There are some partial differential equations which have no solution. However, some of them have something like ‘almost a solution’, which we call a weak solution. Among these there are partial differential equations whose weak solutions model processes in nature, just like solutions of partial differential equations which have a solution.
These weak solutions will be elements of the so-called Sobolev spaces. By proving properties which elements of Sobolev spaces in general have, we will thus obtain properties of weak solutions to partial differential equations, which therefore are properties of some processes in nature.
In this chapter we do show some properties of elements of Sobolev spaces. Furthermore, we will show that Sobolev spaces are Banach spaces (this will help us in the next section, where we investigate existence and uniqueness of weak solutions).
Let be an open set, and . If is a -dimensional multiindex and such that
, we call an th-weak derivative of .
Remarks 12.2: For a function and a -dimensional multiindex exists at most one th-weak derivative. Furthermore, if exists in the classical sense, has a weak derivative and it is equal to the classical derivative.
Notation 12.3 If it exists, we denote the th-weak derivative of by .
Let be open, , and . Assume that have -weak derivatives, which we - consistent with notation 12.3 - denote by and . Then for all :
Definition and first properties of Sobolev spacesEdit
Before proceeding to the definition and first properties of the Sobolev spaces, we recall the following theorem from integration theory:
For an open set and an arbitrary , the space with the norm
is a Banach space.
We omit the proof.
Now we are ready to define the Sobolev spaces:
Definition and theorem 12.6:
Let be open, , and . The Sobolev space is defined as follows:
A norm on is defined as follows:
With respect to this norm, is a Banach space.
In the above definition, denotes the th-weak derivative of .
We show that
is a norm.
We have to check the three defining properties for a norm:
- for every (absolute homogeneity)
- (triangle inequality)
We start with definiteness: If , then , since all the directional derivatives of the constant zero function are again the zero function. Furthermore, if , then it follows that implying that as is a norm.
We proceed to absolute homogeneity. Let .
And the triangle inequality has to be shown:
We prove that is a Banach space.
Let be a Cauchy sequence in . Since for all -dimensional multiindices with and
since we only added non-negative terms, we obtain that for all -dimensional multiindices with , is a Cauchy sequence in . Since is a Banach space, this sequence converges to a limit in , which we shall denote by .
We show now that and with respect to the norm , thereby showing that is a Banach space.
To do so, we show that for all -dimensional multiindices with the th-weak derivative of is given by . Convergence then automatically follows, as
where in the last line all the summands converge to zero provided that for all -dimensional multiindices with .
Let . Since and by the second triangle inequality
, the sequence is, for large enough , dominated by the function , and the sequence is dominated by the function .
incomplete: Why are the dominating functions L1?
, which is why is the th-weak derivative of for all -dimensional multiindices with .
Approximation by smooth functionsEdit
In the chapter about distributions, an example for a bump function was the standard mollifier, given by
, where .
We can also define mollifiers with different support sizes as follows:
With transformation of variables, we have that
Approximation of continuous functionsEdit
Recall the definition of the multi-dimensional mollifiers. Let be a domain and , i. e. is a continuous function with compact support.
Then and uniformly as .
Proof: First, that follows by the definition of the convolution and the Leibniz integral rule (depending on how you define the convolution, you also need to prove that , so that you apply the derivative to ). This shows the first part.
We further know that is a continuous function. Since has compact support, it is also uniformly continuous. Furthermore:
Since is uniformly continuous, we may choose such that . Therefore, we find, since , the following:
Since the smallest possible decreases with decreasing , and is arbitrary, the claim is proven.
Approximation of Lp-functionsEdit
Let be a domain, and , and let furthermore .
Then and uniformly as .
Proof: The first claim, namely , follows exactly the same way as in the subsubchapter about mollifying continuous functions, by the Leibnis rule for integrating under the integral sign.
We will use that is a dense subset of . This means by definition that
i. e. one can approximate functions in arbitrarily well with functions in .
This is proven by approximating the elementary functions, which are used in the definition of the Lebesgue integral, with continuous functions. It is OK that they have only compact support, because all -functions have the property that for arbitrary there exists a bounded set such that
Let' choose an arbitrary . We choose now first such that .
Then we choose such that .
Third, we notice that . With the triangle inequality and the reversed triangle inequality, we furthermore obtain:
If we choose now such that , we obtain, together with the fact that , that
We now choose and obtain with the triangle inequality:
Since was arbitrary, this finishes the proof.
Let be a domain, let , and , such that . Let furthermore . Then is in for and .
Proof: The first claim, that , follows from the fact that if we choose
Then, due to the above section about mollifying -functions, we know that the first claim is true.
The second claim follows from the following calculation, using the one-dimensional chain rule:
Due to the above secion about mollifying -functions, we immediately know that , and the second statement therefore follows from the definition of the -norm.
Partition of unityEdit
Let be an open set, and be an open cover of . Then there exists a sequence of functions such that the following conditions are satisfied:
Proof: We will proof this by explicitly constructing such a sequence of functions. First, we construct a sequence of closed balls with the properties
In order to do this, we first start with the definition of a sequence compact sets:
This sequence obviously has the properties
We now construct our sequence of closed balls such that
We do this in the following way: To meet the first condition, we first cover with balls by choosing for every a ball such that . Since these balls cover , and is compact, we may choose a finite subcover .
To meet the second condition, we do just the same thing, noticing that is compact and is open.
This sequence of open balls fulfills the conditions which we wanted.
We recall now the Definition of the multi-dimensional mollifier and note that we can recenter it to an arbitrary by using the formula
What we do now, is that for each , we choose as the mollifier which is adjusted to the ball, i. e. the ball is it's support. Then we choose
and we have found a sequence of functions which satisfies the conditions we wanted.
Approximation of Wm, p-functions with smooth functions on open setsEdit
Let be an open set. Then for all functions , there exists a sequence of functions in approximating it.
One sees that the are an open cover of . Therefore, we can choose a sequence of functions (partition of the unity) such that
By defining and
- , we even obtain the properties
where the properties are the same as before except the third property, which changed. Let , be a bump function and be a sequence which approximates in the -norm. The calculation
reveals that, by taking the limit on both sides, implies , since the limit of must be in since we may choose a sequence of bump functions converging to 1.
Let's choose now
We may choose now an arbitrary and so small, that
Let's now define
This function is infinitely often differentiable, since by construction there are only finitely many elements of the sum which do not vanish on each , and also since the elements of the sum are infinitely differentiable due to the Leibniz rule of differentiation under the integral sign. But we also have:
Since was arbitrary, this finishes the proof.
Approximation of Wm, p-functions with smooth functions on compact setsEdit
Let be a bounded domain, and let have the property, that for every point , there is a neighbourhood such that
for a continuous function . Then every function in can be approximated by -functions in the -norm.