Sobolev Spaces and the Finite Element Method

Institutionen för naturvetenskap och teknik

Sobolev Spaces and the Finite

Element Method

Örebro universitet

Institutionen för naturvetenskap och teknik

Självständigt arbete för kandidatexamen i matematik, 15 hp

Sobolev Spaces and the Finite Element

Method

Johan Davidsson May 2018

Supervisor: Farid Bozorgnia Examiner: Marcus Sundhäll

Abstract

In this essay we present the Sobolev spaces and some basic properties of them. The Sobolev spaces serve as a theoretical framework for studying solutions to partial differential equations. The finite element method is pre-sented which is a numerical method for solving partial differential equations.

Contents

1 Preliminaries 5

1.1 Notation . . . 5

1.2 Measure, Integration and Almost Everywhere . . . 5

1.3 Functional Analysis . . . 9

1.4 Distributions . . . 10

2 Sobolev Spaces 11 2.1 Hölder Spaces . . . 11

2.2 Weak Derivative . . . 14

2.3 Definitions and Properties of Sobolev Spaces . . . 16

3 Finite Element Method 23 3.1 Weak Solution . . . 23

3.2 Finite Element Method . . . 24

Chapter 1

Preliminaries

In Section 1.1 we present some notations that will be used frequently through-out this text. Section 1.2 deals with some basic concepts in measure theory such as an extension of the length of intervals to open sets. We also present the concept that a relation can hold almost everywhere. For a full review on measure theory the reader can look up Folland [4]. With this it will make sense to consider solutions to partial differential equations satisfying a given equation almost everywhere instead of pointwise. In Section 1.3 we state some functional analysis and at the end we briefly review the concept of distributions. For a review on functional analysis and distributions one can read Debnath and Mikusiński [2].

1.1

Notation

Let Ω denote an open subset of Rn and ∂Ω denote the boundary of Ω. The set N will denote the natural numbers including zero. The closure of a set Ω will be denoted by Ω. The derivative will be denoted by D. A multi-index α is an n-tuple α = (α1, . . . , αn) where αi ≥ 0 for i = 1, . . . n. Let α and β

be multi-indices, below we introduce some useful notations: Sum and difference: α ± β = (α1± β1, . . . , αn± βn).

Sum of components: |α| = α1+ . . . + αn.

Partial derivatives: Dα= ∂α1

∂xα1₁ . . . ∂αn

∂xαnn .

1.2

Measure, Integration and Almost Everywhere

Definition 1.2.1. A σ-algebra is a nonempty family M containing subsets of a set X such that

1. If A₁, A2, . . . ∈M =⇒ ∞

S

i=1

2. If A ∈M =⇒ Ac∈M.

σ-algebras are used as domains for measure. A σ-algebra is intuitively a family of nice sets that we can assign a measure to. The elements ofM are therefore called measurable sets.

Definition 1.2.2. Let X be a set with σ-algebraM. A measure is a function µ :M → [0, ∞] such that

1. µ(∅) = 0.

2. If {Ai}∞1 is a sequence of pairwise disjoint sets in M, the following

must be true µ( ∞ [ i=1 Ai) = ∞ X i=1 µ(Ai).

This function µ relates a set with a number that can be interpreted as the size of that set.

Definition 1.2.3. LetP(X) be the powerset of X. An outer measure is a function µ∗ :P(X) → [0, ∞] such that

1. µ∗(∅) = 0. 2. µ∗(A) ≤ µ∗(B) if A ⊂ B. 3. µ∗( ∞ S i=1 Ai) ≤ ∞ P i=1 µ∗(Ai).

Let µ be a measure defined on a σ-algebra M over X. Then the pair (X,M) is called a measuable space and the triplet (X, M, µ) is called a mea-sure space.

Definition 1.2.4. Let A ⊂ R, then we define outer Lebesgue measure of a set A as m∗(A) = inf ( _∞ X i=1 (bi− ai) : A ⊂ ∞ [ i=1 [ai, bi] ) .

Definition 1.2.5. The family of all Lebesgue measurable sets,L(R), consists of sets A such that the following holds for all E ⊂ R

m∗(E) = m∗(E ∩ A) + m∗(E ∩ Ac), where m∗ is defined in Definition 1.2.4. L(R) is a σ-algebra.

Definition 1.2.6. Let A ∈L(R) then we define the Lebesgue measure m(A) as the outer measure of A,

The Lebesgue measure generalizes the notion of length, area and volume to arbitrary sets in higher dimension. When working with Rn one usually consider µ to be the Lebesgue measure, which is what we will do from now on.

Given a measure space (Ω,M, µ), a null set is a set A ⊂ Ω such that µ(A) = 0. We state a formal definition below.

Definition 1.2.7 (Null set). Let A ⊂ Ω. The set A is called a null set if and only if for every  > 0 there is a countable family of n-cubes {A_i} that covers A such that

∞

X

i=1

µ(Ai) = .

