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Sobolev Spaces

This section of the Bempp Handbook introduces the Sobolev function spaces in which the solutions to the variational boundary integral equation are sought.

Let \(\Omega^-\subset\mathbb{R}^3\) be a bounded domain and let \(\Gamma\) be the boundary of \(\Omega^-\). Let \(\Omega^+=\mathbb{R}^3\setminus\Omega^-\) be the region exterior to \(\Omega^-\).

Scalar function spaces

For Laplace and Helmholtz problems, we use spaces containing scalar functions \(f:\Omega^-\to\mathbb{C}\). We begin by defining the space of square integrable functions:

\[ H^0(\Omega^-):=L^2(\Omega^-):=\left\{v:\Omega^-\to\mathbb{C}\middle|\int_{\Omega^-} |v|^2<\infty\right\} \]

We then define the Sobolev space \(H^1(\Omega^-)\) to be the space of square integrable functions whose first derivatives are also square integrable.

\[ H^1(\Omega^-):=\left\{v\in L^2(\Omega^-)\middle|\frac{\partial v}{\partial x}, \frac{\partial v}{\partial y}, \frac{\partial v}{\partial z}\in L^2(\Omega)\right\} \]

In general, for each positive integer \(k\), we define the space \(H^k(\Omega^-)\) to be the space of square integrable functions whose derivatives of order up to and including \(k\) are also square integrable.

Traces

Next, we define the Dirichlet and Neumann traces of a function on the boundary by

\[ (\gamma^-_\text{D}v)(\mathbf{x}):=\lim_{\Omega^-\ni \mathbf{x}'\to \mathbf{x}\in\Gamma}v(\mathbf{x}'),\]
\[ (\gamma^-_\text{N}v)(\mathbf{x}):=\gamma_\text{D}\nabla v(\mathbf{x'})\cdot\mathbf{n}_\mathbf{x}. \]

We define the space \(H^{1/2}(\Gamma)\) to be the Dirichlet trace of the space \(H^1(\Omega^-)\):

\[ H^{1/2}(\Gamma):=\gamma_\text{D}H^1(\Omega^-)=\left\{\gamma_\text{D}v\middle|v\in H^1(\Omega^-)\right\} \]

In general, for each positive integer \(k\), we define the space \(H^{k-1/2}(\Gamma)\) to be the Dirchlet trace of the space \(H^k(\Omega^-)\).

The space \(H^{-1/2}(\Gamma)\) is defined as the dual space of \(H^{1/2}(\Gamma)\):

\[ H^{-1/2}(\Gamma) = \left\{f:H^{1/2}(\Gamma)\to\mathbb{C}\right\}. \]

Vector function spaces

For Maxwell problems, we use spaces containing vector functions \(\mathbf{f}:\Omega^-\to\mathbb{C}^3\). We begin by defining the space of square integrable functions:

\[ \mathbf{H}^0(\Omega^-):=\mathbf{L}^2(\Omega^-):=\left\{\mathbf{v}:\Omega^-\to\mathbb{C}^3\middle|\int_{\Omega^-} |\mathbf{v}|^2<\infty\right\} \]

We then define the Sobolev space \(\mathbf{H}^1(\Omega^-)\) to be the space of square integrable functions whose first derivatives are also square integrable.

\[ \mathbf{H}^1(\Omega^-):=\left\{\mathbf{v}\in \mathbf{L}^2(\Omega^-)\middle|\frac{\partial \mathbf{v}}{\partial x}, \frac{\partial \mathbf{v}}{\partial y}, \frac{\partial \mathbf{v}}{\partial z}\in \mathbf{L}^2(\Omega)\right\} \]

In general, for each positive integer \(k\), we define the space \(\mathbf{H}^k(\Omega^-)\) to be the space of square integrable functions whose derivatives of order up to and including \(k\) are also square integrable.

Traces

On \(\Gamma\), we define the space of square integrable tangential vector fields:

\[ \mathbf{L}^2_\mathbf{t}(\Gamma):=\left\{\mathbf{v}\in\mathbf{L}^2(\Gamma)\middle|\mathbf{v}\cdot\mathbf{n}=0\right\} \]

Next, we define the tangential and Neumann traces of a function on the boundary by

\[ (\mathbf{\gamma}^-_\textbf{t}\mathbf{v})(\mathbf{x}):=\lim_{\Omega^-\ni \mathbf{x}'\to \mathbf{x}\in\Gamma}\mathbf{v}(\mathbf{x}')\times\mathbf{n}_\mathbf{x}, \]
\[ (\mathbf{\gamma}^-_\textbf{N,k}\mathbf{v})(\mathbf{x}):=\frac1{\mathrm{i}k}\mathbf{\gamma}_\textbf{t}\nabla\times\mathbf{v}(\mathbf{x'}). \]

We define the space \(\mathbf{H}^{1/2}_\times(\Gamma)\) to be the tangential trace of the space \(\mathbf{H}^1(\Omega^-)\):

\[ \mathbf{H}^{1/2}_\times(\Gamma):=\mathbf\gamma_\textbf{t}\mathbf{H}^1(\Omega^-)=\left\{\mathbf\gamma_\textbf{t}\mathbf{v}\middle|\mathbf{v}\in \mathbf{H}^1(\Omega^-)\right\} \]

We define the spaces of div- and curl-conforming functions by:

\[ \mathbf{H}^{-1/2}_\times(\operatorname{div}_\Gamma,\Gamma):=\left\{\mathbf{v}\in\mathbf{H}^{1/2}_\times(\Gamma)\middle|\operatorname{div}_\Gamma\mathbf{v}\in H^{1/2}\times(\Gamma)\right\} \]
\[ \mathbf{H}^{-1/2}_\times(\operatorname{curl}_\Gamma,\Gamma):=\left\{\mathbf{v}\in\mathbf{H}^{1/2}_\times(\Gamma)\middle|\operatorname{curl}_\Gamma\mathbf{v}\in H^{1/2}\times(\Gamma)\right\} \]

In these definitions, \(\operatorname{div}_\Gamma\) and \(\operatorname{curl}_\Gamma\) are the scalar surface div and curl operators. Using the definition of these, it can be seen that

\[ \mathbf{H}^{-1/2}_\times(\operatorname{curl}_\Gamma,\Gamma)=\left\{\mathbf{n}\times\mathbf{v}\middle|\mathbf{v}\in\mathbf{H}^{-1/2}_\times(\operatorname{div}_\Gamma,\Gamma)\right\}. \]