Partial Differential Equations/Sobolev spaces
In this chapter, it will be defined, what Sobolev spaces are. Also, some properties of Sobolev spaces will be shown.
- 1 Weak derivatives and Sobolev spaces
- 2 Approximation by smooth functions
Weak derivatives and Sobolev spaces
Let be a domain, a multiindex, and , where . We call the weak derivative of order of , if and only if:
We denote the weak derivative of with the notation , if it exists.
It can be easily observed that the weak derivatives are linear, i. e. . Furthermore, they are equal to the actual derivatives if these exist. This follows due to integration by parts.
The weak gradient of is defined as follows:
, i. e. it is the sorted d-dimensional vector of all weak derivatives with .
Let , and be a domain. Then the Sobolev space is defined as follows:
The Sobolev spaces also have a norm:
This expression is really a norm.
Proof: To show that the expression is a norm, we have to check the three defining properties for a norm:
- Definiteness, i. e.
- Homogeneity, i. e. for every scalar
- and the triangle inequality, i. e. .
We start with definiteness: If (i. e. f is the function constantly zero), then obviously , since the derivatives of zero (which is a constant function) are zero. Furthermore, if , then in particular it follows that , which implies that since is a norm.
Homogeneity follows directly from the linearity of the derivative and the homogeneity of the -norm.
The triangle inequality follows from the linearity of the derivative, the triangle inequality for the -norm and the commutation law for addition:
All Sobolev spaces are Banach spaces.
Proof: We use the fact that -spaces are Banach spaces. Let be a Cauchy sequence in . Then, since for all multi-indices , we find that:
Therefore, the sequences are all Cauchy sequences in and therefore have a limit . We choose and show that is the limit of in : Let (and therefore also ). Then:
, and due to the dominated convergence theorem,
, which is why is the weak derivative of order of . This finishes the proof, since it shows that with respect to the norm of .
Approximation by smooth functions
Approximation of continuous functions
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-functions
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 unity
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 sets
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 sets
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.