Boundary Value Problems: Higher Order Regularity Data in … · 2012. 7. 25. · This is joint work...

Boundary Value Problems: Higher OrderRegularity Data in Nonsmooth Settings

Dorina Mitrea

University of Missouri

UNC

July 20, 2012

Dorina Mitrea (MU) BVP: Higher Reg Data in Rough Settings 07/20/2012 1 / 38

Consider the following two distinct directions in the development of BVP

pseudodifferential calculus in smooth domains;

the Calderón-Zygmund theory of singular integral operators in thecontext of elliptic boundary value problems in non-smooth domains.

Basic issue left unresolved: reconciling these two theories by developinga theory which contains them as limiting end-points.

Focus on the solvability via BIM of the Dirichlet problem for ellipticsystems under optimal GMT assumptions and with boundary databelonging to higher order Sobolev spaces.

This is joint work with José Maria Martell and Marius Mitrea


Second order elliptic systems

Consider an M ×M system of scalar second-order differential operators(with the usual convention of summation over repeated indices)

Lu :=(∂r (a

αβrs ∂suβ)

)1≤α≤M

with constant complex coefficients, satisfying the Legendre-Hadamardellipticity condition: there exists κ > 0 such that

Re[aαβrs ξrξsηαηβ

]≥ κ|ξ|2|η|2, ∀ ξ = (ξr )1≤r≤n ∈ Rn,

∀ η = (ηα)1≤α≤M ∈ CM .

Call A :=(aαβrs)α,β,r ,s

the coefficient tensor of L.


Nontangential pointwise traces

In the formulation of BVP want traces to the boundary in the pointwisesense. Since we want to treat non-smooth domains, this leads toconsidering pointwise nontangential traces to the boundary.For c > 0 and x ∈ ∂Ω define the nontangential approach region

Γ(x) := Γc(x) := {y ∈ Ω : |y − x | < (1 + c) dist (y , ∂Ω)},

and if u : Ω→ R, the nontangential trace of u

u∣∣∂Ω

(x) := limΓc (x)3y→x

u(y), x ∈ ∂Ω,

and the nontangential maximal function of u

Nu(x) := Ncu(x) := sup {|u(y)| : y ∈ Γc(x)}, x ∈ ∂Ω.


The Dirichlet problem for elliptic systems

If Ω is an open set in Rn, p ∈ (1,∞), ` ∈ N, study the well-posedness ofthe following problem

u ∈ C∞(Ω) and Lu = 0 in Ω,

N(∇mu) ∈ Lp(∂Ω), ∀m ∈ {0, ..., `},

u∣∣∂Ω

= f ∈ Lp` (∂Ω),

satisfying, for some C ∈ (0,∞) independent of f , the estimate

∑̀m=0

‖N(∇mu)‖Lp(∂Ω) ≤ C‖f ‖Lp`(∂Ω)

Here Lp` (∂Ω) are Lp-based Sobolev spaces of order ` on ∂Ω (appropriately

defined).



Interested in boundary SIO methods: Look for a solution u as a doublelayer potential, i.e.,

u(x) := Dg(x) for x ∈ Ω,

with g : ∂Ω→ R yet to be determined where, with ν = outward unitnormal, and σ = surface measure,

Dg(x) :=(−∫∂Ωνs(y)a

βαrs (∂rEγβ)(x − y)gα(y) dσ(y)

)γ, x ∈ Ω.

where E =(Eαβ

)1≤α,β≤M is a fundamental (matrix) solution for L.



Let K be the boundary version of D

Kf (x) :=(− limε→0+

∫y∈∂Ω|x−y |>ε

νs(y)aβαrs (∂rEγβ)(x − y)fα(y) dσ(y)

)γ, x ∈ ∂Ω.

If the following jump relation holds

Dg∣∣∂Ω

=(

12 I + K

)g , a. e. on ∂Ω,

then, at least formally, the solution u of the Dirichlet problem for L isgiven by

u = D[( 12 I + K )

−1f]

In this scenario: need 12 I + K isomorphism of Lp` (∂Ω)...


Suppose one is interested in implementing an approach based on Fredholmtheory. Key ingredient: K is close to being compact on Lp` (∂Ω).Note that if

‖Kf ‖Lp`(∂Ω) ≤ ‖Comp(f )‖+ δ‖f ‖Lp`(∂Ω) (∗)

with δ ≥ 0, then

λI + K is Fredholm with index zero on Lp` (∂Ω) for |λ| > δ.

