Approximation of Functions in Sobolev Spaces by Smooth Functions in ΩFunctions inΩ

Basic Properties of Sobolev Spaces

1.1.5 Approximation of Functions in Sobolev Spaces by Smooth Functions in ΩFunctions inΩ

Let 1≤p <∞. The following two theorems show the possibility of approxi- mating any function inL^l_p(Ω) andW_p^l(Ω) by smooth functions onΩ.

Theorem 1. The spaceL^l_p(Ω)∩C^∞(Ω)is dense in L^l_p(Ω).

Proof. Let {Bk}k≥1 be a locally finite covering of Ω by open balls Bk

with radiirk, ¯Bk ⊂Ω, and let{ϕk}k≥1be a partition of unity subordinate to this covering. Letu∈L^l_p(Ω) and let {k} be a sequence of positive numbers which monotonically tends to zero so that the sequence of balls{(1 +k)Bk} has the same properties as{Bk}. IfBk =B(x), then by definition we put cBk=Bc(x). Letwk denote the mollification ofuk =ϕkuwith radiuskrk. Clearly,w=

w_k belongs toC^∞(Ω). We takeε∈(0,1/2) and choosek to satisfy

u_k−w_kL^l_p(Ω)≤ε^k. On any bounded open setω, ¯ω⊂Ω,we have

u= uk,

where the sum contains a finite number of terms. Hence, u−wL^l_p(Ω)≤ uk−wkL^l_p(Ω)≤ε(1−ε)⁻¹. Therefore,w∈L^l_p(Ω)∩C^∞(Ω) and

u−wL^l_p(Ω)≤2ε.

The theorem is proved.

The next theorem is proved similarly.

Theorem 2.The spaceW_p^l(Ω)∩C^∞(Ω)is dense inW_p^l(Ω)and the space V_p^l(Ω)∩C^∞(Ω)is dense in V_p^l(Ω).

Remark.It follows from the proof of Theorem 1 that the space L^l_p(Ω)∩ C^∞(Ω)∩C( ¯Ω) is dense in L^l_p(Ω)∩C( ¯Ω) if Ω has a compact closure. The same is true ifL^l_pis replaced byW_p^l or byV_p^l.

In fact, letk be such that

uk−wkC( ¯Ω)≤ε^k.

We put

VN =

N

k=1

w_k+

∞ k=N+1

u_k. Then

sup

x∈Ω

w(x)−VN(x)≤ ^∞

k=N+1

uk−wkC( ¯Ω)≤2ε^N+1,

and hencew∈C( ¯Ω) sincewis the limit of a sequence inC( ¯Ω). On the other hand,

u−wC( ¯Ω)≤

∞ k=1

u_k−w_kC( ¯Ω)≤2ε,

which completes the proof.

1.1.6 Approximation of Functions in Sobolev Spaces by Functions in C^∞( ¯Ω)

We consider a domainΩ⊂R²for whichC^∞(Ω) cannot be replaced byC^∞( ¯Ω) in Theorems 1.1.5/1 and 1.1.5/2. We introduce polar coordinates (, θ) with 0≤θ <2π. The boundary of the domainΩ={(, θ) : 1< <2,0< θ <2π} consists of the circles= 1,= 2, and the interval{(, θ) : 1< <2, θ= 0}. The functionu=θ is integrable onΩalong with all its derivatives, but it is not absolutely continuous on segments of straight linesx= const>0, which intersect Ω. According to Theorem 1.1.3/1 the function u does not belong to L^l_p(Ω₁), where Ω₁ is the annulus Ω ={(, θ) : 1 < < 2,0 ≤ θ <2π}. Hence, the derivatives of this function cannot be approximated in the mean by functions inC^∞( ¯Ω).

A necessary and sufficient condition for the density ofC^∞( ¯Ω) in Sobolev spaces is unknown. The following two theorems contain simple sufficient con- ditions.

Definition.A domainΩ⊂Rⁿis calledstarshaped with respect to a point O if any ray with originO has a unique common point with∂Ω.

Theorem 1.Let 1 ≤p <∞. If Ω is a bounded domain, starshaped with respect to a point, thenC^∞( ¯Ω)is dense inW_p^l(Ω)andV_p^l(Ω),p∈[1,∞). The same is true for the space L^l_p(Ω), i.e., for anyu∈L^l_p(Ω)there is a sequence {ui}i≥1 of functions inC^∞( ¯Ω)such that

ui→u inLp(Ω,loc) and ∇l(ui →u)

Lp(Ω)→0.