Example 1.2.1. Let c be a point in R. Let  > 0 and choose Ii = (c − 

2i+1, c +₂i+1 ). We have that

{c} ⊂

∞

[

i=1

Ii

and we get the following

∞ X i=1 µ(Ii) =  2 +  2· 1 2 +  2 · ( 1 2) 2_{+ . . . =} 2 1 −1₂ = . We conclude that µ(c) = 0.

Definition 1.2.8 (Almost everywhere). Let (X,M, µ) be a measure space. We say that a property holds almost everywhere, when it is true everywhere except for a set A ⊂ X such that µ(A) = 0.

Example 1.2.2. Let f, g : R → R, g(x) = x and f (x) =

(

x if x 6= 1; 0 if x = 1.

We can say that f = g almost everywhere since µ(1) = 0.

Next we will develop the Lebesgue integral, we start off by defining a measurable function. Recall that any mapping f : X → Y induces a mapping f−1 defined by

f−1(A) = {x ∈ X : f (x) ∈ A}.

Definition 1.2.9. Let (X,M), (Y, N) be measure spaces. A mapping f : X → Y is called measurable if f−1(A) ∈M for all A ∈ N.

Definition 1.2.10. Let A ⊂ X, the indicator function χA : X → {0, 1} is

defined as

χA(x) =

(

1 if x ∈ A; 0 if x /∈ A.

First off we will present the simple functions, they are easy to construct an integral from. The simple functions has properties that will help us define an integral for more general functions.

Definition 1.2.11. Let A1, . . . , Anbe a sequence of disjoint measurable sets

and a₁, . . . , an∈ R. A simple function f : X → R is a function of the form

f (x) =

n

X

i=1

aiχAi(x).

Definition 1.2.12. For a positive simple function f =Pn

i=1aiχAi we define the integral of f with respect to the measure µ as

Z A f dµ = n X i=1 aiµ(Ai).

So we have an integral for simple functions and from [4] we know that measurable functions can be approximated by simple functions.

Definition 1.2.13. Let f be a positive measurable function. We define the integral of f as Z f dµ = sup φ≤f Z φ dµ,

where the supremum is taken over all simple functions φ such that φ ≤ f . We developed the integral for positive measurable functions. For the general case you split a function f into its positive and negative parts, that is f = f+− f−. We say that a measurable function f is integrable if both R f+ _{dµ and R f}− _{dµ is finite. For details on this the reader can look in [4,}

chapter 2]. From the definition we see that if f, g are integrable and a, b ∈ R. The following properties hold

Z af + bg dµ = a Z f dµ + b Z g dµ, if f ≤ g we get that Z f dµ ≤ Z g dµ.

Theorem 1.2.1. (The dominated convergence theorem) Let {fn} be a

se-quence of measurable functions such that f_n→ f pointwise almost everywhere as n → ∞. Also let |f | ≤ g where g is integrable. Then we have that f is integrable and Z f = lim n→∞ Z fn.

1.3

Functional Analysis

Definition 1.3.1 (Normed space). A vector space Y equipped with a norm is called a normed space.

The norm generalizes the notion of length into abstract vector spaces. Definition 1.3.2 (Bilinear functional). A functional F (x, y) on a vector space E is called a bilinear functional if the following holds:

1. F (ax1+ bx2, y) = aF (x1, y) + bF (x2, y).

2. F (x, ay1+ by2) = conj(a)F (x, y1) + conj(b)F (x, y2).

for any scalars a, b and x, x₁, x2, y, y1, y2 ∈ Y . Where conj denote the

com-plex conjugate.

Definition 1.3.3 (Coercive functional). A bilinear functional F (x, y) on a normed space Y is called coercive if there exist a positive constant C such that

F (x, x) ≥ Ckxk2 ∀x ∈ Y.

Definition 1.3.4 (Complete space). Let (Y, k·k) be a normed space. If every cauchy sequence in Y has a limit that is also in Y , then (Y, k·k) is called a complete space(or Banach space).

Definition 1.3.5 (Inner product space). A vector space equipped with an inner product h·, ·i is called an inner product space.

The inner product space extend the inner product from Rn to vector spaces.

Definition 1.3.6. Let H be an inner product space. H is called a Hilbert space if it is complete with respect to the norm ||x|| =phx, xi.

Definition 1.3.7. Let f be a measurable function on Y and fix 1 ≤ p < ∞, then we define the Lp-norm of f by

kf k_Lp_{(Y )} =  Z Y |f |p _dµ 1/p . For p = ∞, define

kf k_L∞_{(Y )}= inf{C > 0 : |f (x)| ≤ C for almost every x}.

For fixed p the space Lp consists of all functions f that have finite Lp norm. We present two important inequalities:

1. Hölder’s inequality: Let 1 < p < ∞ and 1_p + 1_q = 1. If f, g are measurable functions on Y , then

kf gk_L1_{(Y )}≤ kf k_Lp_{(Y )}kgk_Lq_{(Y )}. For the case p = 1 we have

kf gk_L1_{(Y )} ≤ kf k_L1_{(Y )}kgk_L∞_{(Y )}.