In particular, if (∗) holds for δ < 12 we have

12 I + K invertible on L

p` (∂Ω) ⇔

12 I + K injective on L

p` (∂Ω)

Question: What is the most general GMT context for which Fredholmtheory works?


In light of the approach outlined have a

New Goal: identify the most natural setting in which, for a givenp ∈ (1,∞) and a given ` ∈ N ∪ {0}, the following hold:

the space Lp` (∂Ω) is well-defined

K is well-defined and bounded on Lp` (∂Ω)∑`m=0 ‖N(∇mD(f ))‖Lp(∂Ω) ≤ C‖f ‖Lp`(∂Ω) for every f ∈ L

p` (∂Ω)

K is close to being compact on Lp` (∂Ω)12 I + K is injective on L

p` (∂Ω)


The low regularity case ` = 0

Let Hn−1 be the (n − 1)-dimensional Hausdorff measure in Rn.

A closed set Σ ⊆ Rn is Ahlfors regular if ∃C ≥ 1 s.t.C−1rn−1 ≤ Hn−1(B(x , r) ∩ Σ) ≤ Crn−1, ∀ x ∈ Σ, r > 0.

Definition (G. David and S. Semmes)

Call Σ ⊂ Rn uniformly rectifiable (UR) provided it is Ahlfors regular andthe following holds. There exist ε, M ∈ (0,∞) such that ∀ x ∈ Σ,∀R > 0, there is a Lipschitz map ϕ : Bn−1R → R

n (where Bn−1R is a ball ofradius R in Rn−1) with Lipschitz constant ≤ M, such that

Hn−1(B(x ,R) ∩ Σ ∩ ϕ(Bn−1R )

)≥ εRn−1.

Ω ⊆ Rn open will be called a UR domain if ∂Ω is UR andHn−1(∂Ω \ ∂∗Ω) = 0, where ∂∗Ω, the measure-theoretic boundary, is

∂∗Ω :=

{x ∈ ∂Ω : lim sup

r→0+

Hn(Br (x)∩Ω±

)rn > 0

}Dorina Mitrea (MU) BVP: Higher Reg Data in Rough Settings 07/20/2012 10 / 38


Theorem (Layer potentials corresponding to ` = 0)

Let Ω ⊂ Rn be a UR domain with outward unit normal ν and surfacemeasure σ := Hn−1b∂Ω on ∂Ω. Consider a function

k ∈ C N(Rn \ {0}) with k(−x) = −k(x) andk(λ x) = λ−(n−1)k(x) ∀λ > 0, ∀ x ∈ Rn \ {0},

and define the boundary to domain singular integral operator

Tf (x) :=

∫∂Ω

k(x − y)f (y) dσ(y), x ∈ Ω,

along with the boundary to boundary version

Tf (x) := limε→0+

∫y∈∂Ω, |x−y |>ε

k(x − y)f (y) dσ(y), x ∈ ∂Ω.



Theorem (cont.)

Fix p ∈ (1,∞). Then

T : Lp(∂Ω)→ Lp(∂Ω), is well-defined, linear and bounded.

In addition, ∃C = C (p,Ω) > 0 such that

‖N(Tf )‖Lp(∂Ω) ≤ C‖k |Sn−1‖C N‖f ‖Lp(∂Ω).

and, with ‘hat’ denoting the Fourier transform in Rn, the jump-formula

limΩ3z→xz∈Γκ(x)

Tf (z) = 12√−1 k̂(ν(x))f (x) + Tf (x)

is valid at σ-a.e. x ∈ ∂Ω, if f ∈ Lp(∂Ω).



This result, proved in this form by S. Hofmann, M. Mitrea and M. Taylor,builds on the work of many people, including Calderón,Coifman-McIntosh-Meyer, David, Jerison, Kenig, Semmes, Stein, ...

Applying this theorem to our setting (to operators associated to L), itfollows that K is well-defined and bounded on Lp(∂Ω), D has appropriatenontangential estimates, and the needed jump formula holds.

Next consider the issue of K being close to compact on Lp(∂Ω).



Recall a few definitions introduced byS. Hofmann, M. Mitrea and M. Taylor.

