Trace operator

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A function defined on a rectangle (top figure, in red), and its trace (bottom figure, in red).

In mathematics, the concept of trace operator plays an important role in studying the existence and uniqueness of solutions to boundary value problems, that is, to partial differential equations with prescribed boundary conditions. The trace operator makes it possible to extend the notion of restriction of a function to the boundary of its domain to "generalized" functions in a Sobolev space.

Informal discussion[edit]

Let be a bounded open set in the Euclidean space with C1 boundary If is a function that is (or even just continuous) on the closure of its function restriction is well-defined and continuous on If however, is the solution to some partial differential equation, it is in general a weak solution, so it belongs to some Sobolev space. Such functions are defined only up to a set of measure zero, and since the boundary does have measure zero, any function in a Sobolev space can be completely redefined on the boundary without changing the function as an element in that space. It follows that simple function restriction cannot be used to meaningfully define what it means for a general solution to a partial differential equation to behave in a prescribed way on the boundary of

The way out of this difficulty is the observation that while an element in a Sobolev space may be ill-defined as a function, can be nevertheless approximated by a sequence of functions defined on the closure of Then, the restriction of to is defined as the limit of the sequence of restrictions .

Construction of the trace operator[edit]

To rigorously define the notion of restriction to a function in a Sobolev space, let be a real number. Consider the linear operator

defined on the set of all functions on the closure of with values in the Lp space given by the formula

The domain of is a subset of the Sobolev space It can be proved that there exists a constant depending only on and such that

for all in

Then, since the functions on are dense in , the operator admits a continuous extension

defined on the entire space is called the trace operator. The restriction (or trace) of a function in is then defined as

This argument can be made more concrete as follows. Given a function in consider a sequence of functions that are on with converging to in the norm of Then, by the above inequality, the sequence will be convergent in Define

It can be shown that this definition is independent of the sequence approximating

Application[edit]

Consider the problem of solving Poisson's equation with zero boundary conditions:

Here, is a given continuous function on

With the help of the concept of trace, define the subspace to be all functions in the Sobolev space (this space is also denoted ) whose trace is zero. Then, the equation above can be given the weak formulation

Find in such that
for all in

Using the Lax–Milgram theorem one can then prove that this equation has precisely one solution, which implies that the original equation has precisely one weak solution.

One can employ similar ideas to prove the existence and uniqueness of solutions for more complicated partial differential equations and with other boundary conditions (such as Neumann and Robin), with the notion of trace playing an important role in all such problems.

References[edit]

  • Evans, Lawrence (1998). Partial differential equations. Providence, R.I.: American Mathematical Society. pp. 257–261. ISBN 0-8218-0772-2.