introduction to sobolev spaces

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chapter 1 ........................................................................................................

INTRODUCTION TO SOBOLEV SPACES ........................................................................................................

In this chapter we recall some basics on functional analysis and provide a brief introduction to Sobolev spaces. For a more detailed and comprehensive study, we refer to Adams (1975).

1.1. Banach and hilbert spaces ................................................................................................................

Definition 1.1. Let X be a real linear space. A mapping  ·  : X → R is called a norm on X if • • •

x = 0 ⇔ x = 0 for all x ∈ X, λx = |λ| x for all x ∈ X, λ ∈ R, x + y ≤ x + y for all x, y ∈ X.

The pair (X,  · ) is called a normed space. Setting y = −x in the inequality and using the other two properties we get x ≥ 0 for all x ∈ X. Definition 1.2. A sequence (xn )n∈N is called Cauchy sequence, if for all ε > 0 there exists an index n0 (ε) such that for all m, n > n0 (ε) it holds xm − xn  < ε. Definition 1.3. A sequence (xn )n∈N converges to x ∈ X if for all ε > 0 there is an index n0 (ε) such that for all n > n0 (ε) it holds xn − x < ε. Definition 1.4. A normed space (X,  · ) is called complete if every Cauchy sequence in X converges in X. Definition 1.5. A complete normed space is called Banach space.

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introduction to sobolev spaces

Example 1.1. Let  ⊂ Rd , d is the dimension, be an open and bounded domain. We denote by Lp (), 1 ≤ p < ∞, the set of all measurable functions f :  → R for which  |f (x)|p dx < ∞. 

Similarly, the set of all measurable functions f :  → R satisfying ess sup{|f (x)| : x ∈  } < ∞ will be denoted by L∞ (). Then, Lp (), p ∈ [1, ∞] is a real linear space. Setting for 1 ≤ p < ∞  1/p p and f L∞ () := ess sup{|f (x)| : x ∈  }, f Lp () := |f (x)| dx 

Lp ()

respectively, becomes a normed space for any p ∈ [1, ∞] if we identify functions which are equal up to a set of measure zero. This identification is necessary because we only have for a set M ⊂  of measure zero  |f (x)|p dx = 0 ⇒ f (x) = 0 for all x ∈ \M. 

Strictly speaking, the elements in Lp (), p ∈ [1, ∞] are classes of equivalent functions (equal up to a set of measure zero). Then, the first condition in Definition 1.1 becomes true. The second condition follows directly from the definition of  · Lp () and the third condition is just the inequality stated in Lemma 1.1.

Lemma 1.1 (Minkowski’s inequality). For f , g ∈ Lp (), p ∈ [1, ∞], we have f + gLp () ≤ f Lp () + gLp () . Theorem 1.1. Lp (), p ∈ [1, ∞], is a Banach space. Lemma 1.2 (Hölders’s inequality). For f ∈ Lp () and g ∈ Lq () with p, q ∈ [1, ∞], 1/p + 1/q = 1, we have fg ∈ L1 () and fgL1 () ≤ f Lp () gLq () . Definition 1.6. Two norms  · 1 and  · 2 on a normed space X are called equivalent if there exist positive constants C1 , C2 such that C1 x1 ≤ x2 ≤ C2 x1

for all x ∈ X.

Note that topological properties do not change when switching to an equivalent norm, for example, a sequence is convergent in (X,  · 1 ) iff it is convergent in (X,  · 2 ). Definition 1.7. Let (X,  · X ) be a normed space. A mapping g : X → R is called linear if g(αx + βy) = αg(x) + βg(y)

for all α, β ∈ R, x, y ∈ X.

A linear mapping g : X → R is continuous if there is a constant C such that |g(x)| ≤ CxX

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for all x ∈ X.

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3

1.1. banach and hilbert spaces

Definition 1.8. We define the sum of two continuous linear functionals, g1 and g2 , and the multiplication of a continuous linear functional g with a real number α by (g1 + g2 )(x) := g1 (x) + g2 (x)

and (αg)(x) := αg(x),

α ∈ R, x ∈ X.

Then, it can be shown that the set of all continuous linear functionals create a real linear space, the dual space X ∗ . In the following, for the elements g ∈ X ∗ we use the notation g, x := g(x). Lemma 1.3. The set X ∗ of continuous linear functionals x → g, x on X is a Banach space with respect to the norm | g, x| . 0=x∈X xX

gX ∗ := sup

Proof. Using the definition of the dual norm  · X ∗ , we see gX ∗ = 0

⇔

g, x = 0 for all x ∈ X

⇔

g = 0.

