Partial Differential Equations/Test functions

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MotivationEdit

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:

 

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 functionsEdit

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 if 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 functionsEdit

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:

  1.  
  2.  

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 functionsEdit

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  
  2.  

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 functionsEdit

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. 

ExercisesEdit

  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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