## Archive for May, 2013|Monthly archive page

### Quick Comments on the Trace of a Function

[Most of this is taken from Salsa, Partial Differential Equations in Action and Evans, Partial Differential Equations. It’s not terribly rigorous, mostly because I don’t really know this stuff yet. ]

Consider a general boundary value problem in ${\mathbf{u}(\mathbf{x}),}$ ${\mathbf{u}=(u_1,u_2,\ldots,u_n)}$ and ${\mathbf{x}=(x_1,x_2,\ldots,x_m)}$,

${\mathcal{L}[\mathbf{u}] = \mathbf{F}(\mathbf{u},\mathbf{u}_{x_1},\mathbf{u}_{x_2},\ldots,\mathbf{x}) \qquad \mathbf{x}\in\Omega}$
${\mathbf{u}}$ and its derivatives take on values on ${\partial\Omega}$

for a domain ${\Omega\in\mathbb{R}^m}$ with boundary ${\partial\Omega}$, a function ${\mathbf{F}}$, and a function of differential operators ${\mathcal{L}}$. Under “nice” assumptions the PDE has explicit, strong solutions ${u}$. However, in general, not much can be assumed about ${u}$. In such cases, we search for weak solutions. This can considerably complicate things. In this post, we discuss one such complication – that of defining solutions and their gradients on the boundary.

In what follows, we drop bold text on vectors and assume it will be clear from context. Assume ${u}$ is a weak solution to some PDE defined on a domain ${\Omega}$ with boundary ${\partial\Omega}$. Since ${u}$ is a solution to the weak formulation of the PDE, we have that ${u\in W(D)}$, where ${D\subseteq\Omega}$ and ${W}$ is a Soblev space. In particular, consider the Soblev space of functions in ${L^2(\Omega)}$ whose first derivatives in the sense of distributions are functions in ${L^2(\Omega)}$. That is, ${W(D)=H^1(\Omega)=\{v\in L^2(\Omega):\nabla v\in L^2(\Omega;\mathbb{R}^n)\}.}$ Note that ${H^1(\Omega)}$ is a separable Hilbert space, continuously embedded in ${L^2(\Omega)}$ and that the gradient is continuous from ${H^1(\Omega)}$ to ${L^2(\Omega;\mathbb{R}^n)}$.

In a boundary value problem, we specify the value ${u}$ or its gradient on ${\partial \Omega}$. While this poses no issues in classical PDE theory (where ${\partial \Omega}$ and ${u}$ are usually assumed to be ${C^1}$), problems arise in more general settings; since ${\mu(\partial\Omega)=0}$ for almost any domain (true so long as ${\partial\Omega}$ isn’t fractal), the value of ${u}$ on ${\partial\Omega}$ is completely arbitrary. We need to define an extension to ${u}$ so that the boundary value problem (and its solution) are better-posed. To do this, we define a trace. Essentially, we do this by approximating ${u}$ by a sequence of smooth functions on ${\bar{\Omega}}$. Since these functions are smooth, we remove the obstacle.

Before defining a trace, we define one more space – that of continuous functions with compact support which converge in a suitable way.

Definition Denote by ${C_0^\infty(\Omega)}$ the set of functions in ${C^\infty(\Omega)}$ with compact support. Using the multiindex notation, denote

$\displaystyle D^\alpha = \frac{\partial^{\alpha_1}}{\partial x_1^{\alpha_1}}\ldots\frac{\partial^{\alpha_n}}{\partial x_n^{\alpha_n}}, \ \ \ \ \ (1)$
for ${\alpha=(\alpha_1,\ldots,\alpha_n)}$ and ${|\alpha|=\alpha_1+\ldots+\alpha_n}$. Then, for a sequence of functions ${\{\phi_k\}\subset C_0^\infty(\Omega)}$ and a function ${\phi\in C_0^\infty(\Omega)}$, we say

$\displaystyle \phi_k\rightarrow\phi \qquad \text{in } C_0^\infty(\Omega) \quad \text{as } k\rightarrow \infty \ \ \ \ \ (2)$
if

1) ${D^\alpha\phi_k\rightarrow D^\alpha\phi}$ uniformly in ${\Omega}$ for all ${\alpha}$ and;
2) there exists a compact set ${K\subset \Omega}$ containing the support of every ${\phi_k}$

We denote the space ${C_0^\infty(\Omega)}$ endowed with this definition of convergence by ${\mathcal{D}(\Omega)}$. Moreover, denote by ${\mathcal{D}(\bar{\Omega})}$ the set of restrictions to ${\bar{\Omega}}$ of functions in ${\mathcal{D}(\mathbb{R}^n)}$.

Now, we have sufficient knowledge to define the trace of a function. (Note: Salsa and Evans take this to be a theorem. It seems just as reasonable to define a trace operator as below and state its existence as a theorem.)

Definition Consider ${u}$ and ${\Omega}$ defined as above. The trace operator of ${u}$ on ${\partial\Omega}$ is the linear operator ${T:H^1(\Omega)\rightarrow L^2(\partial\Omega)}$ which satisfies

1) ${T[u] = u|_{\partial\Omega}}$ for ${u\in \mathcal{D}(\bar{\Omega})}$ and;
2) ${\|T[u]\|_{L^2(\partial\Omega)}\leq c(\Omega,n)\|u\|_{H^1(\Omega)}}$, where ${c}$ is a constant and ${n}$ is the size of the space (e.g., ${\mathbb{R}^n}$).

Then, the trace of ${u}$ on ${\partial\Omega}$ is ${T[u]}$, also denoted by ${u|_{\partial\Omega}}$.

It can be shown that the trace operator exists (theorem 7.11 in Salsa). It is constructed via the following process: first, consider ${T:\mathcal{D}(\bar{\Omega})\rightarrow L^2(\partial\Omega)}$. We want ${T}$ to be continuous from ${\mathcal{D}(\Omega)\subset H^1(\Omega)}$ into ${L^2(\Omega)}$. As such, we need to ensure that ${\|T[u]\|_{L^2(\partial\Omega)}\leq c(\Omega,n)\|u\|_{H_1(\Omega)}.}$ These are exactly the two conditions given in the definition of the trace operator, but for ${\mathcal{D}(\Omega)}$ rather than ${H_1(\Omega)}$. Thus, we further need to extend ${T}$ to the whole space ${H^1(\Omega)}$. This can be done by exploiting the fact that ${\mathcal{D}(\bar{\Omega})}$ is dense in ${H^1(\Omega)}$. Specifically, this is done with continuous functions that converge to ${u}$.

Trace functions, operators, and trace spaces form a rather important part of the theory of elliptic PDEs. While this is only a quick note, any book on the subject will yield much more information on the topic.