Further, we have |λ g, x| | g, x| | λg, x| = sup = |λ| sup = |λ| gX ∗ 0=x∈X xX 0=x∈X xX 0=x∈X xX

λgX ∗ = sup

and for the sum of two functionals it holds | g1 + g2 , x| | g1 , x + g2 , x| = sup x xX X 0=x∈X 0=x∈X

g1 + g2 X ∗ = sup

| g2 , x| | g1 , x| + sup = g1 X ∗ + g2 X ∗ 0=x∈X xX 0=x∈X xX

≤ sup

It remains to show that (X ∗ ,  · X ∗ ) is complete. Let (gn )n∈N be a Cauchy sequence in X ∗ , that is, for all ε > 0 there is an index n0 (ε) such that for all m, n > n0 (ε) it holds gm − gn X ∗ < ε. Then, for a fixed x ∈ X, the sequence of real numbers ( gn , x)n∈N is a Cauchy sequence in R due to | gm , x − gn , x| = | gm − gn , x| ≤ gm − gn X ∗ xX < εxX for all m, n > n0 (ε). Any Cauchy sequence of real numbers is convergent in R, that means limn→∞ gn , x = g(x) for all x ∈ X. The linearity of g follows from   g(αx + βy) = lim gn , αx + βy = lim α gn , x + β gn , y n→∞

n→∞

= α lim gn , x + β lim gn , y = αg(x) + βg(y). n→∞

n→∞

Concerning the continuity of g we recall that Cauchy sequences are bounded, thus | gn , x| ≤ gn X ∗ xX ≤ CxX

for all x ∈ X.

The limit n → ∞ shows that the linear mapping g : X → R is continuous.

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introduction to sobolev spaces

Definition 1.9. Let V be a linear space. A mapping (·, ·) : V × V → R is called an inner product on V if (u, v) = (v, u) for all u, v ∈ V , (αu + βv, w) = α(u, w) + β(v, w) for all u, v, w ∈ V , α, β ∈ R, • (u, u) > 0 for u = 0. √ A norm on V is induced by setting u := (u, u). If (V ,  · V ) is complete we call V a Hilbert space. •

•

Lemma 1.4 (Schwarz inequality). Let V be a Hilbert space. Then, |(u, v)| ≤ uV vV

for all u, v ∈ V .

Example 1.2. V = L2 () equipped with the inner product  (f , g) := f (x)g(x) dx 

is a Hilbert space. Theorem 1.2 (Riesz representation theorem). Let V be a Hilbert space. There is a unique linear mapping R : V ∗ → V from the dual space V ∗ into the Hilbert space V such that for all g ∈ V ∗ and v∈V (Rg, v) = g, v

and RgV = gV ∗ .

Fixed point theorems are often used to establish the unique solvability of an abstract operator equation. Let us consider the following problem in a Banach space V with an operator P : V → V: Find u ∈ V such that u = Pu.

(1.1)

Solutions u ∈ V of Problem (1.1) are called fixed points of the mapping P : V → V . Definition 1.10. The mapping P : V → V is called contractive or to be a contraction if there is a positive constant ρ < 1 such that Pv1 − Pv2 V ≤ ρ v1 − v2 V

for all v1 , v2 ∈ V .

Theorem 1.3 (Banach’s fixed point theorem). Let V be a Banach space and P : V → V be a contraction. Then, we have the following statements: (1) There is a unique fixed point u∗ ∈ V solving Problem (1.1). (2) The sequence un = Pun−1 , n ≥ 1, converges to the fixed point u∗ ∈ V for any initial guess u0 ∈ V . (3) It holds the error estimate un − u∗ V ≤

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ρn Pu0 − u0 V 1−ρ

for all n ∈ N.

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1.2.

5

weak derivatives

Proof. We start by estimating the distance of ui ∈ V to ui−1 ∈ V

ui − ui−1 V = Pui−1 − Pui−2 V ≤ ρui−1 − ui−2 V ≤ ρ 2 ui−2 − ui−3 V .. .

.. .

≤

ui − ui−1 V ≤ ρ i−1 Pu0 − u0 V

for all i ∈ N.