Proof.Letu∈W_p^l(Ω). We may assume thatΩis starshaped with respect to the origin. We introduce the notationu_τ(x) =u(τ x) forτ ∈(0,1). We can easily see thatu−u_τL_p(Ω)→0 asτ →1.

1.1 The SpacesLp(Ω),Vp(Ω) andWp(Ω) 11 From the definition of the distributional derivative it follows thatD^α(uτ) = τ^l(D^αu)_τ,|α|=l. Henceu_τ ∈W_p^l(τ⁻¹Ω) and

D^α(u−uτ)

Lp(Ω)≤D^αu

τ−D^α(uτ)

Lp(Ω)+D^αu− D^αu

τ

Lp(Ω)

≤

1−τ^lD^αu

τ

L_p(Ω)+D^αu− D^αu

τ

L_p(Ω). The right-hand side tends to zero asτ→1. Therefore,u_τ →uinW_p^l(Ω).

Since ¯Ω⊂τ⁻¹Ω, the sequence of mollifications ofuτ converges to uτ in W_p^l(Ω). Now, using the diagonalization process, we can construct a sequence of functions inC^∞( ¯Ω) that approximates uin W_p^l(Ω). Thus we proved the density ofC^∞( ¯Ω) inW_p^l(Ω). The spacesL^l_p(Ω) andV_p^l(Ω) can be considered in an analogous manner.

Theorem 2. Let 1 ≤ p < ∞. Let Ω be a domain with compact closure of the class C. This means that every x ∈ ∂Ω has a neighborhood U such that Ω∩U has the representation xn < f(x1, . . . , xn−1) in some system of Cartesian coordinates with a continuous function f. Then C^∞( ¯Ω)is dense in W_p^l(Ω),V_p^l(Ω), andL^l_p(Ω).

Proof.We limit consideration to the spaceV_p^l(Ω). By Theorem1.1.5/2 we may assume thatu∈C^∞(Ω)∩V_p^l(Ω).

Let {U} be a small covering of ∂Ω such that U ∩∂Ω has an explicit representation in Cartesian coordinates and let{η}be a smooth partition of unity subordinate to this covering. It is sufficient to construct the required approximation foruη.

We may specifyΩby Ω=

x= (x, x_n) :x∈G, 0< x_n < f(x) ,

whereG⊂Rⁿ⁻¹andf ∈C( ¯G),f >0 onG. Also we may assume thatuhas a compact support inΩ∪ {x:x ∈G, x_n=f(x)}.

Let ε denote any sufficiently small positive number. Obviously, u_ε(x) = u(x, x_n −ε) is smooth on ¯Ω. It is also clear that for any multi-index α, 0≤ |α| ≤l, we have

D^α(u−u)

Lp(Ω)=D^αu

ε−D^αu

Lp(Ω)→0

as ε→+0. The result follows.

Remark.The domain Ω, considered at the beginning of this section, for whichC^∞( ¯Ω) is not dense in Sobolev spaces, has the property∂Ω=∂Ω. We¯ might be tempted to suppose that the equality∂Ω =∂Ω¯ provides the density ofC^∞( ¯Ω) inL^l_p(Ω). The following example shows that this conjecture is not true.

Example.We shall prove the existence of a bounded domainΩ⊂Rⁿ such that ∂Ω=∂Ω¯ andL¹_p(Ω)∩C( ¯Ω) is not dense inL¹_p(Ω).

We start with the casen= 2. LetK be a closed nowhere dense subset of the segment [−1,1] and let {Bi} be a sequence of open disks constructed on adjacent intervals ofKtaken as their diameters. LetBbe the diskx²+y²<4 and letΩ=B\∪B_i. We can chooseKso that the linear measure ofΓ ={x∈ K:|x|<1/2}is positive. Consider the characteristic functionθof the upper halfplane y > 0 and a function η ∈ C₀^∞(−1,1) which is equal to unity on (−1/2,1/2).

The functionU, defined by

U(x, y) =η(x)θ(x, y),

belongs to the space L¹_p(B) for all p ≥1. Suppose that u_j → U in L¹_p(Ω), where {uj}j≥1 is a sequence of functions inC( ¯Ω)∩L¹_p(Ω). According to our assumption, for almost allx∈Γ and for allδ∈(0,1/2),

uj(x, δ)−uj(x,−δ) = δ

−δ

∂uj(x, y)

∂y dy.

