<b>Initiation into corner singularities</b>

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Initiation into corner singularities

Monique Dauge

To cite this version:

Monique Dauge. Initiation into corner singularities: Course given in School 1 of the RICAM Special

Semester on Computational Methods in Science and Engineering . Doctoral. Analysis and Numerics

of Acoustic and Electromagnetic Problems, Linz, Austria. 2016, pp.58. �cel-01399350�

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Initiation into corner singularities

Monique Dauge

IRMAR, Universit ´e de Rennes 1, FRANCE

Analysis and Numerics of Acoustic and Electromagnetic

Problems

October 10-15, 2016, Linz, Austria

RICAM Special Semester on Computational Methods in Science and Engineering

http://perso.univ-rennes1.fr/monique.dauge

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Outline

1 Brief history of corner problems

2 Planned – Typical examples

3 _{Standard regularity in smooth domains for coercive forms}

4 _{2D polygon – Corner localization}

5 _{2D sector: ∆ Dirichlet}

6 _{2D – ∆ Dirichlet in weighted spaces}

7 _{2D – ∆, miscellaneous in weighted spaces}

8 _{2D, general}

9 _{3D regular cones}

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Outline

1 _{Brief history of corner problems}

2 _{Planned – Typical examples}

3 _{Standard regularity in smooth domains for coercive forms} 4 2D polygon – Corner localization

5 2D sector: ∆ Dirichlet

Changes of variables Mellin transform Mellin symbol Conclusions

6 2D – ∆ Dirichlet in weighted spaces 7 _{2D – ∆, miscellaneous in weighted spaces} 8 _{2D, general}

3D corners

Regular cones in Rn_{and unweighted spaces}

Injectivity modulo polynomials

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What it is about

We address special combinations of classes of

1 _Domains

2 _{Partial Differential Equations} 3 _{Functional spaces}

4 _Questions

General framework of corner studies

1 _{Domains with general and combined corner types} 2 _{Elliptic boundary value problems}

3 _{Weighted and standard Sobolev spaces or H ¨older classes} 4 _{Existence, regularity, expansions of solutions}

A sort of maximal framework would include

1 _{Cones in R}n_{, conical manifolds, edges, manifolds with corners, polyhedra,}

recursively defined corner domains,...

2 _{Multi-order Agmon-Douglis-Nirenberg systems with variable coefficients, possibly}

non-smooth with asymptotics, possibly piecewise smooth on compatible partitions

3 _{Sobolev spaces based on L}2_{(Hilbert) or L}p_{(p > 1), including (or not) weights}

based on distances to singular sets. Possible analytic-type control of derivatives.

4 _{Fredholm, semi-Fredholm, regularity shift, asymptotics of various types}

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A brief history of elliptic BVP with corners: Russian school

V. A. KONDRAT’EV

Boundary-value problems for elliptic equations in domains with conical or angular points.

Trans. Moscow Math. Soc.16(1967)227–313.

1 _{Domains with conical points} 2 _{Scalar elliptic BVP}

3 _{Hilbert Sobolev spaces with or without weights} 4 _{Fredholm, regularity, asymptotics}

V. G. MAZ’YA, B. A. PLAMENEVSKII

Elliptic boundary value problems on manifolds with singularities.

Probl. Mat. Anal.6(1977)85–142.

V. A. KOZLOV, V. G. MAZ’YA, J. ROSSMANN

Elliptic boundary value problems in domains with point singularities.

Mathematical Surveys and Monographs, 52. American Mathematical Society,1997.

V. MAZ’YA ANDJ. ROSSMANN

Elliptic equations in polyhedral domains.

Mathematical Surveys and Monographs, 162. American Mathematical Society,2010.

1 Hierarchy of singular sets

2 Elliptic systems

3 Lp_{Sobolev spaces, Schauder classes, with weights} 4 Fredholm, regularity, asymptotics

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A brief history of elliptic BVP with corners: Ψ-do calculus

S. REMPEL, B. W. SCHULZE

Asymptotics for Elliptic Mixed Boundary Problems.

Akademie-Verlag,1989.

B. W. SCHULZE

Pseudo-differential operators on manifolds with singularities.

Studies in Mathematics and its Applications, Vol. 24. North-Holland,1991.

B.-W. SCHULZE

Boundary value problems and singular pseudo-differential operators.

Pure and Applied Mathematics (New York). John Wiley & Sons Ltd.,1998.

R. B. MELROSE

Pseudodifferential operators, corners and singular limits,

Proc. International Congress of Mathematicians, Math. Soc. Japan,(1991), 217–234.

R. B. MELROSE

Calculus of conormal distributions on manifolds with corners,

Internat. Math. Res. Notices(1992), no. 3, p. 51–61.

R. B. MELROSE

Differential analysis on manifolds with corners ,

http://www-math.mit.edu/ rbm/book.html

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A brief history of elliptic BVP with corners: In France & Belgium

P. GRISVARD

Probl `emes aux limites dans les polygones. Mode d’emploi.

Bull. Dir. Etud. Rech., S ´er. C1(1986)21–59.

P. GRISVARD

Singularit és en élasticit é.

Arch. Rational Mech. Anal.107 (2)(1989)157–180.

P. GRISVARD

Boundary Value Problems in Non-Smooth Domains.

Pitman, London1985.

1 _{Polygonal domains}

2 _{Elliptic BVP (Laplace, Lam ´e, ∆}2₎ S. NICAISE

Le laplacien sur les r ´eseaux deux-dimensionnels polygonaux topologiques.

J. Math. Pures Appl. (9)67(2)(1988)93–113.

S. NICAISE

Polygonal interface problems.

Methoden und Verfahren der Mathematischen Physik, 39. Verlag Peter D. Lang,1993.

1 _{Polygonal domains}

2 _{Elliptic transmission problems (piecewise constant on polygonal subdomains)}

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A brief history of elliptic BVP with corners: Us, before Maxwell

M. DAUGE

Elliptic Boundary Value Problems in Corner Domains.

Lecture Notes in Mathematics, Vol. 1341. Springer-Verlag,1988.