An open set Ω ⊆ Rn satisfies a local John condition if there existθ ∈ (0, 1) and R > 0 (R =∞ if ∂Ω unbounded), called the Johnconstants of Ω, with the following significance.

∀ x ∈ ∂Ω, ∀ r ∈ (0,R) there exists xr ∈ B(x , r) ∩ Ω such that(i) B(xr , θr) ⊂ Ω(ii) for each y ∈ B(x , θr) ∩ ∂Ω one can find a rectifiable path

γy : [0, 1]→ Ω, withlength(γy ) ≤ θ−1r ,γy (0) = y , γy (1) = xr ,

dist (γy (t), ∂Ω) > θ |γy (t)− y |, ∀ t ∈ (0, 1].



An open set Ω ⊆ Rn is said to satisfy a two-sided local John condition ifeach Ω and Rn \ Ω satisfies a local John condition.

For δ > 0 call a bounded domain Ω a δ-SKT domain (acronym forSemmes-Kenig-Toro domain) provided

(1) Ω satisfies a two-sided local John condition,

(2) ∂Ω is Ahlfors regular,

(3) distBMO(∂Ω)(ν,VMO(∂Ω)) < δ

Here BMO is the John-Nirenberg space of functions of bounded meanoscillation and VMO is the closed subspace of BMO of functions ofvanishing mean oscillation introduced by Sarason.



Theorem (S. Hofmann, M. Mitrea, M. Taylor)

Let Ω ⊂ Rn be a bounded open set satisfying a two-sided local Johncondition and with ∂Ω Ahlfors regular. Also, fix p ∈ (1,∞) and supposethat k : Rn \ {0} → R is sufficiently smooth, even, homogeneous of degree−n. Define

Tf (x) := p.v .

∫∂Ω

〈x − y , ν(y)〉k(x − y)︸︷︷︸key algebraic structure

f (y)dσy , x ∈ ∂Ω.

Then for every ε > 0 ∃ δ = δ(Ω, k, p, ε) > 0 such that

Ω is a δ-SKT domain =⇒ dist (T , Comp (Lp(∂Ω))) < ε.

Hence, in our setting, if the kernel of K is of the form〈x − y , ν(y)〉k(x − y), then K is as close to being compact on Lp(∂Ω) asdesired for δ-SKT domains and sufficiently small δ.


Non-uniqueness of coefficient tensor

In general, given a second order operator Lu =(∂r (a

αβrs ∂suβ)

)1≤α≤M

with

constant complex coefficients, there are multiple coefficient tensorsA = (aαβrs )α,β,r ,s that yield the same operator L. In fact,

LA1 = LA2 ⇐⇒ A1 − A2 antisymmetric in the lower indeces.

Each coefficient tensor A yields a different double layer operator KA.

Call A a distinguished coefficient tensor for L if L = LA and the kernel ofKA is of the form 〈x − y , ν(y)〉k(x − y) with k even & homogeneous ofdegree −n.As an example consider the Lamé system

Lu := µ∆u + (λ+ µ)∇div u,

where λ, µ are constants (the Lamé moduli).



For each t ∈ R, set

aαβrs (t) := µ δrsδαβ + (λ+ µ− t) δrαδsβ + t δrβδsα, 1 ≤ α, β, r , s ≤ n,

⇒ aαβrs (t) = aαβrs (0) + t bαβrs , ∀ t ∈ R, ∀α, β, r , s, wherebαβrs := δrβδsα − δrαδsβ, hence B :=

(bαβrs)

1≤r ,s≤n1≤α,β≤n

is antisymmetric in the

lower indices. Bottom line:

µ∆u + (λ+ µ)∇div u =(∂r(aαβrs (t)∂suβ

))1≤α≤n

, ∀ t ∈ R.

Each choice of t ∈ R gives rise to a different layer potential K . Using thestandard fundamental solution for Lamé, the kernel of K (for an arbitraryt) becomes



kernel(x , y) = −L1(t)δjkωn−1

〈x − y , ν(y)〉|x − y |n

algebraically OK

−(1− L1(t))n

ωn−1

〈x − y , ν(y)〉(xj − yj)(xk − yk)|x − y |n+2

algebraically OK

−L2(t)1

ωn−1

(xj − yj)νk(y)− (xk − yk)νj(y)|x − y |n

algebraically not OK

Hence, if L2(t) = 0 ⇒ the kernel of K has the desired algebraic form.