2. Minkowski’s inequality: Let f, g ∈ Lp and 1 ≤ p < ∞ then we have kf + gkp ≤ kf kp+ kgkp.

1.4

Distributions

Definition 1.4.1 (Support & compact support). Let Ω ⊂ R. The support of f , denoted by supp(f ), is defined as

supp(f ) = {x ∈ Ω : f (x) 6= 0}.

We say that a function has compact support in Ω if supp(f ) is a compact subset of Ω.

Let C_c∞ be the set of infinitely differentiable functions that has compact support in Ω. An element of this set is called a test function and they turn out to be quite important.

Definition 1.4.2. Let φ₁, φ2, . . . and φ be test functions. A sequence (φn)

is said to converge to φ ∈ C_c∞, denoted φn Cc∞

−−→ φ, if the following holds: 1. φ₁, φ2. . . and φ vanish outside some bounded set S ⊂ Ω.

2. Dαφn→ Dαφ uniformly on Ω for every multi-index α.

Definition 1.4.3. A distribution T on Ω is a continuous linear functional on C_c∞(Ω). That is, a mapping T : C_c∞(Ω) → C is a distribution if the following conditions are true

1. T (aφ + bψ) = aT (φ) + bT (ψ). 2. φ_n C

∞ c

−−→ φ =⇒ T (φ_n) → T (φ).

A function f on Ω is said to be locally integrable on Ω if f ∈ L1_loc(U ) for all compact subsets U ⊂ Ω. For every locally integrable function f there is a corresponding distribution Tf defined by

Tf(φ) =

Z

Ω

f φ dx, φ ∈ C_c∞(Ω). (1.1) If a distribution T can be expressed in the form of (1.1) we shall call it regular. Otherwise it is singular.

Chapter 2

Sobolev Spaces

Partial differential equations are important due to many real world appli-cations. Sobolev spaces are vector spaces and they are interesting because solutions of partial differential equations, when they exist, belong to some Sobolev space. So understanding the Sobolev spaces will help us to under-stand partial differential equations. We start of by introducing the Hölder spaces in Section 2.1. In Section 2.2 we present the concept of a weak deriva-tive, this is a generalization of the classical derivative. The weak derivative is important because a Sobolev space is a set of functions which has a weak derivative up to some order. In Section 2.3 we introduce the Sobolev spaces and some properties of them. This chapter is based on the material found in Adams [1], Evans [3], Shkoller [6] and Kinnunen [7].

2.1

Hölder Spaces

A function is Lipschitz continuous if there exist a real number K such that that for every pair of points from the graph of the function the absolute value of the slope connecting the line between these points is bounded by K. That is, for positive real constant K we have

|f (x) − f (y)| ≤ K|x − y|, x, y ∈ Ω.

This Lipschitz condition implies that the first derivative exist almost everywhere. We introduce a more general condition, a function is Hölder continuous with exponent γ if there is a positive real constant C and 0 < γ ≤ 1 such that

|f (x) − f (y)| ≤ C|x − y|γ, x, y ∈ Ω.

For γ = 0 the function f need not be continuous, for γ > 0 the condition implies continuity. But for γ > 1 the condition implies that the function f is a constant, therefore we only care about the case 0 < γ ≤ 1. Let

m > 0 be an integer and 0 < γ ≤ 1. The Hölder space Cm,γ(Ω) consists of functions f that are m-times continuously differentiable and whose funcion f and partial derivative are bounded. Also the mth partial derivatives are Hölder continuous with exponent γ. We present the norm and state the definition below:

Definition 2.1.1. For f : Ω → R bounded and continuous we define kf k_C(Ω)= sup

x∈Ω

|f (x)|. We also define the seminorm

[f ]_C0,γ_(Ω) = sup

x,y∈Ω x6=y

|f (x) − f (y)| |x − y|γ .

The Hölder space Cm,γ(Ω) consists of functions f for which the norm kf k_Cm,γ_(Ω) = X |α|≤m kDαf k_C(Ω)+ X |α|=m [Dαf ]_C0,γ_(Ω) is less than infinity.

We show that kf k_Cm,γ_(Ω) is a norm.

Proof. First we show that [f ]_C0,γ_(Ω) is a seminorm:

1. Since everything inside the supremum is greater than or equal to zero we have that [f ]_C0,γ_(Ω) ≥ 0. 2. Let c be a constant. [cf ]_C0,γ_(Ω) = sup x,y∈Ω x6=y |cf (x) − cf (y)| |x − y|γ = |c| sup x,y∈Ω x6=y |f (x) − f (y)| |x − y|γ = |c|[f ]_C0,γ_(Ω). 3. The triangle inequality:

[f + g]_C0,γ_(Ω)= sup x,y∈Ω x6=y |f (x) + g(x) − f (y) − g(y)| |x − y|γ ≤ sup x,y∈Ω x6=y |f (x) − f (y)| + |g(x) − g(y)| |x − y|γ = sup x,y∈Ω x6=y |f (x) − f (y)| |x − y|γ + sup_x,y∈Ω x6=y |g(x) − g(y)| |x − y|γ = [f ]_C0,γ_(Ω)+ [g]_C0,γ_(Ω).

