# Tangential traces We start off from the identity $ \int_{\Omega}\mathbf{u}\cdot\text{curl}\,\mathbf{v} - \mathbf{v}\cdot \text{curl}\, \mathbf{u} = \int_{\Gamma}(\mathbf{u}\times \mathbf{n})\cdot \mathbf{v}|_{\Gamma}\,d\Gamma $ for a bounded domain $\Omega$ with sufficiently smooth boundary $\Gamma$. $u$ and $v$ are vector fields for which we assume that the curl and the above identity is well defined. The above equation motives the definition of the **tangential** **trace operator** $ \gamma_t(\mathbf{u}) = (\mathbf{u}\times \mathbf{n})|_\Gamma. $ Moreover, we can motivate a **tangential component trace operator** $ \pi_t(\mathbf{u}) := \mathbf{u} - (\mathbf{n}\cdot \mathbf{u})\mathbf{n} = \mathbf{n}\times (\mathbf{u}\times \mathbf{n}). $ We have that $ \int_{\Gamma}(\mathbf{u}\times \mathbf{n})\cdot \mathbf{v}|_{\Gamma}\, d\Gamma = \int_{\Gamma}\gamma_t(\mathbf{u})\cdot\pi_t(\mathbf{v})\,d\Gamma $ since $\gamma_t(\mathbf{u})$ is orthogonal to the normal component of $\mathbf{v}|_{\Gamma}$. # Function spaces on smooth domains We denote by $H^s(\Omega)$ the scalar Sobolev space of order $s$ and by $\mathbf{H}^s(\Omega)$ the vectorial Sobolev space of order $s$ with the special case $\mathbf{L}^2(\Omega) = \mathbf{H}^0(\Omega)$. We also require the two domain spaces $ \begin{aligned} \mathbf{H}(\textbf{curl}, \Omega) &:= \{\mathbf{u}\in\mathbf{L}^2(\Omega)|\, \textbf{curl}\, \mathbf{u}\in \mathbf{L}^2(\Omega)\},\\ \mathbf{H}(\textbf{div}, \Omega) &:= \{\mathbf{u}\in\mathbf{L}^2(\Omega)|\, \textbf{div}\, \mathbf{u}\in \mathbf{L}^2(\Omega)\}. \end{aligned} $ On the boundary we define the space of square integrable tangential traces as $ \mathbf{L}_t^2(\Gamma):=\{\mathbf{u}\in \mathbf{L}^2(\Gamma)|\, \mathbf{u}\cdot\mathbf{n} = 0\}. $ For $s\geq 0$ we can also define $ \mathbf{H}_t^s(\Gamma):=\{\mathbf{u}\in \mathbf{H}^s(\Gamma)|\, \mathbf{u}\cdot\mathbf{n} = 0\}. $ For $s< 0$ the spaces $\mathbf{H}_t^s(\Gamma)$ are defined as dual spaces of $\mathbf{H}_t^{-s}(\Gamma)$ with $\mathbf{L}_t^2(\Gamma)$ as pivot space. On $\Gamma$ we also define the two spaces $ \begin{aligned} \mathbf{H}_t^{s}(\textbf{div}_{\Gamma}, \Gamma)&:= \{\mathbf{u}\in \mathbf{H}_t^{s}(\Gamma)|\; \text{div}_{\Gamma}\mathbf{u}\in H^{s}(\Gamma)\}\\ \mathbf{H}_t^{s}(\textbf{curl}_{\Gamma}, \Gamma)&:= \{\mathbf{u}\in \mathbf{H}_t^{s}(\Gamma)|\; \text{curl}_{\Gamma}\mathbf{u}\in {H}^{s}(\Gamma)\}. \end{aligned} $ Here $H^s(\Gamma)$ is the scalar Sobolev of order $s$ with pivot space $L^2(\Gamma)$ defined in the usual way. # Characterising the trace $\gamma_t(\mathbf{u})$ on smooth domains We assume that $\mathbf{u}, \mathbf{v}\in\mathbf{H}^1(\Omega)$ so that the formula $ \int_{\Omega}\mathbf{u}\cdot\text{curl}\,\mathbf{v} - \mathbf{v}\cdot \text{curl}\, \mathbf{u} = \int_{\Gamma}(\mathbf{u}\times \mathbf{n})\cdot \mathbf{v}|_{\Gamma}\,d\Gamma $ makes sense. This motivates that $\gamma_t(\mathbf{u})\in \mathbf{H}^{-1/2}(\Gamma)$ since $\mathbf{v}|_{\Gamma}\in\mathbf{H}^{1/2}(\Gamma)$ by the usual trace theorem. More