This is the case if t :=µ(µ+ λ)

3µ+ λand the corresponding K is the so called

pseudo-stress elastic double layer.


The higher regularity case ` ≥ 1

To successfully handle the case ` ≥ 1 need to:

(1) Define the spaces Lp` (∂Ω) for ` ≥ 1;(2) Prove well-definedness and boundedness for K on Lp` (∂Ω);

(3) Prove nontangential estimates for ∇`D when acting on Lp` (∂Ω);(4) Determine conditions on the domain that imply close to compactness

results for K when acting on Lp` (∂Ω).

Ultimately, the class of domains for which all these things hold shouldconstitute a natural bridge between the δ-SKT domains when ` = 0 andsmooth domains if ` =∞.


The spaces Lp` (∂Ω), ` ≥ 1

Fix: Ω ⊆ Rn open, ∂Ω Ahlfors regular and Hn−1(∂Ω \ ∂∗Ω) = 0.Let σ = Hn−1b∂Ω (note that ν is defined σ-a.e. on ∂Ω).Fix p ∈ (1,∞) and let p′ = p/(p − 1).For ϕ ∈ C10(Rn) set ∂τjkϕ := νj(∂kϕ)

∣∣∂Ω−νk(∂jϕ)

∣∣∂Ω

, j , k = 1, . . . , n,

Lp1(∂Ω) :={f ∈ Lp(∂Ω) : ∃c s.t.

∑j ,k

∣∣∣∫∂Ω

f (∂τjkϕ) dσ∣∣∣ ≤ c∥∥ϕ|∂Ω∥∥Lp′ (∂Ω), ∀ϕ ∈ C 10 (Rn)}

S. Hofmann, M. Mitrea and M. Taylor proved that in this settingf ∈ Lp1(∂Ω) ⇒ for each j , k ∈ {1, ..., n} there exists ∂τkj f ∈ Lp(∂Ω) s.t.∫∂Ω f (∂τjkϕ) dσ =

∫∂Ω(∂τkj f )ϕ dσ, ∀ϕ ∈ C

10 (Rn) and

Lp1(∂Ω) ={

f ∈ Lp(∂Ω) : ∂τjk f ∈ Lp(∂Ω), j , k = 1, . . . , n

},


The spaces Lp` (∂Ω), ` ≥ 1

and Lp1(∂Ω) is Banach when equipped with the norm

‖f ‖Lp1(∂Ω) := ‖f ‖Lp(∂Ω) +n∑

j ,k=1

‖∂τjk f ‖Lp(∂Ω).

Now define higher order Sobolev spaces Lp` (∂Ω) by induction on ` ∈ N:

Lp`+1(∂Ω) :={

f ∈ Lp` (∂Ω) : ∂τjk f ∈ Lp` (∂Ω), ∀ j , k = 1, . . . , n

},

equipped with the natural norm

‖f ‖Lp`+1(∂Ω) := ‖f ‖Lp`(∂Ω) +n∑

j ,k=1

‖∂τjk f ‖Lp`(∂Ω).

Then Lp` (∂Ω) is a Banach space, and ∀ j , k ∈ {1, . . . , n},

∂τjk : Lp` (∂Ω)→ L

p`−1(∂Ω) is bounded.


The operator K on Lp` (∂Ω), ` ≥ 1

Suppose K : Lp`−1(∂Ω)→ Lp`−1(∂Ω) is bounded and fix f ∈ L

p` (∂Ω). For

j , k = 1, ..., n need to estimate ‖∂τjk (Kf )‖Lp`−1(∂Ω). The starting point is

∂τjk (Kf )γ = ∂τjk (12 f + Kf )γ −

12∂τjk fγ

= ∂τjk((Df )γ

∣∣∂Ω

)− 12∂τjk fγ

= νj(∂kDf )γ∣∣∂Ω−νk(∂jDf )γ

∣∣∂Ω−12∂τjk fγ

To proceed

1 ∂k , ∂j , should be absorbed inside Df in a manner that, whenrestricted to ∂Ω, these give rise to operators whose Lp`−1(∂Ω) normwill be bounded by the Lp` (∂Ω) norm of f ,

2 the multiplication with components of ν should preserve themembership to Lp`−1(∂Ω).