1 _{Hierarchy of singular sets} 2 _{Elliptic systems}

3 _{Hilbert Sobolev spaces without weights (interaction with polynomials)} 4 _{Semi-Fredholm, Fredholm, regularity, corner-edge asymptotics}

M. COSTABEL, M. DAUGE

General edge asymptotics of solutions of second order elliptic boundary value problems.

Proc. Royal Soc. Edinburgh123A(1993)109–155 and 157–184.

Construction of corner singularities for Agmon-Douglis-Nirenberg elliptic systems.

Math. Nachr.162(1993)209–237.

Stable asymptotics for elliptic systems on plane domains with corners.

Comm. Partial Differential Equations no9 & 10(1994)1677–1726.

1 _Edges 2 _{Elliptic systems} 3 _{Hilbert Sobolev spaces} 4 _{Structure of singularites}

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A brief history of elliptic BVP with corners: Maxwell

Maxwell and Lam ´e eigenvalues on polyhedra.

Math. Meth. Appl. Sci.22(1999)243–258.

Singularities of electromagnetic fields in polyhedral domains.

Arch. Rational Mech. Anal.151(3)(2000)221–276.

M. COSTABEL, M. DAUGE, S. NICAISE

Singularities of Maxwell interface problems.

M2AN Math. Model. Numer. Anal.33(3)(1999)627–649.

1 3D Polyhedra

2 Harmonic Maxwell

3 Hilbert Sobolev spaces

4 Regularity, structure of singularites

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A brief history of elliptic BVP with corners: Analytic regularity

I. BABUSKA, B. GUOˇ

Regularity of the solution of elliptic problems with piecewise analytic data. I. Boundary value problems for linear elliptic equation of second order,

SIAM J. Math. Anal., 19(1988)172–203

I. BABUSKA, B. GUOˇ

Regularity of the solution of elliptic problems with piecewise analytic data. II. The trace spaces and application to the boundary value problems with nonhomogeneous boundary conditions,

SIAM J. Math. Anal., 20(1989)763–781

1 2D Polygonal domains

M. COSTABEL, M. DAUGE,ANDS. NICAISE

Analytic regularity for linear elliptic systems in polygons and polyhedra,

Math. Models Methods Appl. Sci.,22(2012), 1250015, 63p.

Weighted analytic regularity in polyhedra,

Comput. Math. Appl.,67(2014)807–817.

1 _{3D Polyhedral domains}

GLC project (Grand Livre des Coins),

??(20??)

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Outline

1 _{Brief history of corner problems} 2 _{Planned – Typical examples}

3D corners

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Aim: Introduce corner singularities in acoustic & electromagnetic

... problems in 2D and 3D domains.

Two main steps:

A All (almost all) details in a minimal framework

1 2D: Infinite plane sectors and bounded polygons

2 Model operators (homogeneous with constant coefficients) of acoustics & electromagnetics: Laplace operator ∆, magnetostatic Maxwell system.

3 Weighted L2_{Sobolev spaces} 4 Regularity and singularities, estimates

B Hints of results and proofs in an extended framework

1 3D: Infinite regular cones, infinite edges, infinite polyhedral cones, bounded domains with conical points, edges or corners.

2 More operators of acoustics & electromagnetics (with lower order terms,

variable coefficients, transmission conditions): Helmholtz operator, harmonic Maxwell system, several materials with distinct electromagnetic properties.

3 Ordinary non-weighted L2_{Sobolev spaces (“non-weighted” is not a particular}

case of “weighted”)

4 Regularity and singularities, estimates

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Problems in variational form

We consider problems in simple (coercive, not saddle point) variational form posed on a domain Ω ⊂ Rn. This requires two ingredients:

A real bilinear (or complex sesquilinear) form a of order 1 defined on a space X (Ω) a(u, v ) = d X i=1 d X j=1 X |α|≤1 X |β|≤1 Z Ω aαβ_ij (x ) ∂xαui∂xβ¯vjdx , u, v ∈ X (Ω)

x = (x1, . . . ,xn)is the variable in Ω ⊂ Rnand for α = (α1, · · · , αn)

∂_xα=∂ α1 ∂x1 · · ·∂ αn ∂xn

d is the dimension of the system: u = (u1, . . . ,ud), v = (v1, . . . ,vd)

the coefficients aαβ_ij are smooth (or piecewise smooth) in Ω

for acoustics, elasticity and in many cases, the space X (Ω) is defined as X (Ω) = H1(Ω)d for ∆, Lam ´e, etc... for electromagnetism, X (Ω) is defined as

X (Ω) = H(curl; Ω) ∩ H(div; Ω) for Maxwell

A variational space V ⊂ X (Ω) that determines essential (or Dirichlet) boundary conditions

V = {u ∈ X (Ω), ΠDu = 0 on ∂Ω} where ΠDis a chosen projection operator (examples follow).

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Typical examples [The Laplace operator of acoustics]

Scalar ∇ · ∇ form(here d = 1)

a(u, v ) = Z Ω ∇u · ∇v dx = X |α|=1 Z Ω ∂xαu ∂xαv dx = n X `=1 Z Ω ∂x`u ∂x`v dx

The space X (Ω) is the Sobolev space H1_{(Ω) = {u ∈ L}2_{(Ω), ∇u ∈ L}2_(Ω)n_}.

Two main cases plus a generalization:

1 Dirichlet: The projection operator ΠD_{is the identity on ∂Ω, so}

V = H10(Ω)=

◦

H1(Ω) = {u ∈ H1(Ω), u = 0 on ∂Ω}

2 Neumann: The projection operator ΠD_{is zero on ∂Ω, so V = H}1_(Ω) 2’ Mixed: ∂Ω has a partition in two pieces ∂DΩand ∂Ω \ ∂DΩ:

ΠD(x ) = (

I x ∈ ∂DΩ

0 x ∈ ∂Ω \ ∂DΩ

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Typical vector examples with d = n [Lam ´e and Maxwell]

Lam ´e elastic bilinear formdefined foru, v ∈ H1_(Ω)n

a(u, v ) = Z Ω 2µ n X i=1 n X j=1 eij(u) eij(v ) + λ div u div v dx λ ≥0 and µ > 0 are the Lam ´e coefficients

eij(u) = 12(∂iuj+ ∂jui)are the components of the strain tensor.