From that we get for m > n ≥ 1  m  m      ui − ui−1 V um − un V =  (ui − ui−1 ) ≤   i=n+1

≤

≤

m 

i=n+1 ρn

1−ρ

V

i=n+1

ρ i−1 Pu0 − u0 V ≤ ρ n Pu0 − u0 V → 0

1 − ρ m−n Pu0 − u0 V 1−ρ

for n → ∞.

Therefore, for any ε > 0, there exists an index bound n0 (ε) such that for all m > n ≥ n0 (ε) um − un V ≤

ρn Pu0 − u0 V < ε, 1−ρ

consequently, (un ) is a Cauchy sequence and converges to some u∗ ∈ V . We show that u∗ is a fixed point of P. Indeed, u∗ − Pu∗ V ≤ u∗ − un V + Pun−1 − Pu∗ V

≤ u∗ − un V + ρ un−1 − u∗ V → 0 for n → ∞.

Next we prove that there is at most one fixed point. Assume that there are two fixed points u1∗ = u2∗ . Then, we conclude u1∗ − u2∗ V = Pu1∗ − Pu2∗ V ≤ ρu1∗ − u2∗ V

⇒

(1 − ρ) u1∗ − u2∗ V ≤ 0.

Since ρ < 1 we get u1∗ = u2∗ . Finally, we recall the estimate for m > n ≥ 1 un − u∗ V ≤ un − um V + um − u∗ V ≤ from which the last statement follows by m → ∞.

ρn Pu0 − u0 V + um − u∗ V 1−ρ



1.2. Weak derivatives ................................................................................................................

First we introduce a generalization of the integration by parts formula known for smooth functions u, v : [a, b] → R:  b  b u′ v dx = uv|ba − uv ′ dx. a

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We start with a characterization of geometric properties of a domain  ⊂ Rd . Definition 1.11. The domain  ⊂ Rd is said to have a Lipschitz continuous boundary, if for every point of the boundary Ŵ = ∂ there exists a local coordinate system in which the boundary corresponds to some hypersurface with the domain  lying on one side of that surface and ∂ can locally be represented by the graph of a Lipschitz continuous mapping. Theorem 1.4 (Gauss theorem). Let  ⊂ Rd be a bounded domain with Lipschitz continuous boundary Ŵ and outer normal n. Then, for w ∈ C 1 ()d we have 



div w dx =

   d ∂wi dx = w · n dγ . ∂xi  Ŵ i=1

Let j ∈ {1, 2, . . . , d} be a fixed index. The formula of integration by parts follows by setting wi = uv for i = j and wi = 0 for i = j, and using the product rule. Lemma 1.5 (Integration by parts). For u, v ∈ C 1 () we have    ∂u ∂v v dx = uvnj dγ − u dx,  ∂xj Ŵ  ∂xj

j = 1, . . . , d.

Now we have the tools to generalize the concept of differentiability. Definition 1.12. The support of a function f :  → R is defined by supp f = {x ∈  : f (x) = 0}. The set of all functions that are differentiable of any order with compact support in  will be denoted by C0∞ (). For  = (0, 1), the function x → x(1 − x) does not belong to C0∞ () because its support is [0, 1] ⊂ (0, 1). Indeed, functions in C0∞ () are zero in the neighbourhood of the boundary ∂. As a consequence, a function f ∈ C0∞ () can be extended to a smooth function defined on Rd by setting f = 0 outside of . Definition 1.13. We define 1 Lloc () := {f :  → R : f ∈ L1 (A) for all compact A ⊂ }. 1 (). For bounded domains , we have L2 () ⊂ L1 () ⊂ Lloc 1 () but not to L1 (). Example 1.3. Consider  = (0, 1). The mapping x → 1/x belongs to Lloc √ 1 2 The mapping x → 1/ x belongs to L () but not to L ().

Definition 1.14. Let α = (α1 , . . . , αd ) be a multi-index, that is, αi are non-negative integers and 1 () has a weak derivative v = D α f ∈ L1 () if for any |α| = α1 + · · · + αd . A function, f ∈ Lloc loc ϕ ∈ C0∞ ()   v ϕ dx = ( − 1)|α| f Dα ϕ dx. 

