78. Traces spaces

We are going to show that we can pose Dirichlet boundary conditions of an \(H^1\)-elliptic problem in the space \(H^{1/2}(\partial \Omega)\), and Neumann boundary conditions are allowed exactly in its dual.

78.1. Natural trace space

Let \(\operatorname{tr} u\) be the trace operator taking boundary values. We already know that the trace operator is well defined on \(H^1\), and its range is contained in \(L_2(\partial \Omega)\). We can define the trace norm in abstract form as follows: For a given \(w \in W := \operatorname{tr} H^1\), we set

\[ \| w \|_W = \inf_{v \in H^1 \atop \operatorname{tr} v = w } \| v \|_{H^1} \]

This minimizer is exactly realized by the solution of the Dirichlet problem: Find \(u \in H^1\) such that \(\operatorname{tr} u = w\), and $\( \int \nabla u \nabla v = 0 \qquad \forall \, v \in H_0^1 \)$

The proper norm for Neumann boundary condition is the dual-norm:

\[\begin{eqnarray*} \| g \| & = & \sup_{v \in H^1(\Omega)} \frac{\int_{\Gamma} g \operatorname{tr} v }{\| v\|_{H^1}} \\ & = & \sup_{w \in W} \sup_{v \in H^1 \atop \operatorname{tr} v = w} \frac{\int_{\Gamma} g w }{\| v\|_{H^1}} \\ & = & \sup_{w \in W} \frac{\int_{\Gamma} g w }{\inf_{v \in H^1 \atop \operatorname{tr} v = w} \| v\|_{H^1}} \\ & = & \sup_{w \in W} \frac{\int_{\Gamma} g w }{\| w \|_W} \\ & = & \| g \|_{W^{\ast}} \end{eqnarray*}\]

79. Interpolation space \(H^s\)

Let \(I = (0,\pi)\). For \(s \in (0,1)\) we define the interpolation norm \(H^s(I)\) as interpolation norm between \(H^0(I) = L_2(I)\) and \(H^1(I)\).

We pose the eigen-value problem: Find \(z \in H^1\) such that

\[ (z, v)_{H^1(I)} = \lambda \, (z,v)_{L_2(I)} \qquad \forall \, v \in H^1 \]

We can easily compute the \(L_2\)-orthonormal eigen-functions \(z_k = \frac{\sqrt{2}}{\pi} \cos (k x)\) for \(k \in {\mathbf N}\) and \(z_0 = \frac{1}{\pi}\), and corresponding eigen-values \(\lambda_k = 1 + k^2\). A given function \(u \in L_2(I)\) can be expanded in this eigen-system

\[ u = \sum u_k z_k \]

and its norm is $\( \| u \|_{L_2}^2 = \sum u_k^2. \)$

If \(u\) is also in the sub-space \(H^1 \subset L_2\), then

\[ \| u \|_{H^1}^2 = \sum \lambda_k u_k^2. \]

Indeed, the series on the right hand side converges if and only if \(u \in H^1\).

For \(s \in (0,1)\), we define the \(s\)-norm as

\[ \| u \|_{H^s}^2 = \sum \lambda_k^s u_k^2. \]

This is consistent with the limits \(L_2\) for \(s \rightarrow 0\) and \(H^1\) for \(s \rightarrow 1\).

79.1. Trace norm on bottom edge

On the domain \(\Omega = (0,\pi)^2\) we consider the Laplace equation \(\Delta u = 0\), with Dirichlet boundary conditions \(u = u_D\) on the bottom edge \(I = (0,\pi)\), and homogeneous Neumann boundary conditions else. The canonical trace norm on \(I \subset \partial \Omega\) is

\[ \| u_D \|_W^2 = \inf_{u \in H^1(\Omega) \atop \operatorname{tr} u = u_D} \| u \|_{H^1(\Omega)}^2 \]

We expand the Dirichlet data in the Fourier-cos series:

\[ u_D = \sum_{k=0^\infty} u_k \cos (k x) \]

We use a separation Ansatz to compute the solution.

Exercise: Show that a boundary data \(\cos (kx)\) has the solution

\[ \tilde u_k(x,y) = \cos(kx) \frac{\cosh k(y-\pi)}{\cosh k \pi} \]

Compute the \(H^1\) norm, and verify

\[ \| \tilde u_k \|_{H^1(\Omega)}^2 \approx 1 + k \]

The functions \(\tilde u_k\) are also orthogonal with respect to \((.,.)_{H^1(\Omega)}\).

We have computed that the trace norm of \(u_D\) is

\[\begin{eqnarray*} \| u_D \|_W^2 & = & \sum_{k=0}^\infty u_k^2 \| \tilde u_k \|_{H^1(\Omega)}^2 \\ & \approx & u_k^2 (1+k) \approx u_k^2 (1+k^2)^{1/2} \\ & = & \| u \|_{H^{1/2}(I)}^2 \end{eqnarray*}\]

The trace space has norm \(H^{1/2}(I)\), and the space is the closure of \(H^1(I)\) with respect to this norm.

79.2. Trace norm on boundary sub-domains

For \(\Omega = \Omega_1 \cup \Omega_2\) with disjoint \(\Omega_1\) and \(\Omega_2\), there holds \(\| u \|_{L_2(\Omega)}^2 = \| u \|_{L_2(\Omega_1)}^2 + \| u \|_{L_2(\Omega_2)}^2\), and the same for the \(H^1\)-norm. However, the \(H^{1/2}\)-norm is not additive.

We modify the example on the square \(\Omega = (0,\pi)^2\) such that we prescribe Dirichlet boundary conditions left and right. Then we have to use also some kind of homogeneous boundary conditions for the boundary data on the bottom edge. We use the eigen-system from the Dirichlet-eigenvalue problem: find \(z \in H_0^1(I)\) such that

\[ (z, v)_{H^1} = \lambda (z, v) \qquad \forall \, v \in H_0^1(I) \]

Eigenfunctions are now \(z_k = \sin (k x)\). We can now define the interpolation space by the expansion with \(\sin\)-functions, and the interpolation space as closure of \(H_0^1\) with respect to the interpolation norm.

One obtains that

\[ \| u \|_{[L_2, H_0^1]_{1/2}}^2 = \| u \|_{[L_2, H^1]_{1/2}}^2 + \int_I \frac{1}{\operatorname{dist}(x, \partial I) } u^2 \]

For \(I_1 = (-1,0)\) and \(I_2 = (0,1)\), the \(H^{1/2}\)-norm on \(I=(-1,1)\) is

\[ \| u \|_{H^{1/2}(I)}^2 = \| u \|_{H^{1/2}(I_1)}^2 + \| u \|_{H^{1/2}(I_2)}^2 + \int_0^1 \frac{1}{x} (u(x)-u(-x))^2 \, dx \]

If \(u\) is smooth on both sub-intervals, and can be continuously extended to \(0\), the function must be continuous on \(I\).