∂j(Df)γ

= aβαrs ∂rSγβ(νj(∇tanfα)s

)+(D((∇tanf )j

))γ

, where

Sγβ g(x) :=

∫∂Ω

Eγβ(x − y)g(y) dσ(y), x ∈ Rn \ ∂Ω

and ∇tanf :=(νk∂τkj f

)1≤j≤n

νk(∂jDf )γ∣∣∂Ω

= 12 aβαrs νkνrνj

(Sym (L; ν)−1

)γ,β

(∇tanfα)s

+ aβαrs νk(∂rSγβ)(νj(∇tanfα)s

)+ νk(

12 I + K )

((∇tanf )j

)γ

where Sym (L; ν) := −(νiνja

µγji

)γ,µ

is the principal symbol of L at ν.



Return to ∂rSγβ g(x) =

∫∂Ω∂rEγβ(x − y)g(y) dσ(y)

Idea: decompose the operator D : u 7→ (∂juk)k,j into a tangential andnormal component on ∂Ω, analogously to the standard decomposition

∇ = ∇tan + ν (ν · ∇)

Given A =(aαβrs)α,β,r ,s

define the conormal derivative of u as

∂Aν u :=(νra

αβrs ∂suβ

)α

An inspection of

Dg(x) =(−∫∂Ωνs(y)a

βαrs (∂rEγβ)(x − y)gα(y) dσ(y)

)γ

x ∈ Ω

reveals that

Dg(x) = −∫∂Ω〈∂A>ν Eγ·(x − y) , g(y)〉γ dσ(y) x ∈ Ω



Key decomposition adapted to L: Given any tensor coefficient Aassociated to L there holds:

Du = ∂τD u +√−1Sym (D; ν)Sym (L>; ν)−1∂ A>ν u

where∂τD u := −Sym (L

>; ν)−1(νr a

βαsr ∂τsk uβ

)α,k

Based on this, we obtain the following key formula

∂r(Sγβ f

)= −Sγη

(∂τsr(νia

ηµsi

(Sym (L; ν)−1

)µ,β

f))

+(D(((

Sym (L; ν)−1)α,βνr f)α

))γ

in Ω

Moral: derivative of single layer ≡ combinations of single and double layer.



At the level of boundary to boundary operators the final identities in acondensed form are as follows. First,

∂τ (Kf ) ≡ K (∂τ f ) + MνS(∂τMν∂τ f ) + MνK (Mν∂τ f )

+S(∂τMν∂τ f ) + K (Mν∂τ f ), ∀ f ∈ Lp1(∂Ω)

Above, Mν denotes (possibly repeated) compositions of multiplicationoperators with components of ν, or entries in Sym (L; ν)−1, whileu ≡ v means that each component of u is a linear combination of thecomponents of v , with coefficients depending only on A.Second,

∂τ (Sf ) ≡ MνS(∂τMν f

)+ MνK

(Mν f

), ∀ f ∈ Lp(∂Ω)


Multiplier conditions on ν

Definition

Given X ,Y Banach spaces of functions, and given a function b, writeb ∈M (X → Y ) if the operator of pointwise multiplication by b iswell-defined & bounded from X to Y . If X = Y , abbreviateM X := M (X → X ). The operator of pointwise multiplication by b isdenoted by Mb.

When solving the BVP at level `, the multiplier condition on ν is

ν ∈⋂

1≤k≤`−1M Lpk(∂Ω).

Under this condition the Sobolev space Lp` (∂Ω) is quite rich, for example{ϕ|∂Ω : ϕ ∈ C `0 (Rn)

}⊆ Lp` (∂Ω).

Also, ν ∈⋂

1≤k≤`−1M Lpk(∂Ω) =⇒ Sym (L; ν)

−1 ∈⋂

0≤k≤`−1M Lpk(∂Ω).


Boundedness of layer potentials on Lpk(∂Ω)

Theorem (Martell, M. Mitrea, D.M.)

Let Ω ⊆ Rn be a UR domain with compact boundary, outward unit normalν, and surface measure σ := Hn−1b∂Ω. Let L be any second-order,constant coefficient elliptic system and K , D be any double layerpotentials associated to L. Fix p ∈ (1,∞). If there exists ` ∈ N such that

ν ∈`−1⋂k=0

M Lpk(∂Ω) then

K : Lpk(∂Ω)→ Lpk(∂Ω), well-defined, linear, bounded ∀ k ∈ {0, ..., `}.