It is a seminorm because there are functions f 6= 0 such that [f ]_C0,γ_(Ω) = 0. Consider for example a constant function f 6= 0, this will make the numerator zero and thus [f ]_C0,γ_(Ω) = 0.

Because of the above reasoning it is obvious that kf k_C(Ω)= sup

x∈Ω

|f (x)|

is a norm, the reader is encouraged to verify it as an exercise. Now we show that kf k_Cm,γ_(Ω) is a norm:

1. We show that kf k_Cm,γ_(Ω)= 0 ⇐⇒ f = 0:

Assume f = 0, this implies kf k_Cm,γ_(Ω) = 0 since k0k_C(Ω) = 0 and [0]_C0,γ_(Ω)= 0.

Assume kf k_Cm,γ_(Ω) = 0, this implies X

|α|≤m

kDαf k_C(Ω)+ X

|α|=m

[Dαf ]_C0,γ_(Ω)= 0.

Since all terms are positive, we get that in particular kDαf k_C(Ω)= 0

should hold for all multi-indices |α| ≤ m. This certainly includes the case α = 0 and we get that f = 0.

2. Since kf k_Cm,γ_(Ω) is a sum of norms and seminorms, it is obvious that we can bring out the constant.

3. To show the triangle inequality we use the properties we now know: kf + gk_Cm,γ_(Ω)= X |α|≤m kDα_{f + D}α_gk C(Ω)+ X |α|=m [Dαf + Dαg]_C0,γ_(Ω) ≤ X |α|≤m kDαf k + X |α|=m [Dαf ]_C0,γ_(Ω)+ X |α|≤m kDαgk_C(Ω) + X |α|=m [Dαg]_C0,γ_(Ω) = kf k_Cm,γ_(Ω)+ kgk_Cm,γ_(Ω).

The Hölder space Cm,γ(Ω) equppied with the norm kf k_Cm,γ_(Ω) is a Ba-nach space [3, page 241]. We will return to the Hölder continuity in the end of this chapter.

2.2

Weak Derivative

The classical definition of the derivative is stated pointwise since it tells us whether the derivative of a function exist in a specific point or not. In this section we state a more general definition that gives us a derivative even if the function, to some degree, is not smooth at certain points. Below follow some reasoning that motivates the new definition of the derivative.

Let f ∈ C1_{(Ω) and φ ∈ C}∞

c (Ω), since φ = 0 on the boundary of Ω

integration by parts gives us the following Z Ω f ∂φ ∂xi dx = − Z Ω ∂f ∂xi φ dx.

In general, for f ∈ Ck(Ω) and some multi-index α such that |α| ≤ k we have Z Ω f Dαφ dx = (−1)|α| Z Ω Dαf φ dx (2.1)

This is true for f ∈ Ck_{, but there is something interesting going on here.}

Let us consider (2.1) for f ∈ L1_loc(Ω), the left hand side still makes sense since the integral is guaranteed to exist. In the right hand side we have a problem, since f is no longer assumed to be differentiable, Dαf does not make any sense. What we will do is replace Dαf with g and ask ourselves if there is a function g ∈ L1_loc(Ω) that satisfies (2.1). This is our new definition of a derivative, we state it below:

Definition 2.2.1 (Weak derivative). Let f, g ∈ L1_loc(Ω). If Z Ω f Dαφ dx = (−1)|α| Z Ω gφ dx

holds for all φ ∈ C_c∞(Ω), then we say that g is the weak derivative of order α of f .

We use the same notation for the weak and classical derivative, it will be clear from the context which interpretation is used.

Theorem 2.2.1. Let g be the weak derivative of f , then g is unique up to a set of measure zero.

Proof. Let f, g, h ∈ L1_loc, g 6= h and assume that both g and h is the weak derivative of f . By definition we get the following,

Z Ω f Dαφ dx = (−1)|α| Z Ω gφ dx = (−1)|α| Z Ω hφ dx, ∀φ ∈ C_c∞(Ω). Which gives us Z Ω gφ dx = Z Ω hφ dx,

so we get that Z

Ω

(g − h)φ dx = 0 =⇒ g = h a.e. ∀φ ∈ C_c∞(Ω).

Example 2.2.1. Consider f (x) = |x| on Ω = (−1, 1). It is clear that the classical derivative does not exist at x = 0, therefore we will find f0 in the weak sense. Integration by parts give

Z 1 −1 f φ0 dx = Z 0 −1 (−x)φ0 dx + Z 1 0 xφ0 dx = − Z 0 −1 (−1)φ dx − Z 1 0 φ dx = − Z 1 −1 gφ dx. where g(x) = ( −1 if −1 ≤ x < 0; 1 if 0 ≤ x ≤ 1. By Definition 2.2.1 we get that f0= g.