To define essential boundary conditions, introducen as the exterior unit normal to ∂Ω. 3 Simply supported: The projection operator ΠD_{is defined as Π}D_{u = u − (u · n)n,}

so ΠD_{u = 0 means that the tangential component of u is 0 on the boundary ∂Ω.}

V = HN := {u ∈ H1(Ω)n, u − (u · n)n = 0 on ∂Ω}

Maxwell regularized electric formdefined foru, v ∈ H(curl; Ω) ∩ H(div; Ω)=R ∩ D

a(u, v ) = Z

Ω

µ−1curl u curl v + s div εu div εvdx

with electric permittivity ε > 0 and magnetic permeability µ > 0. Real parameter s > 0.

4 Electric perfect conductor condition: The projector ΠD_{is defined as above.}

V = XN := {u ∈ H(curl; Ω) ∩ H(div; Ω), u − (u · n)n = 0 on ∂Ω}

NB: If n = 3, u − (u · n)n = 0 if and only if u × n = 0.

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Outline

3 _{Standard regularity in smooth domains for coercive forms}

4 2D polygon – Corner localization 5 2D sector: ∆ Dirichlet

3D corners

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Coercivity

Definition

Let a be a sesquilinear form defined on V ⊂ X (Ω) with compact embedding in L2_(Ω).

a is said continuous if

(1) ∃C > 0, ∀u, v ∈ V , |a(u, v )| ≤ Ckuk

X (Ω)kv kX (Ω)

a is said V -coercive if

(2) ∃C > 0, c > 0 ∀u ∈ V , Re a(u, u) ≥ ckuk2

X (Ω)− Ckuk 2 L2(Ω)

a is said strongly V -coercive if C can be taken to 0 in (2).

A continuous sesquilinear form a defined on V is associated with an operator A : V → V0 u 7−→ v 7→ a(u, v )

Theorem S.1

If a is continuous and strongly V -coercive, then A is an isomorphism V → V0

If a is continuous and V -coercive, then A is Fredholm of index 0 from V into V0

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Integration by parts

Assumption: Au = f with f ∈ L2_(Ω)d u ∈ V , ∀v ∈ V , a(u, v ) = Z Ω f ¯v dx

Therefore, in the distributional sense (we take v ∈D(Ω))

Lu = f with (Lu)j= d X i=1 X |α|≤1 X |β|≤1 (−1)|β|∂βxaαβij (x ) ∂ α xui, j = 1, . . . , d

Integrating by parts gives a sense to “natural boundary conditions” that complete the essential boundary conditions. Revisit examples.

Here A is the second order operator associated with the general a. It is distinct from generic first order operators by Dirk

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Integration by parts: The four examples

1 Dirichlet for scalar Laplacian(strongly coercive): (

−∆u = f in Ω

u = 0 on ∂Ω.

2 Neumann for scalar Laplacian(coercive): (

−∆u = f in Ω

∂nu = 0 on ∂Ω.

3 Simply supported for Lam ´e(strongly coercive): Settinguτ =u − (u · n)n

     −(µ∆ + (λ + µ)∇ div)u = f in Ω uτ = 0 on ∂Ω µ(∂nu) · n + (λ + µ) div u = 0 on ∂Ω

4 Electric perfect conductor condition for Maxwell(strongly coercive): (µ = ε = 1) 

 

 

(curl curl − s∇ div)u = f in Ω uτ = 0 on ∂Ω

divu = 0 on ∂Ω

NB: In any open set where ∂Ω is flat anduτ=0, we have divu = ∂n(u · n) = (∂nu) · n 14/58

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The regular case [general]

Assume

Ωis a bounded open set with a smooth boundary in Rn

The coefficients aαβ_ij of the form a are smooth functions on Ω The variational space is a subset of H1_{(Ω), i.e. X (Ω) = H}1_(Ω)

The form a is continuous and V -coercive

u ∈ V is solution of Au = f with f ∈ L2_(Ω), h_{i.e. ∀v ∈ V , a(u, v ) =}R

Ωf ¯v dx

i

Theorem S.2 Resolvent estimates

Under the previous assumptions:

u belongs to the Sobolev space H2_(Ω)

There exists a constant C independent of u such that

(1) kuk H2_(Ω)≤ C kf k L2_(Ω)+ kukH1_(Ω)

Let U and U0_{two open sets in R}n_{such that U ⊂ U}0_{; Set V = Ω ∩ U and}

V0_{= Ω ∩ V}0_{. We have the local a priori estimates with a constant independent of u}

(2) kuk H2_(V)≤ C kf k L2_(V0₎+ kuk_H1_(V0₎

If f belongs to Hm−2_(Ω)_{with an integer m > 2, then u belongs to H}m_(Ω)_{and the}

estimates (1) and (2) hold with Hm_{and H}m−2_{instead of H}2_{and L}2_{, respectively.}

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The regular case [Maxwell]

Assume

Ωis a bounded open set with a smooth boundary in Rn

The coefficients µ and ε of the regularized Maxwell form a are smooth positive functions on Ω. Recall

a(u, v ) = Z

Ω

µ−1curl u curl v + s div εu div εvdx

Theorem S.3 See Martin’s course Convex polygons

Under the previous assumptions:

The variational space XN(Ω)is a subset of H1(Ω)d, which means that XN(Ω)

coincides with the variational space HN(Ω)of elasticity.

The same holds with the “magnetic” spaces XT(Ω)and HT(Ω)for which the

essential boundary condition isu · n = 0 on ∂Ω.

There exists a constant C such that for allu ∈ XN(Ω) (oru ∈ HT(Ω))

(3) kuk H1_(Ω)≤ C k curl uk L2_(Ω)+ kdivuk_L2_(Ω)+ kuk_L2_(Ω)

As a consequence of this theorem, the regularized Maxwell form satisfies the assumptions of the general case.

So the conclusions of Theorem S.2 apply.

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Outline

3D corners

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Dirichlet Laplacian on a polygon – Corner localization

We are going to study in detail solutions of

−∆u = f in Ω, u ∈ H01(Ω)

when Ω is apolygonin the plane R2_{, i.e. Ω is a bounded open set and its boundary is a}

finite union of segments. The ends of these segments are thecornersof Ω. Denote byC the set of corners c.