Also, ∃C = C (n, p,Ω) ∈ (0,∞) such that

‖N(∇kDf )‖Lp(∂Ω) ≤ C‖f ‖Lpk (∂Ω),

for every f ∈ Lpk(∂Ω) and every k ∈ {1, . . . , `}.


Domination by compact operators

Definition

Given X1 and X2 Banach spaces and δ ≥ 0, a bounded linear operatorT : X1 → X2 is Dominated by Compact operators with error δ, and writeT ∈ DCδ(X1 → X2), if there exist a Banach space Y and a compactoperator C : X1 → Y such that

‖Tf ‖X2 ≤ ‖Cf ‖Y + δ ‖f ‖X1 , ∀ f ∈ X1.

Remarks:

Comp (X1 → X2) = DC0(X1 → X2)dist(T ,Comp(X1 → X2)) < δ =⇒ T ∈ DCδ(X1 → X2)DCδ(X1 → X2) is the “correct” class (both for Fredholm theory andour induction scheme)


Our class of domains

Definition

Let p ∈ (1,∞), ` ∈ N ∪ {0}, δ ∈ [0,∞), and CJ ,CA,CN ∈ (0,∞) begiven. Define Mp` (CJ ,CA,CN , δ; n) to be the family of open, nonempty,proper subsets Ω of Rn such that:(a) Ω satisfies a two-sided local John condition with character controlled

by CJ ;

(b) ∂Ω is compact, Ahlfors regular with character controlled by CA;

(c) if σ := Hn−1b∂Ω and ν=outward unit normal, then

ν ∈M Lpk(∂Ω) for k ∈ {1, . . . , `− 1} with max1≤k≤`−1 ‖ν‖MLpk (∂Ω)

≤ CN ,

M∇tanν ∈ DCδ(Lpk(∂Ω)→ L

pk−1(∂Ω)

)for k ∈ {1, . . . , `− 1},

and distBMO (∂Ω)(ν , VMO (∂Ω)

)≤ δ.


Our class of domains

(i) By design, this class of domains is monotonically decreasing in `:

Mp`+1(CJ ,CA,CN , δ; n) ⊆Mp` (CJ ,CA,CN , δ; n), ∀ ` ∈ N

and monotonically increasing in each of CJ , CA, CN and δ.

(ii) Corresponding to ` = 1, we have (M1 ≡ δ-SKT)

∀CA,CJ > 0, ∃ δo > 0 and ∃C ≥ 1 such that, for all δ ∈ (0, δo),(δ/C )-SKT domains ⊆M1(CJ ,CA, δ; n) ⊆ (Cδ)-SKT domains.

(iii)

Ω is a C∞ domain in Rnwith compact boundary

}⇐⇒

Ω ∈∞⋂`=1

Mp` (CJ ,CA,CN , 0 ; n)

∀ p ∈ (1,∞), ∀CJ ,CA,CN > 0.


An example

Let Ω ⊂ Rn be a bounded Lipschitz domain and suppose that {ϕj}1≤j≤N isa family of Lipschitz functions whose graphs (up to rotations) describe ∂Ω.If there exist p ∈ (1,∞), ` ∈ N and δ > 0 such that for every 1 ≤ j ≤ Nwe have

ϕj ∈ Lp` (Rn−1) if p > max

{n−1`−1 , 1

},

∇ϕj ∈M Lp`−1(Rn−1) and ‖∇ϕj‖MLp`−1(Rn−1) ≤ δ if 1 < p <

n−1`−1 ,

then there exist constants C ∈ (0,∞) and c ∈ [0,∞) depending on n, p, `such that Ω ∈Mp` (C ,C ,C , cδ; n).


Domination by compact operators for K

Theorem (Martell, M.Mitrea, D.M.)

Let p ∈ (1,∞), ` ∈ N, ε > 0, and CJ ,CA,CN ,CL ∈ (0,∞) be given. Thenthere exists δ = δ(n, p, `, ε,CJ ,CA,CN ,CL) > 0 doing the following job.

Suppose L is a second order elliptic operator which has a distinguishedcoefficient tensor A with |A| ≤ CL, and suppose Ω ∈Mp` (CJ ,CA,CN , δ; n).Let K be the double layer potential associated to A on ∂Ω. Then

K ∈ DCε(Lpk(∂Ω)→ L

pk(∂Ω)

)for each k ∈ {0, 1, ..., `}.