It is obvious from the definition that if a function is differentiable in the normal sense, that is f ∈ Ck _{for k ∈ N ∪ {∞}. Then the weak derivative of} f will align with the classical one. Just like the classical derivative, the weak derivative do not necessarily exist which is shown in the next example. Example 2.2.2. Consider Ω = [−1, 1] and the Heaviside step function,

H(x) = (

0 if −1 ≤ x < 0; 1 if 0 ≤ x ≤ 1.

Let φ ∈ C_c∞(Ω), since φ = 0 on the boundary we get the following, Z Ω Hφ0 dx = Z 1 0 φ0 dx = φ    1 0 = −φ(0).

If we can find a function g such that Z

gφ dx = φ(0) ∀φ ∈ C_c∞,

then g is the weak derivative of H. Assume that such a function g exist, we get that lim h→0 Z h −h |g| dx = lim h→0 Z R |g|χ_[−h,h](x) dx.

The sequence of functions |g|χ_[−h,h] tends to zero almost everywhere as h tends to zero. The same sequence is also dominated by |g|χ_[−1,1] on the interval [−1, 1]. So by the Lebesgue’s dominated convergence theorem we have that lim h→0 Z h −h |g| dx = 0. Therefore we can choose δ > 0 such that

Z δ

−δ

|g| dx ≤ 1 2.

Pick an infinitely differentiable φ : R → [0, 1] such that φ(0) = 1 with support contained in the interval [−δ, δ]. We get the following

1 = φ(0) = Z Ω gφ dx = Z δ −δ gφ dx ≤ max x |φ(x)| Z δ −δ |g| dx ≤ 1 2, a contradiction. We conclude that there is no such function g and that H has no weak derivative.

The following familiar properties for the classical derivative also hold for the weak one, if we let a, b ∈ R and f, g be m times differentiable. Then

1. Dα(Dβf ) = Dβ(Dαf ) = Dα+βf , for all |α| + |β| ≤ m. 2. Dα(af + bg) = aDαf + bDαg, for all |α| ≤ m.

2.3

Definitions and Properties of Sobolev Spaces

Definition 2.3.1. Let m ∈ N, for multi-index α and 1 ≤ p < ∞ we define the norm kf k_Wm,p_(Ω)= X |α|≤m kDαf kp_Lp_(Ω) !1/p , and for p = ∞ kf k_Wm,∞_(Ω)= max |α|≤mkD α_{f k} L∞_(Ω).

Example 2.3.1. Let 1 ≤ p < ∞, for m = 1 and m = 2 we get the following norms respectively, kf k_W1,p_(Ω)= Z Ω |f |p+ n X k=1 Z Ω     ∂f ∂xk     p!1/p , kf k_W2,p_(Ω)= Z Ω |f |p+ n X k=1 Z Ω     ∂f ∂xk     p + n X k=1 n X l=1 Z Ω     ∂f ∂xk∂xl     p!1/p .

For p = ∞, multi-index α = (α₁, . . . , αn) and m = 0, m = 1 respectively, kf k_W0,∞_(Ω)= max n ||D(0,...,0)f ||L∞ o , kf k_W1,∞_(Ω) = max n ||D(0,...,0)f ||L∞, ||D(1,0,...,0)f ||_L∞, . . . , ||D(0,...,0,1)f ||_L∞ o . Since the order m just adds more positive terms to the sum and in the case of p = ∞, m adds more alternatives to maximize, we get the following useful estimate

kDαf k_Lp_(Ω) ≤ kf k_Wm,p_(Ω), |α| ≤ m. (2.2) Lemma 2.3.1. The functional k·k_Wm,p,_(Ω) is a norm.

Proof. Let f, g ∈ Wm,p_{(Ω), m ∈ N, p ∈ [1, ∞) and λ ∈ R.}

1. kf k_Wm,p_(Ω) = 0 implies kf k_Lp_(Ω) = 0 which implies that f = 0 almost everywhere in Ω.

2. kλf k_Wm,p_(Ω)= |λ|kf k_Wm,p_(Ω) is clear from the definition.

3. The triangle inequality follow from applying Minkowski’s inequality twice, kf + gk_Wm,p_(Ω)= X |α|≤m kDαf + Dαgkp_Lp_(Ω) !1/p ≤ X |α|≤m  kDα_{f k} Lp_(Ω)+ kDαgk_Lp_(Ω) p !1/p ≤ X |α|≤m kDαf kp_Lp_(Ω) !1/p + X |α|≤m kDαgkp_Lp_(Ω) !1/p = kf k_Wm,p_(Ω)+ kgk_Wm,p_(Ω). For p = ∞ the proof is similar.

Before we state the definition of the Sobolev spaces we will talk about the space Cm. If we consider this space equipped with the norm k·k_Wm,p it will be an incomplete space. Meaning that if we have a cauchy sequence inside Cm, the limit can be found outside of Cm. We will show two examples of this to convince the reader.

Example 2.3.2. Let C(R) be equipped with the L2_{-norm, consider the}

function fm(x) =      0 if x ≤ 0; mx if 0 < x ≤ _m1; 1 if x > _m1.

