Universal Extension for Sobolev Spaces of Differential ...hiptmair/org/Oberwolfach/... ·...

Outline

Universal Extension for Sobolev Spaces ofDifferential Forms and Applications

Jingzhi Li1

Joint work with Ralf Hiptmair and Jun Zou

1Seminar fur Angewandte MathematikETH Zurich

Oberwolfach Workshop15 Feb., 2010

Jingzhi Li Oberwolfach Workshop 2010

Outline

Outline

1 Motivation

2 Preliminaries

3 Universal extensionMain resultTechnical lemmataUniversal extension formulaSketch of the proof

4 Application : regular decomposition


Outline

Outline

1 Motivation

2 Preliminaries




Outline

Outline

1 Motivation

2 Preliminaries




Outline

Outline

1 Motivation

2 Preliminaries




MotivationPreliminaries

Universal extensionApplication : regular decomposition

Classical Extension

Stein extension operator [Stein 70]

E : C∞(Ω) 7→ C∞(Rd) , E u(x) = u(x) ∀x ∈ Ω ,

which fulfills that for any m ∈ N0, 1 ≤ p <∞,

∃ C = C(m, p,Ω) > 0 : ‖E u‖W m,p(Rd ) ≤ C‖u‖W m,p(Ω) ∀ u ∈ C∞(Ω) .




Review and applications

Extension operators for various function spaces on Lipschitzdomains and beyond, e.g, [Rychkov99,ROG06,HEK92,JON81]

Interpolation spaces and regularity estimates in PDEs,[Mitrea08b,Mitrea07]

Fundamental to the theoretical analysis of, e.g., electromagneticphenomena governed by Maxwell’s equation[Hiptmair02,Monk03,Mitrea04,Douglas06], the Navier-Stokesequation [Girault86].




Notations (1)

Exterior and interior products

(ω∧η)(v1, . . . , vl+k) =∑

σ

sgn(σ)ω(vσ(1), . . . , vσ(l))η(vσ(l+1), . . . , vσ(l+k)),

ay ω =l∑

k=1

(−1)k−1ajk dxj1∧ · · · ∧dx jk∧ · · · ∧dxjl ∈∧l−1

,

Differential forms

A differential form ω of degree l, l ∈ N0, and class Cm, m ∈ N0, in Ω isa l-form valued mapping

ω =∑

IωIdxI : x ∈ Ω ⊂ R

d 7→ ω(x) ∈∧l

,




Notations (2)

Sobolev spaces of differential forms

Hs(Ω;∧l

(Rd )) (s ∈ R+0 ) denotes the space consisting of all

differential forms with each component in Hs(Ω) with norm

‖ω‖2Hs(Ω;

∧ l (Rd )) :=∑

I

‖ωI‖2Hs(Ω)

Pullback

If T : Ω 7→ Ω, is a diffeomorphism between two manifolds in Rd , then

T ∗ : DF l,∞(Ω) 7→ DF l,∞(Ω) is given by

((T ∗ω)(x))(v1, . . . ,vl) = (ω(T (x)))(DT (x)v1, . . . ,DT (x)vl),

where v1, . . . ,vl ∈ Rd and the linear map DT (x) : R

d 7→ Rd is the

derivative (Jacobian) of T at x.




Notations (3)

exterior derivative and CD property

dω :=

d∑

i=1

∑

I

∂ωI

∂xidxi ∧ dxI ∈ DF l+1,∞(Ω) ,

T∗(dω) = d(T ∗ω), ∀ ω ∈ DF l,∞(Ω)

Sobolev spaces of differential forms

Hs(d ,Ω,∧l

(Rd)) :=

ω ∈ Hs(Ω;

∧l(Rd )) | dω ∈ Hs(Ω;

∧l+1(Rd))

with the graph norm

‖ω‖2Hs(d,Ω,

∧ l (Rd )) := ‖ω‖2Hs(Ω,

∧ l (Rd )) + ‖dω‖2Hs(Ω,

∧ l+1(Rd )) .




Main resultTechnical lemmataUniversal extension formulaSketch of the proof

Outline

1 Motivation

2 Preliminaries







Main result

Theorem

Let Ω be a Lipschitz epigraph in Rd , k ∈ N0 and 0 ≤ l ≤ d. Then

there exists an extension operator El satisfing

‖Elω‖Hk (d,Rd ,∧ l (Rd )) ≤ C ‖ω‖Hk (d,Ω,

∧ l (Rd )) ∀ ω ∈ DF l,∞(Ω) ,

with a constant C = C(Ω, d , k , l) > 0. Thus, El can be extended to acontinuous extension operator

El : Hk (d ,Ω,∧l

(Rd)) 7→ Hk (d ,Rd ,∧l

(Rd )) .