generally, for $\mathbf{u}\in\mathbf{H}^1(\textbf{curl}, \Omega)$ it can be shown that $\gamma_t(\mathbf{u})\in \mathbf{H}^{-1/2}(\Gamma)$ and that this mapping is continuous. We will also require the following result from the [[Divergence Theorem]] $ \int_{\Omega}\text{div}\,(\mathbf{u}p)\,d\Omega = \int_{\Omega}(\,\text{div}\,\mathbf{u} + \mathbf{u}\cdot\nabla p)\,d\Omega = \int_{\Gamma}p|_{\Gamma}\mathbf{u}\cdot\mathbf{n}\,d\Gamma. $ This motivates that $\mathbf{u}\cdot \mathbf{n}\in{H}^{-1/2}(\Gamma)$ for $\mathbf{u}\in \mathbf{H}(\textbf{div}, \Omega)$. We now again use the divergence theorem to obtain $ \int_{\Omega} \text{curl}\, \mathbf{u}\,\cdot \nabla\phi\,d\Omega = \int_{\Gamma}\phi|_{\Gamma}\,\mathbf{n}\cdot\textbf{curl}\,\mathbf{u}\,d\Gamma $ for $u\in \mathbf{H}(\textbf{curl},\Omega)$ and $\phi\in H^1(\Omega)$. Hence, $\mathbf{n}\cdot\textbf{curl}\,\mathbf{u}\in H^{-1/2}(\Gamma)$. We furthermore have that $ \int_{\Omega}\text{curl}\,\mathbf{u}\cdot\nabla\phi\,d\Omega = -\int_{\Gamma}(\mathbf{u}\times\mathbf{n})\cdot\nabla_{\Gamma}\phi\,d\Gamma. $ Hence, $ -\int_{\Gamma}(\mathbf{u}\times\mathbf{n})\cdot\nabla_{\Gamma}\phi\,d\Gamma = \int_{\Gamma}\phi|_{\Gamma}\,\mathbf{n}\cdot\textbf{curl}\,\mathbf{u}\,d\Gamma $ for $\phi\in H^1(\Omega)$ and $\mathbf{u}\in \mathbf{H}(\textbf{curl}, \Omega)$. We now use duality of $\text{div}_{\Gamma}$ and $\nabla_{\Gamma}$ (see [[Stokes Identities]]) to obtain $ \int_{\Gamma}\text{div}_{\Gamma}\,\gamma_t(\mathbf{u})\cdot \phi|_{\Gamma}\, d\Gamma = \int_{\Gamma}\phi|_{\Gamma}\,\mathbf{n}\cdot\textbf{curl}\,\mathbf{u}\,d\Gamma. $ This shows that $\text{div}_{\Gamma}\,\gamma_t(\mathbf{u})=\mathbf{n}\cdot\textbf{curl}\,\mathbf{u}\in H^{-1/2}(\Gamma)$. Hence, we have that $ \gamma_t(\mathbf{u})\in \mathbf{H}^{-1/2}(\textbf{div}_{\Gamma}, \Omega). $ # Characterising the trace $\pi_t(\mathbf{u})$ on smooth domains From Green’s theorem in its curl form we have with $u=\nabla\phi$ and relabelling $\mathbf{v}$ as $\mathbf{u}$ that $ \begin{aligned} \int_{\Omega}\nabla\phi\cdot\text{curl}\,\mathbf{u}\,d\Omega &= \int_{\Gamma}(\nabla\phi\times \mathbf{n})\pi_t(\mathbf{u})\,d\Gamma\\ &=\int_{\Gamma}\overrightarrow{\text{curl}}_{\Gamma}\,\phi| \,\pi_t(\mathbf{u})\,d\Gamma\\ &=\int_{\Gamma}\phi|_{\Gamma}\,\text{curl}_{\Gamma}\pi_t(\mathbf{u})\,d\Gamma. \end{aligned} $ The second equality is just the definition of the tangential curl and the third equality is the duality of tangential and surface curl. Using as above that $ \int_{\Omega} \text{curl}\, \mathbf{u}\,\cdot \nabla\phi\,d\Omega = \int_{\Gamma}\phi|_{\Gamma}\,\mathbf{n}\cdot\textbf{curl}\,\mathbf{u}\,d\Gamma $ we can see that $ \text{curl}_{\Gamma}\,\pi_t(\mathbf{u})=\text{div}_{\Gamma}\gamma_t(\mathbf{u})\in H^{-1/2}(\Gamma). $ It hence follows that $\text{curl}_{\Gamma}\,\pi_t(\mathbf{u})\in H^{-1/2}(\Gamma)$ and therefore $ \pi_t(\mathbf{u})\in \mathbf{H}^{-1/2}(\textbf{curl}_{\Gamma}, \Omega). $ # Duality of $\mathbf{H}^{-1/2}(\textbf{curl}_{\Gamma}, \Omega)$ and $\mathbf{H}^{-1/2}(\textbf{div}_{\Gamma}, \Omega)$ The duality of these two spaces is a consequence of $ \int_{\Omega}\mathbf{u}\cdot\text{curl}\,\mathbf{v} - \mathbf{v}\cdot \text{curl}\, \mathbf{u} = \int_{\Gamma}\gamma_t(\mathbf{u})\cdot\pi_t(\mathbf{v})\,d\Gamma $ for $\mathbf{u}, \mathbf{v}\in\mathbf{H}(\mathbf{curl}, \Omega)$. For sufficiently smooth boundaries the two trace operators operators are continuous and surjective. Moreoever, the two spaces are identical. # Non-smooth boundaries If $\Gamma$ is not sufficiently smooth the above derivations run into problems. If $\Gamma$ is a polyhedral domain then we only have that $n\in L^{\infty}(\Gamma)$ and the scalar product $\mathbf{n}\cdot \mathbf{u}$ is not defined for $u\in\mathbf{H}^{-1/2}(\Gamma)$. In the following I try to give a rough idea of the ideas presented in beautiful paper by [Buffa, Costabel and Sheen](https://www.sciencedirect.com/science/article/pii/S0022247X02004559) on tangential trace spaces for Lipschitz domains. We define $V= \mathbf{H}^{1/2}(\Gamma)$ and $V’ = \mathbf{H}^{-1/2}(\Gamma)$. Let $\gamma:\mathbf{H}^1(\Omega)\rightarrow V$ be the usual trace operator and denote by $\gamma^{-1}$ its right inverse. We can extend $\gamma_t$ and $\pi_t$ as operators on $V$ by applying $\gamma_t\circ\gamma^{-1}$ and $\pi_t\circ\gamma^{-1}$. This defines the two spaces $V_{\gamma} :=\gamma_t(V)$ and $V_{\pi}:=\pi_t(V)$. We endow them with the usual trace norms $ \begin{aligned} \|\mathbf{\lambda}\|_{V_{\Gamma}} &= \inf_{\mathbf{u}\in V}\{\|\mathbf{u}\|_V: \gamma_t(\mathbf{u}) = \mathbf{\lambda}\},\\ \|\mathbf{\lambda}\|_{V_{\pi}} &= \inf_{\mathbf{u}\in V}\{\|\mathbf{u}\|_V: \pi_t(\mathbf{u}) = \mathbf{\lambda}\}. \end{aligned} $ The spaces $V_{\gamma}$ and $V_{\pi}$ are dense subspaces of $\mathbf{L}^2_t(\Gamma)$. They take the role of tangential trace spaces of order $1/2$, respectively $-1/2$ in the case of $V_{\pi}'$ and $V_{\gamma}’$. Now let $(\mathbf{\tau}_1, \mathbf{\tau}_2, \mathbf{n})$ be a system of orthonormal vectors with $\mathbf{\tau}_1$ and $\mathbf{\tau}_2$ belonging to the tangent plane at a boundary point. This coordinate system can be defined almost everywhere on the boundary thanks to the Lipschitz assumption. We have $ \begin{aligned} \pi_t(\mathbf{u}) &= (\mathbf{u}|_{\Gamma}\cdot\tau_1)\tau_1 + (\mathbf{u}|_{\Gamma}\cdot\tau_2)\tau_2\\ \gamma_t(\mathbf{u}) &= (\mathbf{u}|_{\Gamma}\cdot\tau_2)\tau_1 - (\mathbf{u}|_{\Gamma}\cdot\tau_1)\tau_2.\\ \end{aligned} $ The first equation is a consequence of $\pi_t$ simply preserving the tangential components while $\gamma_t$ rotates the tangential components an angle of $\pi/2$ clockwise. Let us now consider the adoint operation. Since $\pi_t$ is an identity on the tangent plane, its adjoint $i_{\pi}$ also acts like an identity, while the adoint $i_{\gamma}$ of $\gamma_t$ is a rotation by $\pi/2$ anti-clockwise. We hence have that $ \begin{aligned} i_{\pi}(\mathbf{u}) &= (\mathbf{u}|{\Gamma}\cdot\tau_1)\tau_1 + (\mathbf{u}|{\Gamma}\cdot\tau_2)\tau_2\\ i_{\gamma}(\mathbf{u}) &= -(\mathbf{u}|{\Gamma}\cdot\tau_2)\tau_1 + (\mathbf{u}|{\Gamma}\cdot\tau_1)\tau_2\\ \end{aligned} $ for $u_2\in \mathbf{L}_t^2(\Gamma)$. We can now on $\mathbf{L}_t^2$ define a rotation operator $r:=i_{\pi}^{-1}\circ i_{\gamma}$. This rotation operator describes a rotation by $\pi/2$ anti-clockwise, which is identical to the operation $\mathbf{n}\times \mathbf{u}$ for $\mathbf{u}\in L_t^2(\Gamma)$ where $\mathbf{n}$ is well defined. The operator $r$ can now restricted and extended to operators $r:V_{\pi}\rightarrow V_{\gamma}$ and $r:V_{\pi}’\rightarrow V_{\gamma}’$. Note that this makes sense to define the domain and range of $r$ like this. The space $V_{\pi}$ just has the tangential components. The space $V_{\gamma}$ is a rotation of those tangential components and $r$ is just -1 times this rotation. We hence have $ \begin{aligned} \gamma_t(\mathbf{u}) &= -r(\pi_t(\mathbf{u})),\\ \pi_t(\mathbf{u}) &= r(\gamma_t(\mathbf{u}). \end{aligned} $ The technical advantage of this construction is that we have an abstract representation of tangential vector fields and of rotations on those fields through operators that correspond to geometric operations when those are well defined. With suitable definitions of the operators $\text{div}_{\gamma}$ and $\text{curl}_{\Gamma}$ on Lipschitz boundaries we can now define $ \begin{aligned} \mathbf{H}_t^{-1/2}(\textbf{div}_{\Gamma}, \Gamma)&:= \{\mathbf{u}\in V_{\pi}'|\; \text{div}_{\Gamma}\,\mathbf{u}\in H^{-1/2}(\Gamma)\}\\ \mathbf{H}_t^{-1/2}(\textbf{curl}_{\Gamma}, \Gamma)&:= \{\mathbf{u}\in V_{\gamma}'|\; \text{curl}_{\Gamma}\,\mathbf{u}\in {H}^{-1/2}(\Gamma)\}. \end{aligned} $ and obtain that $\gamma_t:\mathbf{H}(\text{curl},\Omega)\rightarrow\mathbf{H}^{-1/2}_t(\mathbf{div},\Gamma)$ $\pi_t:\mathbf{H}(\text{curl},\Omega)\rightarrow\mathbf{H}^{-1/2}_t(\mathbf{curl},\Gamma)$ are linear, continuous and surjective operators. Moreover, we obtain the following useful Hodge decompositions $ \begin{aligned} \mathbf{H}^{-1/2}_t(\mathbf{div},\Gamma) = V_{\gamma} + \overrightarrow{\text{curl}}_{\Gamma}(H^{1/2}(\Gamma)),\\ \mathbf{H}^{-1/2}_t(\mathbf{curl},\Gamma) = V_{\pi} + \nabla_{\Gamma}(H^{1/2}(\Gamma)). \end{aligned} $ # De Rham Complex The full de Rham complex is shown in [[De Rham Sequence for Trace Spaces]] # References - Jean-Claude Nédélec, _[Acoustic and electromagnetic equations](https://link.springer.com/book/10.1007/978-1-4757-4393-7)_ Chapter 5.4.1 has a summary of the main results for smooth domains. - A. Buffa, M. Costabel, D. Sheen, _[On traces for $\mathbf{H}(\textbf{curl}, \Omega)$ spaces in Lipschitz domains](https://www.sciencedirect.com/science/article/pii/S0022247X02004559)_The main reference for results on arbitrary Lipschitz domains - A. Alonso, a. Valli, _[Some remarks on the characterization of the space of tangential traces of $H(\text{rot};\Omega)$ and the construction of an extension operator](https://link.springer.com/article/10.1007/BF02567511)_ Has some good short proofs of some of the results on smooth domains.