Let m > n, kfm− fnk2_L2_(R)= Z R |fm− fn|2 dx = Z 1/m 0 (fm− fn)2 dx + Z 1/n 1/m (fm− fn)2 dx = (m − n)2 Z 1/m 0 x2 dx + Z 1/n 1/m 1 − 2nx + n2x2 dx = 1 3m− 2n 3m2 + 1 3n− 1 m + n m2 → 0 when m, n → ∞.

So fmis a cauchy sequence in our space. It is clear that limm→∞fm= H(x)

and since H(x) is not continuous we have that H(x) /_{∈ C(R). We can} conclude that our space is not complete.

x f

1

1/m

Figure 2.1: Function fm

Example 2.3.3. Let C1([0, 1]) be equipped with the norm k·k_W1,1_([0,1]) and consider the sequence

fm= r x + 1 m. Let m > n kf_n− f_mk_W1,1_([0,1]) = Z 1 0 fn− fm dx + Z 1 0 f_n0 − f_m0 dx = 2 3  1 n+ x 2/3 −1 m+ x 2/3    1 0 +r 1 n+ x − r 1 m+ x     1 0 → 2 3(1 − 1) + 1 − 1 = 0 when m, n → ∞.

So fm is a cauchy sequence in our space. But again we have that the limit

limm→∞fm = f =

√

x does not lie in our space since f0 = ₂√1

x can not be

With both the weak derivative and this incompletion of Cm in our mind, we are ready to define the Sobolev spaces.

Definition 2.3.2. For 1 ≤ p ≤ ∞, m ∈ N and multi-index α we consider the following spaces equipped with the norm in definition 2.3.1 to be the Sobolev spaces.

1. Wm,p(Ω) = {f ∈ Lp(Ω) : Dαf ∈ Lp(Ω) ∀|α| ≤ m}, where Dα is the weak derivative.

2. The completion of Cm(Ω) with respect to the norm k·k_Wm,p.

So according to the first definition the elements of a Sobolev space are functions in Lp such that the weak partial derivatives up to some order are also in Lp_{. These spaces generalize the space L}p _{because W}0,p _{= L}p_.

Func-tions in Sobolev spaces are identified if they are equal almost everywhere. By definition 2 we take the completion of Cm with respect to the Sobolev norm, meaning we consider the space that has all of the elements in Cm and the limit points.

Definition 2.3.3 (Convergence). We say the the sequence {f_i} converges to f , written f_i → f , in Wm,p_{(Ω) if}

lim

i→∞kfi− f kWm,p(Ω)= 0.

Theorem 2.3.1 (Completeness). The Sobolev space Wm,p(Ω) with respect to the norm in Definition 2.3.1 is a Banach space.

Proof. Let {fi} be a cauchy sequence in Wm,p(Ω), the estimate (2.2) gives

us the following

kDαfi− Dαfjk_Lp_(Ω)≤ kfi− fjk_Wm,p_(Ω), |α| ≤ m.

Which implies that {Dαfi} is a cauchy sequence in Lp(Ω). Since the space

Lp_{(Ω) is complete, this implies that there exist functions f}

α, f ∈ Lp(Ω) such that lim i→∞kD α_f i → fαk_Lp_(Ω)= 0, lim i→∞kfi → f kLp(Ω)= 0, |α| ≤ m.

Now we will show that Dαf = fαby using the result above and the definition

of the weak derivative. Z Ω f Dαφ dx = lim i→∞ Z Ω fiDαφ dx = (−1)|α| lim i→∞ Z Ω Dαfiφ dx = (−1)|α| Z Ω fαφ dx ∀φ ∈ Cc∞(Ω).

Now we know that Dαf exist and that Dαf = fα. Since f ∈ Lp(Ω) and

Dαf ∈ Lp(Ω) for |α| ≤ m we have that f ∈ Wm,p(Ω). The only thing left to show is that f is the limit of fi, which follows by

lim i→∞kfi− f k p Wm,p_(Ω)= lim i→∞ X |α|≤m kDαfi− Dαf kp_Lp = 0.

It should be noted that Wm,p(Ω) is a separable space for 1 ≤ p < ∞ [1, page 61]. The case p = 2 is of special interest because we can define an inner product, hf, gi_Wm,2_(Ω)= X |α|≤m hDα_{f, D}α_gi L2_(Ω). As expected, q hf, gi_Wm,2_(Ω) = kf k_Wm,2_(Ω).

Since our norm induced by the inner product yields a complete space, Wm,2(Ω) is a Hilbert space and is therefore denoted by Hm(Ω).

Example 2.3.4. Let Ω = B(0, 1), the open unit ball in Rn. Consider f = |x|−a for x ∈ Ω, x 6= 0.

We want to find out for which values a > 0, n, p does the function f belong to W1,p(Ω). First will check for which values f belong to Lp(Ω) with the help of spherical coordinates. For positive constant M1 we get

kf kp_Lp_(Ω) = Z Ω |x|−apdx = Z 2π ξn−1=0 Z π ξn−2=0 · · · Z π ξ1=0 Z 1 r=0 r−apdnV ≤ M₁ Z 1 0 rn−1−ap dr = M1  1 n − ap− 0n−ap n − ap  < ∞ when a < n p, where

dnV = rn−1sinn−2(ξ1) sinn−3(ξ2) · · · sin (ξn−2) dr dξ1 dξ2· · · dξn−1.

