Partial Differential Equations/Test functions

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Partial Differential Equations
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Motivation[edit]

Before we dive deeply into the chapter, let's first motivate the notion of a test function. Let's consider two functions which are piecewise constant on the intervals and zero elsewhere; like, for example, these two:

Example for step functions 1.svg

Example for step functions 2.svg

Let's call the left function , and the right function .

Of course we can easily see that the two functions are different; they differ on the interval ; however, let's pretend that we are blind and our only way of finding out something about either function is evaluating the integrals

and

for functions in a given set of functions .

We proceed with choosing sufficiently clever such that five evaluations of both integrals suffice to show that . To do so, we first introduce the characteristic function. Let be any set. The characteristic function of is defined as

With this definition, we choose the set of functions as

It is easy to see (see exercise 1), that for , the expression

equals the value of on the interval , and the same is true for . But as both functions are uniquely determined by their values on the intervals (since they are zero everywhere else), we can implement the following equality test:

This obviously needs five evaluations of each integral, as .

Since we used the functions in to test and , we call them test functions. What we ask ourselves now is if this notion generalises from functions like and , which are piecewise constant on certain intervals and zero everywhere else, to continuous functions. The following chapter shows that this is true.

Bump functions[edit]

In order to write down the definition of a bump function more shortly, we need the following two definitions:

Definition 3.1:

Let , and let . We say that is smooth iff all the partial derivatives

exist in all points of and are continuous. We write .

Definition 3.2:

Let . We define the support of , , as follows:

Now we are ready to define a bump function in a brief way:

Definition 3.3:

is called a bump function iff and is compact. The set of all bump functions is denoted by .

These two properties make the function really look like a bump, as the following example shows:

The standard mollifier in dimension

Example 3.4: The standard mollifier , given by

, where , is a bump function (see exercise 2).

Schwartz functions[edit]

As for the bump functions, in order to write down the definition of Schwartz functions shortly, we first need two helpful definitions.

Definition 3.5:

Let be an arbitrary set, and let be a function. Then we define the supremum norm of as follows:

Definition 3.6:

For a vector and a -dimensional multiindex we define , to the power of , as follows:

Now we are ready to define a Schwartz function.

Definition 3.7:

We call a Schwartz function iff the following two conditions are satisfied:

By we mean the function .

Example 3.8: The function

is a Schwartz function.

Theorem 3.9:

Every bump function is also a Schwartz function.

This means for example that the standard mollifier is a Schwartz function.

Proof:

Let be a bump function. Then, by definition of a bump function, . By the definition of bump functions, we choose such that

, as in , a set is compact iff it is closed & bounded. Further, for arbitrary,

Convergence of bump and Schwartz functions[edit]

Now we define what convergence of a sequence of bump (Schwartz) functions to a bump (Schwartz) function means.

Definition 3.10:

A sequence of bump functions is said to converge to another bump function iff the following two conditions are satisfied:

  1. There is a compact set such that

Definition 3.11:

We say that the sequence of Schwartz functions converges to iff the following condition is satisfied:

Theorem 3.12:

Let be an arbitrary sequence of bump functions. If with respect to the notion of convergence for bump functions, then also with respect to the notion of convergence for Schwartz functions.

Proof:

Let be open, and let be a sequence in such that with respect to the notion of convergence of . Let thus be the compact set in which all the are contained. From this also follows that , since otherwise , where is any nonzero value takes outside ; this would contradict with respect to our notion of convergence.

In , ‘compact’ is equivalent to ‘bounded and closed’. Therefore, for an . Therefore, we have for all multiindices :

Therefore the sequence converges with respect to the notion of convergence for Schwartz functions.

The ‘testing’ property of test functions[edit]

In this section, we want to show that we can test equality of continuous functions by evaluating the integrals

and

for all (thus, evaluating the integrals for all will also suffice as due to theorem 3.9).

But before we are able to show that, we need a modified mollifier, where the modification is dependent of a parameter, and two lemmas about that modified mollifier.

Definition 3.13:

For , we define

.

Lemma 3.14:

Let . Then

.

Proof:

From the definition of follows

.

Further, for

Therefore, and since

, we have:

In order to prove the next lemma, we need the following theorem from integration theory:

Theorem 3.15: (Multi-dimensional integration by substitution)

If are open, and is a diffeomorphism, then

We will omit the proof, as understanding it is not very important for understanding this wikibook.

Lemma 3.16:

Let . Then

.

Proof:

Now we are ready to prove the ‘testing’ property of test functions:

Theorem 3.17:

Let be continuous. If

,

then .

Proof:

Let be arbitrary, and let . Since is continuous, there exists a such that

Then we have

Therefore, . An analogous reasoning also shows that . But due to the assumption, we have

As limits in the reals are unique, it follows that , and since was arbitrary, we obtain .

Remark 3.18: Let be continuous. If

,

then .

Proof:

This follows from all bump functions being Schwartz functions, which is why the requirements for theorem 3.17 are met.

Exercises[edit]

  1. Let and be constant on the interval . Show that

  2. Prove that the standard mollifier as defined in example 3.4 is a bump function by proceeding as follows:
    1. Prove that the function

      is contained in .

    2. Prove that the function

      is contained in .

    3. Conclude that .
    4. Prove that is compact by calculating explicitly.
  3. Let be open, let and let . Prove that if , then and .
  4. Let be open, let be bump functions and let . Prove that .
  5. Let be Schwartz functions functions and let . Prove that is a Schwartz function.
  6. Let , let be a polynomial, and let in the sense of Schwartz functions. Prove that in the sense of Schwartz functions.
Partial Differential Equations
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