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1.3. sobolev spaces

The notion of a weak derivative generalizes the classical partial derivatives. Indeed, a partial derivative in the classical sense satisfies the above condition by partial integration. √ Example 1.4. Let f : ( − 1, +1) → R with f (x) = 2 − |x|. Then, the generalized derivative 1 ( − 1, +1) is given by Df ∈ Lloc sgn x Df (x) = − √ 2 |x|

x ∈ ( − 1, 0) ∪ (0, +1).

Proof. Applying integration by parts separately in the two subintervals, we get for ϕ ∈ C0∞ ( − 1, +1)  0  +1  +1 1 1 sgn x − √ ϕ dx = √ ϕ dx − √ ϕ dx 2 |x| −1 2 |x| 0 −1 2 |x|   0

 = 2 − |x| ϕ|0x=−1 − 2 − |x| ϕ ′ dx



+ 2 − |x| ϕ|+1 x=0 − =−



+1

−1



−1 +1



0



 2 − |x| ϕ ′ dx

2 − |x| ϕ ′ dx,

where we used the continuity of ϕ at x = 0 and ϕ( ± 1) = 0.



1.3. Sobolev spaces ................................................................................................................

Definition 1.15. We denote by W m,p (), 1 ≤ p ≤ ∞, m ≥ 0, the set of all functions, f ∈ Lp () with weak derivatives Dα f ∈ Lp () up to the order |α| ≤ m. The sets W m,p () are called Sobolev spaces. Lemma 1.6. The Sobolev space W m,p () equipped with the norm ⎛ ⎞1/p   |Dα f (x)|p dx ⎠ if 1 ≤ p < ∞ f W m,p () := ⎝  |α|≤m

and   f W m,p () := max ess supx∈ |Dα f (x)| |α|≤m

if p = ∞

is a Banach space. In particular, the Sobolev space H m () := W m,2 () is a Hilbert space with respect to the inner product    α α (u, v)H m () := (D u, D v) = Dα u Dα v dx for all u, v ∈ H m (). |α|≤m

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introduction to sobolev spaces

Proof. Here, we give the arguments for 1 ≤ p < ∞. For the case p = ∞, we refer to Adams (1975). The sets W m,p () are real linear spaces (below we show that the sum (f + g) of two elements f , g ∈ W m,p () is an element of W m,p ()). Next we have to show that the mapping f → f W m,p () is a norm, that is, the three conditions in Definition 1.1 hold. If f = 0, then f W m,p () = 0 by definition of  · W m,p () . Further, since 0 = f W m,p () ≥ f Lp () we conclude f = 0 in the sense of Lp (), that is, functions which are equal up to a set of measure zero are identified. Thus, the first condition in Definition 1.1 holds. The second condition follows directly from the definition ⎛ ⎞1/p ⎛ ⎞1/p     λf W m,p () = ⎝ |Dα λf (x)|p dx ⎠ = ⎝ |λ|p |Dα f (x)|p dx ⎠  |α|≤m

⎛

= |λ| ⎝







 |α|≤m

⎞1/p

|Dα f (x)|p dx ⎠

|α|≤m

= |λ| f W m,p () .

Let f , g ∈ W m,p () and |α| ≤ m. The Minkowski’s inequality, Lemma 1.1, implies that Dα (f + g) = Dα f + Dα g ∈ Lp () and Dα (f + g)Lp () ≤ Dα f Lp () + Dα gLp () . Using the Minkowski’s inequality for sums, we get    p p p Dα (f + g)Lp () ≤ Dα f Lp () + Dα gLp () f + gW m,p () = |α|≤m

⎡⎛

⎢ ≤ ⎣⎝



|α|≤m

|α|≤m

p

⎞1/p

Dα f Lp () ⎠

⎛

+⎝



|α|≤m

 p ≤ f W m,p () + gW m,p ()

⎞1/p ⎤p ⎥ p Dα gLp () ⎠ ⎦

from which the third condition follows. It remains to prove the completeness of the normed space W m,p (). Let (fn )n∈N be a Cauchy sequence in W m,p (). Then, (Dα fn )n∈N is a Cauchy sequence in Lp () for any multi-index α with |α| ≤ m. Since Lp () is a Banach space, there is f α ∈ Lp () with Dα fn → f α in Lp (). It remains to show that f α = Dα f where f = f 0 . We show this for α = (1, 0, . . . , 0). For any ϕ ∈ C0∞ () we have           (f α ϕ + fDα ϕ)dx  ≤ |f α − Dα fn | |ϕ|dx +  (Dα fn ϕ + fn Dα ϕ)dx      





+





|f − fn | |Dα ϕ|dx.