So we have that f ∈ Lp(Ω) for a < n_p. Next we must investigate the deriva-tive. The function f is differentiable away from x = 0 with derivative

∂f (x) ∂xi

= −axi

In the same fashion we get that kDf kp_Lp_(Ω)= |a| Z Ω |x|−(a+1)p dx ≤ M2 Z 1 0 r−(a+1)p+n−1 dr = M2  1 −(a + 1)p + n− 0−(a+1)p+n −(a + 1)p + n  < ∞ for a < n p − 1.

This implies that Df ∈ Lp(Ω) for a < n_p − 1. Let φ ∈ C_c∞(Ω) and  > 0, we get that Z Ω−B(0,) f ∂φ ∂xi dx = − Z Ω−B(0,) ∂f ∂xi φ dx + Z ∂B(0,) f φvi dx,

where v = (v1, . . . , vn) denote the inward pointing normal on ∂B(0, ). Let

p = 1 and α < n − 1. In this case we get      Z ∂B(0,) f φvi dS      ≤ kφk_L∞ Z ∂B(0,) −adS ≤ Cn−1−a→ 0. Thus Z Ω f ∂φ ∂xi dx = − Z Ω ∂f ∂xi φ dx ∀φ ∈ C_c∞(Ω). We conclude that f = |x|−a∈ W1,p_{(Ω) if and only if a <} n

p − 1.

Whether a function belong to a certain Sobolev space or not is obviously depending on the integrability p and regularity m of the function. This example show that it can also depend on the dimension n of the domain Ω. We will show some smoothness properties that hold for certain sobolev spaces.

Consider n = 1 and the space W1,∞_{(R). A function belonging to this} space is Lipschitz continuous which is shown below by using Hölder’s in-equality: |f (x) − f (y)| =     Z x y f0(t) dt     ≤ Z x y  f0(t)   dt ≤ f0(t) L∞k1k_L1 = f0(t) _L∞|x − y|.

Since f0 ∈ L∞_{(R), the following holds} |f (x) − f (y)| |x − y| ≤ f0(t) L∞ = C < ∞.

Now consider 1 < p < ∞, 1_p + 1_q = 1 and f ∈ W1,p(R). Again using Hölder’s inequality: |f (x) − f (y)| =     Z x y f0(t) dt     ≤ Z x y  f0(t) dt ≤ f0(t) Lpk1kLq = f0(t) Lp|x − y| 1 q_.

In the same way as before, since f0 ∈ Lp_{(R), the following holds}

|f (x) − f (y)| |x − y|1q

≤ f0(t)

Lp = C < ∞.

We conclude that if f ∈ W1,p_{(R) then f is Hölder continuous with} expo-nent γ = 1 − 1_p.

Chapter 3

Finite Element Method

We present the concept of a weak solution and the finite element method, Section 3.2 is based on the material in Logg and Mardal [5].

3.1

Weak Solution

A weak solution to a differential equation is a function for which the deriva-tives may not all exist but still satisfy the equation in some precisely defined way. Consider the partial differential operator

L = X

|α|≤m

AαDα,

where α is a multi-index and Aαfunctions in Rn. If a function u is sufficiently

differentiable in the classical sense so that Lu is well defined pointwise, then it is called a classical solution if it satisfies the equation

Lu = f, (3.1)

for some function f on Rn. By a weak solution of 3.1 we mean a function u on Rn that does not need to be sufficiently differentiable to make Lu meaningful in a classical sense. In this case we will have some other precise way of considering u to be a solution anyway. As for example by considering the weak derivative instead of classical.

Example 3.1.1. Given a bounded open subset Ω ⊂ Rn, the distance func-tion is defined as dist(x, ∂Ω) =    inf y∈∂Ωkx − yk if x ∈ Ω; 0 if x ∈ ∂Ω.

Consider the problem of finding u such that (

k∇uk = 1 in Ω;

The distance function satisfies this problem except at the point where the slope changes from 1 to −1. To get some idea of how this looks consider the case when we have Ω = (−1, 1) and k·k = |·|. We draw the distance function in this case below:

x f

−1 1

1

Figure 3.1: The function dist(x,{-1,1}).

It is clear that dist(x,{-1,1}) does not satisfy the problem in classical sense. But if we consider using the weak derivative instead it does satisfy (3.2) in the weak sense. We show it below:

Z Ω dist(x, {−1, 1})φ0 dx = Z 0 −1 (x + 1)φ0 dx + Z 1 0 (−x + 1)φ0 dx = ((x + 1)φ)    0 −1− Z 0 −1 1φ dx + ((−x + 1)φ)    1 0− Z 1 0 (−1)φ dx = φ(0) − Z Ω gφ dx − φ(0) = − Z Ω gφ dx where g(x) = ( 1 if −1 ≤ x < 0; −1 if 0 ≤ x ≤ 1.

