## Archive for the ‘pdes’ Tag

### 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 and ,

and its derivatives take on values on

for a domain with boundary , a function , and a function of differential operators . Under “nice” assumptions the PDE has explicit, strong solutions . However, in general, not much can be assumed about . 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 is a weak solution to some PDE defined on a domain with boundary . Since is a solution to the weak formulation of the PDE, we have that , where and is a Soblev space. In particular, consider the Soblev space of functions in whose first derivatives in the sense of distributions are functions in . That is, Note that is a separable Hilbert space, continuously embedded in and that the gradient is continuous from to .

In a boundary value problem, we specify the value or its gradient on . While this poses no issues in classical PDE theory (where and are usually assumed to be ), problems arise in more general settings; since for almost any domain (true so long as isn’t fractal), the value of on is completely arbitrary. We need to define an extension to 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 by a sequence of smooth functions on . 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 the set of functions in with compact support. Using the multiindex notation, denote

for and . Then, for a sequence of functions and a function , we say

if

1) uniformly in for all and;

2) there exists a compact set containing the support of every

We denote the space endowed with this definition of convergence by . Moreover, denote by the set of restrictions to of functions in .

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 and defined as above. The trace operator of on is the linear operator which satisfies

1) for and;

2) , where is a constant and is the size of the space (e.g., ).

Then, the trace of on is , also denoted by .

It can be shown that the trace operator exists (theorem 7.11 in Salsa). It is constructed via the following process: first, consider . We want to be continuous from into . As such, we need to ensure that These are exactly the two conditions given in the definition of the trace operator, but for rather than . Thus, we further need to extend to the whole space . This can be done by exploiting the fact that is dense in . Specifically, this is done with continuous functions that converge to .

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.