For eachc ∈C , there exists an infinite plane sector Γcthat coincides with Ω in a

neighborhood B(c, Rc)ofc.

Denote by ωctheopeningof Γc, which defines the opening of Ω at the cornerc.

Introduce local polar coordinates (rc, θc)such that

Γc=x ∈ R2, rc>0, θc∈ (0, ωc)

Introduce a smooth cut-off χc(x ) =

(

0 if x 6∈ B(c, Rc)

1 if x ∈ B(c, Rc/2)

We have −∆(χcu) = χcf + 2∇χc· ∇u + (∆χc)u. Therefore, by extension by 0

∆(χcu) ∈ L2(Γc), χcu ∈ H01(Γc)

Regularity outside corners Theorem L.2

For any cut-off χ with support U0_{disjoint from the corners we find by Theorem S.2 that}

χu ∈ H2_{(U ∩ Ω)} _with _{U = χ}−1_(1).

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Outline

3D corners

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Changes of variables

Dirichlet Laplacian on a plane sector – Change of variables

We have reduced our problem on a polygon to problems of the type (we re-baptize as u the localized function χcu and we drop the indexc)

(

∆u = f in Γ, f ∈ L2_(Γ), _{supp f ⊂ B(0, R)}

u ∈ H1

0(Γ), supp u ⊂ B(0, R)

Use polar coordinates (r , θ). We have x1=r cos θ, x2=r sin θ and

(

r ∂r = r cos θ ∂x₁+r sin θ ∂x₂

∂θ = −r sin θ ∂x₁+r cos θ ∂x₂

Seteu(r , θ) = u(x ) andeg(r , θ) = r2f (x ). The equation ∆u = f is equivalent to (r ∂r)2+ ∂2θ

e

u =eg in R+× I with I = (0, ω).

Then set t = log r (i.e. r = et_{) – Euler change of variables –}

r ∂r= ∂t and dx = r dr dθ = e2tdt dθ

Set ˘u(t, θ) =eu(r , θ) and ˘g(t, θ) =eg(r , θ). The equation ∆u = f is equivalent to ∂t2+ ∂θ2˘u = ˘g in R × I with I = (0, ω).

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Dirichlet Laplacian on a sector – Exponential weights

Lemma

(1) If f ∈ L2_(Γ)_{with supp f ∈ B(0, R), then ˘}_{g = e}2t˘_{f satisfies} ∀η ≤ 1, e−ηt˘g ∈ L2(R × I) and ke−ηt˘gk

L2_(R×I)≤ Ckf k_L2_(Γ) (2) If u ∈ H1

0(Γ)with supp u ∈ B(0, R), then ˘u satisfies

∀η ≤ 0, e−ηtu ∈ H˘ ₀1(R × I) and ke−ηtuk˘

H1_(R×I)≤ Ckuk_H1_(Γ)

because of the support condition and 1 − η ≥ 0.

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Dirichlet Laplacian on a sector – Exponential weights

Proof of(2) Z R Z I |e−ηt_∂ tu(t, θ)|˘ 2dt dθ = Z R Z I |e−ηt_e−t_∂ tu(t, θ)|˘ 2e2t dt dθ ≤ Z Γ |r−η∂x1u(x )| 2_{+ |r}−η_∂ x2u(x )| 2_dx ≤ R−η Z Γ |∇u(x)|2dx because of the support condition and −η ≥ 0. The same for ∂θ˘u.

and may conclude if we know that r−1_{u belongs to L}2_(Γ).

The latter is true thanks to Poincar ´e’s inequality (due to boundary conditions) keu(r , ·)k L2_(I)≤ Ck∂θeu(r , ·)k L2_(I) that implies kr−1uk L2_(Γ)≤ Ck∇ukL2_(Γ). 20/58

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Mellin transform

Dirichlet Laplacian on a sector – Laplace and Mellin transform

Definition

If r−η−1v belongs to L2_{(Γ), we define the Mellin transform Mv as}

Mv [λ](θ) = Z ∞ 0 r−λev (r , θ) dr r , λ ∈ C, Re λ = η

If e−ηt˘v belongs to L2_{(R × I), we define the Fourier-Laplace transform L(˘}_{v ) as}

L(˘v )[λ](θ) = Z

R

e−λt˘v (t, θ) dt, λ ∈ C, Re λ = η

Note the equivalence

r−η−1v ∈ L2(Γ) ⇐⇒ e−ηt˘v ∈ L2(R × I) and the identities, with F the standard Fourier transform,

λ = η +iξ : Mv [λ] = L(˘v )[λ] = F(e−ηt˘v )[ξ]

We mainly use it in its Mellin form.

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Mellin transform

Dirichlet Laplacian on a sector – Laplace and Mellin transform

Properties

Let X be a Hilbert space of functions on the interval I, for example X = Hm(I). The function e−ηt_˘_{v belongs to L}2_{(R, X), if and only if the function}

ξ 7−→ Mv [η + iξ] belongs to L2_{(R, X)}

Let η0< η1. If e−ηt˘v ∈ L2(R, X) for η = η0and η = η1, then e−ηt˘v ∈ L2(R, X) for

all η ∈ [η0, η1]and the function

λ 7−→ Mv [λ] is holomorphic with values in X Recall that u ∈ H01(Γ), supp u ⊂ B(0, R), ∆u = f ∈ L2(Γ).

With g := r2_{f and I = (0, ω) we have}

Mu isholomorphicin the complex half-plane Re λ < 0 with values in H1 0(I)

Mg isholomorphicin the complex half-plane Re λ < 1 with values in L2_(I)_and

ξ 7→ Mg[η + iξ] belongs to L2(R × I) for all η ≤ 1.

Owing to formulas L(∂t˘u)[λ] = λL(˘u)[λ] and L(∂θ˘u)[λ] = ∂θL(˘u)[λ] we have

(λ2+ ∂_θ2) Mu[λ] = Mg[λ], ∀λ, Re λ ≤ 0.