Hence

Γ

u_j(x, δ)−uj(x,−δ)dx≤

Γ(δ)

gradu_j(x, y)dxdy, where Γ(δ) =Γ×(−δ, δ).

Sinceuj →U inL¹₁(Ω), the integrals

Γ(δ)

graduj(x, y)dxdy, j≥1,

are uniformly small. Therefore, for each ε >0 there exists aδ0>0 such that for allδ∈(0, δ0)

Γ

uj(x, δ)−u_j(x,−δ)dx < ε.

Applying Fubini’s theorem, we obtain that the sequence in the left-hand side

converges to

Γ

U(x, δ)−U(x,−δ)dx=m1(Γ)

as j → ∞ for almost all small δ. Hence m1(Γ) ≤ ε which contradicts the positiveness ofm1(Γ). Since∂Ω=∂Ω, the required counterexample has been¯ constructed forn= 2.

In the casen >2, let Ω2 denote the plane domain considered previously, put Ω=Ω2×(0,1)ⁿ⁻², and duplicate the above argument.

1.1.7 Transformation of Coordinates in Norms of Sobolev Spaces LetH andGbe domains inRⁿ and let

1.1 The SpacesLp(Ω),Vp(Ω) andWp(Ω) 13 T :y→x(y) =

x1(y), . . . , xn(y) , be a homeomorphic map ofH ontoG.

We say thatT is aquasi-isometric map, if for anyy₀∈H, x₀∈G, lim sup

y→y0

|x(y)−x(y0)|

|y−y₀| ≤L, lim sup

x→x0

|y(x)−y(x0)|

|x−x₀| ≤L, (1.1.3) and the Jacobian detx(y) preserves its sign inH.

We can check that the estimates (1.1.3) are equivalent to x(y) ≤L a.e. on H, y(x) ≤L a.e. onG,

where x, y are the Jacobi matrices of the mappings y → x(y), x → y(x) and · is the norm of the matrix. This immediately implies that the quasi- isometric map satisfies the inequalities

L⁻ⁿ ≤detx(y)≤Lⁿ. (1.1.4) By definition, the map T belongs to the class C^l−1,1( ¯H), l ≥ 1, if the functions y →x_i(y) belong to the classC^l−1,1( ¯H). It is easy to show that if T is a quasi-isometric map of the class C^l−1,1( ¯H), then T⁻¹ is of the class C^l−1,1( ¯G).

Theorem. Let T be a quasi-isometric map of the classC^l−1,1( ¯H),l≥1, which mapsH ontoG. Letu∈V_p^l(G) andv(y) =u(x(y)). Then v∈V_p^l(H) and for almost ally∈Hthe derivativesD^αv(y),|α| ≤lexist and are expressed by the classical formula

D^αv(y) =

1≤|β|≤|α|

ϕ^α_β(y) D^βu

x(y)

. (1.1.5)

Here

ϕ^α_β(y) =

s

c_s n i=1

j

D^sijx_i (y),

and the summation is taken over all multi-indices s = (sij) satisfying the conditions

i,j

si,j=α, |sij| ≥1,

i,j

|sij| −1

=|α| − |β|. Moreover, the normsvV_p^l(H) anduV_p^l(G) are equivalent.

Proof. Let u ∈ C^∞(G)∩V_p^l(G). Then v is absolutely continuous on al- most all straight lines that are parallel to coordinate axes. The first partial derivatives ofv are expressed by the formula

∂v(y)

∂ym

=

n

i=1

∂xi(y)

∂ym

∂u

∂xi

x(y)

, (1.1.6)

for almost ally. Since

∇vLp(H)≤c∇uLp(G),

it follows by Theorem1.1.3/2 thatv∈V_p¹(H). After the approximation of an arbitrary u∈ V_p^l(G) by functions in C^∞(G)∩V_p¹(G) (cf. Theorem 1.1.5/2) the result follows in the case l= 1.

Forl >1 we use induction. Let (1.1.5) hold for|α|=l−1. SinceD^βu∈ V_p¹(G), the functionsy→(D^βu)(x(y)) belong to the space V_p^l(H). This and the inclusion ϕ^α_β ∈ C^0,1( ¯H) imply that each term on the right-hand side of (1.1.4) with |α| =l−1 belongs toV_p¹(H). Applying (1.1.6) to (1.1.5) with

|α|=l, we obtain

∇lvL_p(H)≤cuV_p^l(G).

The result follows.