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).
The fundamental lemma of the calculus of variations
But first we shall repeat the definition of the standard mollifier defined in chapter 3.
Example 3.4: The standard mollifier , given by
, where , is a bump function (see exercise 3.2).
For , we define
Lemma 12.1: (to be replaced by characteristic function version)
Let be a simple function, i. e.
where are intervals and is the indicator function. If
The following lemma, which is important for some theorems about Sobolev spaces, is known as the fundamental lemma of the calculus of variations:
Let and let be functions such that and . Then almost everywhere.
Let be a set, and . If is a -dimensional multiindex and such that
, we call an th-weak derivative of .
Remarks 12.2: If is a function and is a -dimensional multiindex, any two th-weak derivatives of are equal except on a null set. Furthermore, if exists, it also is an th-weak derivative of .
1. We prove that any two th-weak derivatives are equal except on a nullset.
Let be two th-weak derivatives of . Then we have
Notation 12.3 If it exists, we denote the th-weak derivative of by , which is of course the same symbol as for the ordinary derivative.
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 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 functions
We shall now prove that for any function, we can find a sequence of bump functions converging to that function in norm.
approximation by simple functions and lemma 12.1, ||f_eps-f|| le ||f_eps - g_eps|| + ||g_eps - g|| + ||g - f||
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.
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.
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.
Hölder spaces and Morrey's inequality
The Gagliardo–Nirenberg–Sobolev inequality
Sobolev embedding theorems