Archive for the ‘analysis’ 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.