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Mellin symbol

Dirichlet Laplacian on a sector – Mellin symbol

Definition

TheMellin symbolof ∆ with Dirichlet conditions is defined for any λ ∈ C as A[λ] : H1

0(I) −→ H−1(I)

U 7−→ (λ2_{+ ∂}2 θ)U

The symbol A[λ] can also be considered as acting from (H2∩ H1

0)(I)into L 2_(I).

Denote by P the positive Laplace operator −∂2 θfrom (H

2_{∩ H}1

0)(I)into L 2_(I).

The operator P is self-adjoint on L2_{(I), positive, with compact resolvent. Its}

spectrum σ(P) is a discrete subset of (0, +∞). Explicit calculation shows that σ(P) = n µ`:= `π ω 2 , ` ∈ N∗ o Since A[λ] = −P + λ2

we deduce that the spectrum σ(A) of the Mellin symbol A is σ(A) =nλ`:=

`π

ω, ` ∈ Z∗ o

The symbol λ 7→ A[λ] isholomorphicand its inverse λ 7→ A[λ]−1_is_meromorphic_.

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Mellin symbol

Dirichlet Laplacian on a sector – Meromorphic extension

Recall that

∀λ s.t. Re λ ≤ 0, A[λ] Mu[λ] = Mg[λ]. We define a meromorphic extension U[λ] of Mu[λ] by setting

∀λ s.t. Re λ ∈ (0, 1], U[λ] = A[λ]−1 _Mg[λ] We are going to prove

Proposition

(1) Exists u1such that e−tu˘1∈ H2(R × I) and Mu1[λ] =U[λ] for λ s.t. Re λ = 1 (2) The difference between u and u1is given by the residue formula

u1− u = X λ0∈σ(A) Re λ0∈(0,1) Res λ=λ0 rλ_U[λ]

Using that σ(A) =π_ωZ∗, we obtain as a corollary

   u1=u if ω < π u1− u = Res λ=π_ω rλ_U[λ] _{if ω > π} 24/58

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Mellin symbol

Dirichlet Laplacian on a sector – Resolvent estimates

The proof of the proposition is based on estimates of A[λ]−1_{in operator norm.}

Define the parameter norm kUk

Hm_(I;λ)for m ∈ N as kUk2 Hm(I;λ)= m X k =0 (|λ| +1)2k_kUk2 Hm−k(I) Lemma General 2D

Let η0≤ η1and δ > 0. Define the set

Λ = n λ ∈ C, Re λ ∈ [η0, η1] o \ [ λ0∈σ(A) B(λ0, δ).

Let m ≥ 2. Exists a constant C such that

(1) ∀λ ∈ Λ, ∀G ∈ Hm−2_(I), _kA[λ]−1_Gk

Hm_(I;λ)≤ CkGkHm−2_(I;λ)

Proof: Based on two steps.

Step 1 If K is a compact set in C disjoint from the spectrum of A the resolvent estimate (1) holds with a constant C depending on K by continuity of A[λ]−1_{with respect to λ.} Step 2 It remains to prove (1) for a set Λ that is the intersection of the strip

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Mellin symbol

Dirichlet Laplacian on a sector – Resolvent estimates, continued

(1) ∀λ ∈ Λ, ∀G ∈ L2_(I), _kA[λ]−1_Gk

Hm(I;λ)≤ CkGkHm−2(I;λ)

To perform Step 2 of the proof of (1), we choose λ, G, and set V = A[λ]−1G. Then introduce on R × I ˘ v (t, θ) = eλtV (θ) and g(t, θ) = e˘ λtG(θ). We can check: ˘ v ∈ H1((−2, 2) × I), ˘v = 0 on R × ∂I ∆˘v = ˘g

Local estimates (Theorem S.2) give k˘v k Hm_{((−1,1)×I)}≤ C k˘gk Hm−2_{((−2,2)×I)}+ k˘v k_H1_{((−2,2)×I)}

Coming back to V and G —here C depends on η0and η1,

kV k Hm_(I;λ)≤ C kGk Hm−2_(I;λ)+ kV k_H1_(I;λ)

We conclude because for |λ| large enough, CkV k

H1_(I;λ)≤ 1

2kV kHm_(I;λ).

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Conclusions

Dirichlet Laplacian on a sector – Regular part

Proposition, point(1), more precise

Let u1be defined as the inverse Mellin transform

u1(x ) = 1 2iπ Z Re λ=1 rλU[λ](θ) dλ Then e−t˘u1∈ H2(R × I), which means for u1:

r−2+|α|∂xαu1∈ L2(Γ), |α| ≤ 2.

U[λ] = A[λ]−1_Mg[λ]

The Mellin transform of g = r2_{f satisfies “ξ 7→ Mg[1 + iξ] belongs to}

L2

(R, L2(I))”

The resolvent estimate with m = 2 on the line Re λ = 1 (disjoint from σ(A)!) yields ξ 7→ |1 + iξ|kU[1 + iξ] belongs to L2(R, H2−k(I)), k = 0, 1, 2.

Since ₁ 2iπ Z Re λ=1 rλU[λ] dλ = 1 2π e tZ R

eitξU[1 + iξ] dξ we find by inverse Fourier transform that e−t˘u1∈ H2(R × I)).

Back to Cartesian coordinates with r = et_{gives the weighted regularity for u} 1.

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Conclusions

Dirichlet Laplacian on a sector – Residue formula

Proposition, point(2) u1− u = X λ0∈σ(A) Re λ0∈(0,1) Res λ=λ0 rλU[λ] U[λ] = A[λ]−1Mg[λ]

The poles of U[λ] are in the set of poles of A[λ]−1since Mg[λ] is holomorphic For any simple rectifiable curve γ that is contained in the open strip

Re λ ∈ (−∞, 1) and surrounds the pole λ0=πωwhen ω > π, we have

1 2iπ Z γ rλU[λ](θ) dλ = X λ0∈σ(A) Re λ0∈(0,1) Res λ=λ0 rλU[λ]

Take γ as the rectangle Re λ = 0, Im λ = −ξ, Re λ = 1 − δ, Re λ = ξ, with δ > 0 small enough and ξ > 0. We may push ξ to infinity and δ to 0 using the resolvent estimates to bound U[λ].

