Partial Differential Equations/Sobolev spaces

< Partial Differential Equations
Partial Differential Equations
 ← Characteristic equations Sobolev spaces Calculus of variations → 

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 variationsEdit

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).

Definition 3.13:

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


then  .

The following lemma, which is important for some theorems about Sobolev spaces, is known as the fundamental lemma of the calculus of variations:

Lemma 12.2:

Let   and let   be functions such that   and  . Then   almost everywhere.


We define


Weak derivativesEdit

Definition 12.1:

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.

Theorem 12.4:

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

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:

  •   (definiteness)
  •   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

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.


Let's choose




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.


to follow

Hölder spaces and Morrey's inequalityEdit

Continuous representativesEdit

The Gagliardo–Nirenberg–Sobolev inequalityEdit

Sobolev embedding theoremsEdit