The second term on the right hand side equals zero since Dα fn is the weak derivative of fn . The convergence of fn to f and Dα fn to f α in Lp (), respectively, imply the convergence in L1 (), thus the first and third term on the right hand side tend to zero for n → ∞. The left hand side

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1.3. sobolev spaces

is non-negative and independent of n, consequently we conclude that it is zero, which means f α is the weak derivative of f . The properties for the inner product in H m (), see Definition 1.9, follow from its definition  and the observation that (v, v)H m () = v2W m,2 () for all v ∈ W m,2 (). m,p

Definition 1.16. We denote by W0 () the closure of C0∞ () in the norm  · W m,p () . Analogously, H0m () is the closure of C0∞ () in H m (). In the following, for f ∈ W m,p (ω), m ≥ 0, 1 ≤ p ≤ ∞, the norm will be shortly denoted by f m,p,ω := f W m,p (ω) and a seminorm is introduced by ⎛

|f |m,p,ω := ⎝





ω |α|=m

⎞1/p

|Dα f (x)|p dx ⎠

if 1 ≤ p < ∞

and   |f |m,p,ω := max ess supx∈ |Dα f (x)| |α|=m

if p = ∞.

In case, ω =  and/or p = 2 we omit  and p, respectively. Thus, f m = f m,2, ,

f m,p = f m,p, ,

|f |m = |f |m,2, ,

|f |m,p = |f |m,p, .

Lemma 1.7 (Poincaré inequality). There is a positive constant cP ≤ 2a such that v0,p ≤ cP |v|1,p

1,p

for all v ∈ W0 ().

Here, ‘a’ denotes the diameter of  in an arbitrary but fixed direction. Proof. Suppose the Poincaré inequality is true for functions from C0∞ (). Then, for any 1,p 1,p v ∈ W0 () there is a sequence vn ∈ C0∞ () with vn → v in W0 (). We have v0,p ≤ v − vn 0,p + vn 0,p ≤ v − vn 0,p + cP |vn |1,p ≤ (1 + cP )v − vn 1,p + cP |v|1,p . Since v − vn 1,p tends to zero for n → ∞, the Poincaré inequality is true for functions from 1,p W0 (). In order to prove the Poincaré inequality for functions v ∈ C0∞ () we extend v :  → R by setting v(x) = 0 for all x ∈ Rd \. Let  ⊂ [ − a, +a] × Rd−1 , then v(x) =

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x1

−a

∂v (s, x2 , . . . , xd ) ds ∂x1

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and Hölder’s inequality with 1/p + 1/q = 1 implies p 1/p  x1 1/q  x1   ∂v  q   |v(x)| ≤ , 1 ds (s, x2 , . . . , xd ) ds  −a −a ∂x1 p  +a   ∂v  p p/q   ds, |v(x)| ≤ (2a) (s, x , . . . , x ) 2 d  ∂x  1 −a p  +a   +a   ∂v p 1+p/q   |v(x)| dx1 ≤ (2a)  ∂x (s, x2 , . . . , xd ) ds, 1 −a −a p     ∂v  p p  |v(x)| dx ≤ (2a) (x) dx.  Rd ∂x1 Rd Using v(x) = 0 for x ∈ Rd \ the inequality follows.



The Poincaré inequality allows to show that the seminorm | · |m,p is equivalent to the norm m,p m,p 1,p  · m,p on W0 (). Let v ∈ W0 (), then Dα v ∈ W0 () for |α| ≤ m − 1. We apply successively the inequality |Dα v|0,p ≤ cP |Dα v|1,p

for |α| = 0, 1, . . . , m − 1

to estimate the lower derivatives by the highest derivative and obtain ⎛ ⎞1/p  p |Dα v|0,p ⎠ ≤ C|v|m,p , |v|m,p ≤ vm,p = ⎝ |α|≤m

that is, the equivalence of the seminorm to the norm. An important tool will be the Sobolev embedding theorems which state that functions from ∗ W m,p () belong to W m−1,p for some p∗ > p. Moreover, it can be shown that for suitable numbers m, p functions from W m,p () belong to classical spaces of continuous and continuously differentiable functions, respectively. Definition 1.17. We introduce the space of Hölder continuous functions with Hölder exponent β ∈ (0, 1]   |u(x) − u(y)| 0 0,β C () := u ∈ C () : sup