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Conclusions

Dirichlet Laplacian on a sector – The residue

If ω > π, we have one residue in the relevant region Re λ ∈ [0, 1].

Recall that A[λ] = −P + λ2_{with P = −∂}2 θon H

1 0(I).

Let φ`be an orthonormal spectral basis for P associated with eigenvalues µ`

µ`= λ2` with λ`= `π ω, and φ`(θ) = r 2 ω sin `π ω θ, ` ≥1 Then PU = X `∈N∗ µ`hφ`,Ui φ` hence (−P + λ2)−1G = X `∈N∗ 1 λ2_{− λ}2 ` hφ`,Gi φ`

With G[λ] the Mellin transform of g = r2_{f , we find}

Res λ=λ₁r λ_{U[λ] = Res} λ=λ₁r λ_{(−P + λ}2₎−1_G[λ] =rλ1 1 2λ1 hφ1,G[λ1]i φ1 = γ1r π ω sinπ ωθ with γ1= 1 π Z Γ r−ωπsinπ ωθf (x ) dx Finally u = u1− γ1r π ω sinπ

ωθand the proposition is proved and the theorem

follows...

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Conclusions

Dirichlet Laplacian on a sector – Theorem with rhs in L

2

Theorem L.1 Theorem L.3 Theorem N.1

Let Γ be a plane sector of opening ω ∈ (0, π) ∪ (π, 2π) and u ∈ H1

0(Γ)with compact

support such that

∆u = f in Γ, f ∈ L2(Γ).

If ω < π, then u belongs to H2_(Γ)_{and moreover satisfies r}−2_{u, r}−1_{∇u in L}2_(Γ)

If ω > π, then (L.1) u = u1− γ1r π ω sinπ ωθ with γ1= 1 π Z Γ r−πωsinπ ωθf (x ) dx and u1satisfying r−2+|α|∂xαu1∈ L2(Γ), |α| ≤ 2

Case of a crack Optimality

If ω = 2π, Theorem L.1 does not apply because the Mellin symbol has a pole in λ = 1. Nevertheless it is possible to prove that (L.1) holds with the weaker regularity for u1

∂xαu1∈ L2(Γ), |α| =2, and r−2+δ+|α|∂αxu1∈ L2(Γ), |α| ≤1, δ > 0.

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Outline

6 2D – ∆ Dirichlet in weighted spaces

7 _{2D – ∆, miscellaneous in weighted spaces} 8 _{2D, general}

3D corners

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Dirichlet Laplacian on a polygon – Theorem with rhs in L

2

Theorem L.2 Theorem L.4

Let Ω be a plane polygon with corners setC 3 c associated with openings ωc∈ (0, π) ∪ (π, 2π). Let χcsmooth cut-off functions separating the corners.

Let u ∈ H1 0(Ω)be solution of ∆u = f in Ω, f ∈ L2(Ω). Then (L.2) u = u1,reg+ X c∈C , ωc>π χc(x ) γcr π ωc c sinωπcθc with

constants γcdepending continuously on f ∈ L2(Ω)

a regular part u1,reg∈ H2(Ω)satisfying

rc−2+|α|∂xαu1∈ L2(Ω), |α| ≤ 2, c ∈C

Proof: Localize around each corner as specified inSection 4

Then apply Theorem L.1 at each corner, multiply the expansion (L.1) by χcand sum on c ∈C .

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Weighted Sobolev spaces of Kondrat’ev type

The same tools allow to prove results in a general class of weighted Sobolev spaces (those introduced by Kondrat’ev, with a different notation).

Definition

Let m ∈ N and β ∈ R.

Let Γ be a plane sector, with r the distance to its vertex.

K_βm(Γ) = {u ∈ L2_loc(Γ), rβ+|α|∂xαu ∈ L2(Γ), ∀α, |α| ≤m}

Let Ω be a polygon with corner set_{C , and r}c=rc(x ) the distance of x toc. Let ρ

be the minimum of the rc, c ∈C ., i.e. the distance function to the set of corners.

Kβm(Ω) = {u ∈ L2loc(Ω), ρβ+|α|∂xαu ∈ L2(Ω), ∀α, |α| ≤m}

Straightforward properties (or already seen): If u ∈ Km

β(Ω), then χcu ∈ Kβm(Γc).

∆is continuous from K_βm(Γ)into K_β+2m−2(Γ) Km β(Γ) ⊂K m−1 β (Γ) ⊂ · · · ⊂K 0 β(Γ) Km β(Ω) ⊂K m0 β0(Ω)for m0≤ m and β0≥ β Km −m(Ω) ⊂Hm(Ω) ⊂K0m(Ω) H1 0(Ω) ⊂K−11 (Ω) 32/58

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Weighted Sobolev spaces and Mellin transform

Like in the particular case m = 2, β = −2, we can prove Lemma

Let m ∈ N and β ∈ R. Let Γ be a plane sector of opening ω, and I = (0, ω). Set η = −β −1. The following three propositions are equivalent

1 _{u ∈ K}m β(Γ)

2 _e−ηt˘u ∈ Hm_{(R × I), with u(x) = ˘}_{u(t, θ).}

3 _{The Mellin transform U[λ] := Mu[λ] is well defined for Re λ = η and the function}

(1) ξ 7−→ kU[η + iξ]k

Hm_(I;η+iξ) belongs to L 2_(R)

Moreover, if the function U satisfies (1), then the inverse Mellin transform u(x ) = 1 2iπ Z Re λ=η rλ_{U[λ](θ) dλ} defines an element of Km β(Γ). 33/58

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Dirichlet Laplacian in weighted Sobolev spaces on a sector

The same tools as used forTheorem L.1, including the general form of our resolvent estimates, allows to prove:

Theorem L.3 Theorem L.5

Let Γ be a plane sector of opening ω. Let m ≥ 2 and β < −1. Set η = −β − 1. We assume that

The line Re λ = η is disjoint from the Mellin spectrum σ(A) (here this means that η 6∈π_ωZ∗)

Let u ∈ H1

0(Γ)with compact support such that

∆u = f in Γ, f ∈ K_β+2m−2(Γ). Then (L.3) u = uη+ X `∈N∗, `πω<η γ`r` π ω sin `π_ωθ with uη∈ Kβm(Γ) and γ`= 1 π Z Γ r−`πωsin `π_ωθf (x ) dx 34/58

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Dirichlet Laplacian in weighted Sobolev spaces on a polygon

Theorem L.4 Theorem L.2 Theorem N.2 Theorem G.1

Let Ω be a plane polygon with corners setC 3 c. Let χcsmooth cut-off functions

separating the corners.