Outline

1 Motivation

2 Preliminaries







Technical lemma (1)

Lemma

[Regularized distance [Thm. 2, pp. 171, Stein70] For a closed domainΩ ∈ R

d , there exists a regularized distance function ∆(x) = ∆(x,Ω)

such that for x ∈ Ωc

i). cδ(x) ≤ ∆(x) ≤ Cδ(x);

ii). ∆(x) is C∞-smooth in Ωc

and∣∣ ∂α

∂xα ∆(x)∣∣ ≤ Cα(δ(x))1−|α|,

where c > 0 and C > 0 are constants independent of Ω and Cα > 0depends on the multi-index α.





Technical lemma (2)

Lemma

[Weighting function [Lemma 1, pp. 182, Stein70] The weightingfunction

ψ(λ) :=eπλ

ℑ(

exp(12

√2(−1 + i)(λ − 1)1/4)

)

is defined in [1,∞), and satisfies the decay property

ψ(λ) = O(λ−n) as λ→ ∞, ∀ n ∈ N,

and all its higher moments vanish

∫ ∞

1λkψ(λ) dλ =

1, for k = 0 ,

0, for k ∈ N .





Outline

1 Motivation

2 Preliminaries







Universal extension formula : special case (1)

x

y

x = (x, y)

Rλ

Rλ(x) = (x, y + λδ∗(x))

∂Ω = (x, y) | y = φ(x)

Ω

Ωc





Universal extension formula : special case (2)

Parametrized reflection

Rλ(x) = (x, y + λδ∗(x)) = x + λδ∗(x)ed .

Universal extension formula

(Elω)(x) :=

ω(x), x ∈ Ω;∫ ∞

1(R∗

λω)(x)ψ(λ) dλ, x ∈ Ωc.





Universal extension formula : Avatars in 3D

E (f )(x) =

∫ ∞

1f (x, y + λδ∗(x))ψ(λ) dλ .

E1u(x) =

∫ ∞

1( DRλ(x) )T u(x, •)ψ(λ) dλ

=

∫ ∞

1

(u(x, •) + λu3(x, •) grad δ∗(x)

)ψ(λ) dλ ,

E2u(x) =

∫ ∞

1( DRλ(x) )

−1 det ( DRλ(x) )u(x, •)ψ(λ) dλ

=

∫ ∞

1

((1 + λ

∂δ∗

∂x3(x))u(x, •) −

(00

λ grad δ∗(x) · u(x, •)

) )ψ(λ) dλ .

E3u(x) =

∫ ∞

1det ( DRλ(x) )u(x, •)ψ(λ) dλ

=

∫ ∞

1

(1 + λ

∂δ∗

∂x3(x)

)u(x, •)ψ(λ) dλ .





Outline

1 Motivation

2 Preliminaries







Sketch of the proof

Proof.

1 Special Lipschitz epigraph case for smooth differential forms;1 The extended differential form is well-defined;2 The norm of such extended differential form is bounded;

2 Apply the density argument;3 Generalize to Lipschitz domains by partition of unity.




Regular potential in Rd

Lemma (Existence of regular potentials in Rd )

For 1 ≤ l ≤ d, l ∈ N and every k ∈ N0 there is a continuous liftingmapping

L : H(d0,Rd ,∧l

(Rd)) ∩ Hk (Rd ,∧l

(Rd )) 7→ Hk+1loc (Rd ,

∧l−1(Rd))

such that for all ω ∈ H(d0,Rd ,∧l

(Rd)) ∩ Hk (Rd ,∧l

(Rd)),

dL ω = ω.

ω(ξ) := F (ω)(ξ) =∑

I

ωI(ξ)dξI , F (dω) = ıξ ∧ F (ω)

ωI(ξ) := F (ωI)(ξ) =1

(2π)d/2

∫

Rdexp(−ıξ · x)ωI(x) dx ,




Lifted regular decompositions

Theorem

(Lifted regular decompositions) For every k ∈ N0, 1 ≤ l ≤ d, thereexist continuous maps R : Hk (d ,Ω,

∧l(Rd)) 7→ Hk+1(Ω,∧l(Rd )) and

N : Hk (d ,Ω,∧l

(Rd)) 7→ Hk+1(Ω,∧l−1

(Rd)) such that

R + d N = Id on Hk (d ,Ω,∧l

(Rd)) . (1)

In addition, there are continuous mapsR0 : Hk

0(d ,Ω,∧l

(Rd )) 7→ Hk+10 (Ω,

∧l(Rd)) and

N0 : Hk0(d ,Ω,

∧l(Rd )) 7→ Hk+10 (Ω,

∧l−1(Rd)) such that

R0 + d N0 = Id on Hk0(d ,Ω,

∧l(Rd)) . (2)




Thank you




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