde {H}^{1/2}(\Gamma _0)\).
The inner product is \begin{equation} ( u,v )_{H^{1/2}(\Gamma ),\Gamma _0} := \lVert u \rVert _{L_2(\Gamma \backslash \Gamma _0)} \cdot \lVert v \rVert _{L_2(\Gamma \backslash \Gamma _0)} + \lvert u \rvert _{H^{1/2}(\Gamma )}\cdot \lvert v \rvert _{H^{1/2}(\Gamma )}. \end{equation} Here \(u\) and \(v\) belongs to \(H^{1/2}(\Gamma )\). When they are only defined on \(\Gamma _0\) and belong to \(\tilde {H}^{1/2}(\Gamma _0)\), we need to apply the canonical extension to them, i.e. extension by zero while preserving the resulted function within the space \(H^{1/2}(\Gamma )\). \begin{equation} \tilde {v} = \begin {cases} v(x) & x\in \Gamma _0 \\ 0 & x\in \Gamma \backslash \Gamma _0 \end {cases}. \end{equation} Then we define the inner product of \(u\) and \(v\) to be equal to \(( \tilde {u},\tilde {v} )_{H^{1/2}(\Gamma ),\Gamma _0}\).
The corresponding norm is \begin{equation} \lVert v \rVert _{H^{1/2}(\Gamma ),\Gamma _0} := \left \{ \lVert v \rVert _{L_2(\Gamma \backslash \Gamma _0)}^2 + \lvert v \rvert _{H^{1/2}(\Gamma )}^2 \right \}^{1/2}. \end{equation} This norm is equivalent to the Sobolev norm for \(\tilde {H}^{1/2}(\Gamma _0)\).
The space \(\tilde {H}^{1/2}(\Gamma _0)\) is orthogonal to \(\mathrm {ker}(D)\) and \(D\) is \(\tilde {H}^{1/2}(\Gamma _0)\)-elliptic.
To prove the ellipticity of \(D\) either on the whole space or a subspace, the Sobolev norm should be used. The above definitions of norm play the role of scaffold during such a proof.
Orthogonality defined based on different definitions of inner product lead to ellipticity of \(D\) with respect to different subspaces which are orthogonal to \(\mathrm {ker}(D)\)
5 Steklov-Poincaré operator
The symmetric form of the Steklov-Poincaré operator is \begin{equation} S := D + (\sigma I + K') V^{-1} (\sigma I + K). \end{equation} \(V^{-1}\) is \(H^{1/2}(\Gamma )\)-elliptic and the ellipticity of \(S\) is the same as \(D\).
References
Olaf Steinbach. Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements. Springer Science & Business Media. ISBN 978-0-387-31312-2.