Let m ≥ 2 and β < −1. Set η = −β − 1. We assume that

∀c ∈C , the line Re λ = η is disjoint from the Mellin spectrum σ(Ac) (here this means that η 6∈ π_ω

cZ∗for allc ∈C ) Let u ∈ H1

0(Ω)such that ∆u = f in Ω, f ∈ K m−2 β+2(Ω). Then (L.4) u = ureg+ X c∈C χc(x ) X `∈N∗, `π ωc<η γc,`r `π ωc c sin `ωπcθc with ureg∈ K_βm(Ω)

Relations with standard Sobolev spaces

If f ∈ K_−m+2m−2 (Ω), then ureg∈ Hm(Ω). But the case f ∈ Hm−2(Ω)is not straightforward.

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Outline

6 2D – ∆ Dirichlet in weighted spaces 7 _{2D – ∆, miscellaneous in weighted spaces}

8 _{2D, general} 9 _{3D regular cones}

3D corners

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Neumann Laplacian

Let us consider now a solution H1_(Ω)_{of the Neumann problem}

(

−∆u = f in Ω

∂nu = 0 on ∂Ω.

with f ∈ L2(Ω). What is different wrt Dirichlet?

1 _{The Mellin symbol(s)}

A[λ] :U 7→ (λ2U + U00) : nU ∈ H2(I), U0(0) = U0(ω) =0o−→ L2_(I) 2 _{The “initial regularity” of u, which does not belong to K}1

−1(Ω)in general Remark: Alternative definitions of the Mellin symbol

A[λ]can also be defined in variational form from H1_(I)_{into its dual H}1_(I)0_.

Or, equivalently, H2_{(I) →}_L2_{(I) × R}2_{, with U 7→ (λ}2_{U + U}00_,_U0_{(0), U}0_(ω)).

These choices are more adapted to non-homogeneous Neumann boundary conditions. Like for Dirichlet, we can write A[λ] = −P + λ2_{with the Neumann 1D Laplacian P on}

I = (0, ω). An orthonormal spectral basis φ`for P associated with eigenvalues µ`is

given by µ`= λ2` with λ`= `π ω, ` ≥0 and φ`(θ) =    q 1 ω if ` = 0 q 2 ω cos `π ωθ if ` ≥ 1 36/58

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Neumann Laplacian on a plane sector

The novelties are

1 _{The pole at λ = 0 for A[λ]}−1_{. This pole is}_double_:

(−P + λ2)−1G = 1 λ2hφ0,Gi φ0+ X `∈N∗ 1 λ2_{− λ}2 ` hφ`,Gi φ`

Calculating the residue of rλ_{(−P + λ}2₎−1_{G[λ] at λ}

0=0 uses that

rλ₌_{1 + λ log r +}1 2λ

2_log2_{r + · · ·} _{λ →}_0. 2 _{For all positive δ, the variational solution u belongs to K}1

−1+δ(Ω). This is due to Hardy’s inequality“at infinity”.

We can analyze the solutions on Γ in the same way as before. If u ∈ K1

−1+δ(Γ), our

integration contour in C is Re λ = −δ, Re λ = 1. The resolvent estimates are still valid for the Neumann Mellin symbol as a consequence of Theorem S.2.

Theorem N.1 Theorem L.1 Theorem M.1

Let Γ be a plane sector of opening ω ∈ (0, π) ∪ (π, 2π) and u ∈ K1

−1+δ(Γ)for all δ > 0

with compact support such that

∆u = f in Γ, f ∈ L2(Γ) and ∂nu = 0 on ∂Γ Then (N.1) u = u1+ γ0+ γ00log r +If ω > π(γ1r π ω cosπ ωθ) with u1∈ K 2 −2(Γ) 37/58

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Hardy’s inequality

Hardy’s inequality Back to Taylor

For any f ∈C∞ 0 (R+)and γ 6= 1: Z∞ 0 rγ−2|f (r )|2_{dr ≤} 4 (1 − γ)2 Z ∞ 0 rγ|f0_{(r )|}2_{dr .}

When γ < 1, this inequality still holds for any function f which is zero at 0“Hardy’s inequality at zero”in the following sense

∀R > 0, f0∈ L1_{(0, R)} _and _{f (r ) =}

Zr 0

f0(s) ds

When γ > 1, this inequality still holds for any function f which is zero at infinity “Hardy’s inequality at infinity”in the following sense

∀R > 0, f0∈ L1(R, ∞) and f (r ) = Z ∞

r

f0(s) ds

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Neumann Laplacian in weighted Sobolev spaces on a polygon

Theorem N.2 Theorem L.4 Theorem M.2

∀c ∈C , the line Re λ = η is disjoint from the Mellin spectrum σ(Ac) (here this means that η 6∈ π_ω

cZ∗for allc ∈C ) Let u ∈ H1_(Ω)_{s. t.} _{∆u = f in Ω, f ∈ K}m−2 β+2(Ω) and ∂nu = 0 on ∂Ω Then (N.2) u = ureg+ X c∈C χc(x ) γc,0+ X `∈N∗, `_ωπ c<η γc,`r `_ωπ c c cos `ωπcθc with ureg∈ Kβm(Ω) Corollary

If f ∈ L2_(Ω)_{and Ω is}_convex_{then u ∈ H}2_{(Ω). Same consequence for Dirichlet.}

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Dirichlet-Neumann Laplacian on a plane sector

Let ∂0Γand ∂ωΓbe the two sides of the sector Γ of opening ω. We consider the

Laplace equation on Γ with the mixed boundary conditions u = 0 on ∂0Γ and ∂nu = 0 on ∂ωΓ

The Mellin symbol A[λ] incorporates these boundary conditions:

A[λ] :U 7→ (λ2U + U00) : nU ∈ H2(I), U(0) = 0, U0(ω) =0o−→ L2(I) The spectrum of the corresponding operator P is positive (as for Dirichlet). An orthonormal spectral basis φ`for P associated with eigenvalues µ`is given by

µ`= λ2` with λ`= ` −1 2 π ω, and φ`(θ) = r 2 ω sin λ`θ, ` ≥1 Any u ∈ H1_(Γ)_{such that u = 0 on ∂}

0Γ, belongs to K₋₁1 (Γ).