Moreover, if Ω ∈Mp` (CJ ,CA,CN , 0 ; n) then one actually has

K ∈ Comp(Lpk(∂Ω)→ L

pk(∂Ω)

)for each k ∈ {0, 1, ..., `}.

This generalizes (to the higher regularity case) results proved by Hofmann,M. Mitrea and Taylor in the case when ` = 0, ` = 1.


DC operators for K : ideas from the proof

Let [B,C ] := BC − CB be the commutator between operators B, C .Then, further work on earlier identities yields

∂τ (Kf ) ≡ K (∂τ f ) + [Mν , ∂S ](Mν∂τ f ) ∀ f ∈ Lp1(∂Ω)

Proceed by induction on ` to establish the main claim in theorem

For ` = 1 use result by Hofmann, M. Mitrea, Taylor: X space of hom.type, T a C-Z operator bdd. on L2(X ), p ∈ (1,∞) ⇒ ∃C > 0 s.t

infR comp

∥∥[Mb,T ]− R∥∥Lp(X )→Lp(X ) ≤ C distBMO(X )(b,VMO(X ))In parallel, in the induction proof we also need to show that[Mν , ∂S ] ∈ DCε

(Lpk(∂Ω)→ L

pk(∂Ω)

)for each k ∈ {0, ..., `− 1}.

∂τ[Mν , ∂S

]f ≡ (∂τν)S

(∂τMν f

)+ Mν

[Mν , ∂S

](∂τMν f

)+(∂τν)K

(Mν f

)+ Mν∂τK

(Mν f

)+ ∂τK

(Mν f

),


The Dirichlet problem for arbitrary `

Theorem (Martell, M.Mitrea, D.M.)

Let p ∈ (1,∞), ` ∈ N, and CJ ,CA,CN ,CL > 0 be given. Then there existsδ = δ(n, p, `,CJ ,CA,CN ,CL) > 0 having the following significance.

Suppose Ω ∈Mp` (CJ ,CA,CN , δ; n) is bounded with connectedcomplement, and that L is a second order, constant coefficient ellipticoperator which has a distinguished coefficient tensor which is positivedefinite. Let D, K be the double layer potential operators associated tothis coefficient tensor relative to Ω.

Then, for every k ∈ {0, 1, . . . , `} and every f ∈ Lpk(∂Ω), the BVP

(DL)

u ∈ C∞(Ω) and Lu = 0 in Ω,

N(∇mu) ∈ Lp(∂Ω), ∀m ∈ {0, ..., k},

u∣∣∂Ω

= f ∈ Lpk(∂Ω),∑km=0 ‖N(∇mu)‖Lp(∂Ω) ≤ C‖f ‖Lpk (∂Ω),


The Dirichlet problem for arbitrary `

Theorem (cont.)

admits the solution u(x) = D((

12 I + K )

−1f)

(x), x ∈ Ω.

Moreover, under the additional assumption that L∗ also has adistinguished coefficient tensor which is positive definite, the function u isthe only solution of (DL).

Comments:

Corresponding to the low regularity cases ` = 0, ` = 1 we recover theresults proved by Hofmann, M. Mitrea, Taylor

corresponding to ` =∞ we recover the classical results in C∞domains

our theorem can target any space Lp` , ` = 0, 1, . . . ,∞, as wanted


The Dirichlet problem for the Lamé operator

Corollary

Assume µ > 0 and λ > −n+2n µ and let p ∈ (1,∞), ` ∈ N, andCJ ,CA,CN ∈ (0,∞) be given.Then there exists δ = δ(n, p, `,CJ ,CA,CN , µ, λ) > 0 so that ifΩ ∈Mp` (CJ ,CA,CN , δ; n) is bdd. and has connected complement, theBVP

u ∈ C∞(Ω),µ∆u + (λ+ µ)∇div u = 0 in Ω,N(∇mu) ∈ Lp(∂Ω), ∀m ∈ {0, ..., k},u∣∣∂Ω

= f ∈ Lpk(∂Ω),

is well-posed for every k ∈ {0, 1, . . . , `}.


Boundary Value Problems: Higher Order Regularity Data in … · 2012. 7. 25. · This is joint work...

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