We concude that |dist(x, {−1, 1})0| = 1 in weak sense on Ω.

3.2

Finite Element Method

The finite element method is a numerical method for solving partial differ-ential equations. When the problem has been formulated according to the

method you end up with a system of algebraic equations. A partial differen-tial equation has the general form

(

Lu = f in Ω;

Boundary conditions on ∂Ω.

This is called the strong form of the problem, the first thing is to turn it into the weak form. The weak form is obtained by taking the inner product of the two sides together with some test function. So let V be a Hilbert space, we have

hLu, vi = hf, vi. Define

a(u, v) = hLu, vi, b(v) = hf, vi.

Where a : V × b_{V → R and b : bV → R. The spaces V and b}V are some specific Sobolev spaces, the choice of which Sobolev space to use depends on the problem. In many problems these spaces will be chosen so that they are the same space, but in general they can be chosen to be different. The weak formulation of our problem is finding u ∈ V such that

a(u, v) = b(v) ∀v ∈ bV . (3.3)

We present a result regarding the existence and uniqueness of weak solutions for the case V = bV :

Theorem 3.2.1 (Lax-Milgram Theorem). Let a(x, y) be a bounded, coercive, bilinear functional on a Hilbert space H. For every bounded linear functional b(y) on H, there exist a unique xb such that

a(xb, y) = b(y) ∀y ∈ H.

So assume a(u, v) is bounded, coercive and bilinear and b(v) bounded and linear. We have that there is a unique u satisfying (3.3). There are generalizations of this theorem, for the case V 6= bV the interested reader can look up Babuška–Lax–Milgram Theorem.

The space V can be infinite so when we want to solve this in a computer we need to make it finite. Pick a Vk such that

1. dim(V_k) = k. 2. V_k⊂ V_k+1. 3. ∪Vk= V .

Let the vectors {φ₁, . . . , φk} span Vk. We are now looking for an

approxi-mation uk ∈ Vk to our solution u ∈ V . So instead of solving (3.3) we will

solve

a(uk, v) = b(v) ∀v ∈ bVk. (3.4)

Since we want to put this in a computer it is nice to have it on matrix form. If we let {wk+1, . . . , wn} span V \ Vk, we can describe our solution as u =

Pk

i=1ciφi+

Pn

i=kciwi. We will write our approximation as uk =

Pk

i=1ciφi

in order to get the matrix form. Also let {ψ₁, . . . , ψk} be a basis for Vk. We

get the following

a(

k

X

i=1

ciφi, ψj) = b(ψj) for j = 1, . . . , k,

which can be written as

k X i=1 cia(φi, ψj) = b(ψj) for j = 1, . . . , k. Let c =    c1 .. . ck   , and b =    b(ψ1) .. . b(ψk)   .

Also let A = (ai,j) where ai,j = a(φi, ψj). We now have our equation (3.4)

on the form

Ac = b, which is solved by a suitable method.

Chapter 4

Future Work

Here is an informal discussion of the possible continuations on this essay. If I had more time I would write about the relation between the two definitions of the Sobolev spaces stated in definition 2.3.2. They are in fact equivalent, which is far from trivial to show. According to [1, page 60] this fact was discovered in 1964, having in mind that people have been working on Sobolev spaces since at least late 1930’s, it is quite interesting that this elementary result was discovered so late. Another result is the Trace Theorem, which asserts that we can talk about the boundary of a function in a Sobolev space. This is not trivial since in a Sobolev space a function is defined almost everywhere and we have that the n-dimensional measure of the boundary is zero. So it is not clear how to describe values at the boundary for a function in a Sobolev space. I consider this topic relevant since when we talk about a partial differential equation we often state some criteria on the boundary values. For a more theoretical topic, there is a big result called the Sobolev Embedding Theorem stating that certain Sobolev spaces can be embedded in various spaces such as, for example, certain Hölder spaces. This could be interesting if one want to write more about the connection between Sobolev and Hölder spaces, since this relation was only mildly investigated in this essay. The last suggestion is looking into the interesting generalizations of Sobolev spaces, such as when the space Wm,p is extended to include spaces where m need not be an integer. If these spaces have some relevance to partial differential equations I do not know, but that would be interesting to investigate.

Bibliography

[1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2 ed. Elsevier, 2005. [2] L. Debnath and P. Mikusiński, Hilbert Spaces with Applications, 3 ed.

Elsevier, 2005.

[3] L. C. Evans, Partial Differential Equations, 1 ed. Vol. 19. American Mathematical Society, 2002.

[4] G. B. Folland, Real Analysis, 2 ed. John Wiley & Sons, 1999. [5] A. Logg and K-E. Mardal, Lectures on the Finite Element Method. [6] S. Shkoller, Notes on Lpand Sobolev Spaces, Department of Mathematics,

University of California, 2009.

[7] J. Kinnunen, Sobolev Spaces, Department of Mathematics and Systems Analysis, Aalto University, 2017.