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)

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.


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