Theorem M.1 Theorem N1

Γplane sector of opening ω 6∈ {π₂,3π₂}. Let u ∈ H1_(Γ)_{with compact support s. t.}

∆u = f in Γ, f ∈ L2_(Γ) _and _{u = 0 on ∂}

0Γ, ∂nu = 0 on ∂ωΓ

Then

(M.1) u = u1+If ω >π₂(γ1rλ1 sin λ1θ) +If ω > 3π₂(γ2rλ2 sin λ2θ)

with ureg∈ K−22 (Γ)

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Dirichlet-Neumann Laplacian on a polygon

Theorem M.2 Theorem N.2

separating the corners. Let {λc,`, ` ≥1} be the positive part of the spectrum σ(Ac).

We assume that

∀c ∈C , the line Re λ = 1 is disjoint from the Mellin spectrum σ(Ac) Let u ∈ H1_(Ω)_{such that}

∆u = f in Ω, f ∈ L2(Ω) and u = 0 on ∂DirΩ, ∂nu = 0 on ∂NeuΩ

Then (M.2) u = ureg+ X c∈C χc(x ) X `∈N∗, λc,`<1 γc,`r λc,` c φc,`(θc) with ureg∈ H2(Ω) Remark

“Corner” has the extended sense of any transition point between Dir. and Neu. bc’s. The opening π is thus admissible, and associated with the singularity r12sin1₂θ.

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A last slide on the 2D Laplacian

Terminology

In expansions(L.2),(L.4),(N.2),(M.2)

the functions uregrepresents aregular part,

the sum of terms in rλc,`

c is the singular part, each term is asingular functionorsingularity. Two more things.

1 _{A comment on the condition}

∀c ∈C , the line Re λ = 1 is disjoint from the Mellin spectrum σ(Ac)

This condition is necessary if one wants a regular part in K−22 (Ω)(in this case one has

to consider constants functions at corners as singularities).

If one is satisfied with a regular part in H2_{(Ω), the constant functions are no more}

“singular”, and the condition can be relaxed to the new condition of“Injectivity modulo polynomials”that we will see later on.

2 _{These various bc’s can be combined with}_{transmission conditions}_{for which}

a(u, v ) =XK

k =1

Z

Ωk

ak∇u · ∇v dx, ak >0, Ωk polygonal partition of Ω

The collection of corners is the union of the corners of all the Ωk.

Model pbs are posed on sectors (possibly R2_{!) with partition by sectors of same vertex.} 42/58

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Outline

3D corners

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Homogeneous form with constant coefficients

The result generalizesTheorem L.4and proved in a very similar way (next slides)

Theorem G.1 Theorem G.3 Maxwell

Let Ω be a plane polygon with corners setC 3 c, with χcsmooth cut-off functions

Let a be a V -coercive form homogeneous with constant coefficients (aαβ· · ∈ C are 0 if

|α| + |β| < 2). Assume that X (Ω) = H1_(Ω)d_{. Recall that A is the op. associated with a.}

∀c ∈C , the line Re λ = η is disjoint from the Mellin spectrum σ(Ac) Let u ∈ V s. t. Au = f , f ∈ K_β+2m−2(Ω)d. Zero natural bc’s are included! Then (G.1) u = ureg+ X c∈C χc(x ) X λ∈σ(Ac),Re λ∈[0,η) Φc,λ with ureg∈ Kβm(Ω) d

Here Φcare singular functions associated with the cornerc Φc,λ(x ) = XQ q=1r λ c logqrc φc,λ,p(θc) 43/58

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Mellin symbol, in general

The problem Au = f with f ∈ L2_(Ω)d_{(or K}0

β0(Ω)dwith β0<1) is equivalent to a pde

Lu = f with L homogeneous of deg. 2 with constant coeffs At each cornerc, the Euler change of variables x 7→ (t, θ) transforms Lu = f into

Lc(θ; ∂t, ∂θ)˘u = e2t˘f

In the full Dirichlet case V = H1 0(Ω)

d_{, the Mellin symbol A}

c[λ]is defined on H₀1(Ic)das

U 7−→ Lc(θ; λ, ∂θ)

A suitable modification has to be performed for more general bc’s.

Theorem 2D Laplace

The corner Mellin symbol A associated with a coercive form (homogeneous with constant coefficients) has a meromorphic resolvent λ 7→ A[λ]−1that satisfies the uniform parameter estimates

(1) ∀λ ∈ Λ, ∀G ∈ Hm−2_(I)d_, _kA[λ]−1_Gk

Hm_(I;λ)≤ CΛkGkHm−2_(I;λ)

on any set Λ of the form Λ =λ ∈ C, Re λ ∈ [η0, η1] \ Sλ0∈σ(A)B(λ0, δ)

The proof is based on elliptic estimates (Th S.2) and the Analytic Fredholm Theorem. 44/58

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Residues, in general

The general resolvent estimates in the previous lemma allows to justify the residue formula related with inverse Mellin transform (after localization at each cornerc):

uη− u = X λ0∈σ(Ac) Re λ0∈[0,η) Res λ=λ0 rλ_A c[λ]−1Gc[λ] = X λ∈σ(Ac) Re λ∈[0,η) Φc,λ L Φc,λ=0 in Γc

and Φc,λsatisfies zero bc’s on ∂Γc. See also: Residues with polynomials Proof. We have, for a suitable closed contour γ around λ

Φc,λ= 1 2iπ Z γ rλAc[λ]−1Gc[λ]dλ Therefore L Φc,λ= 1 2iπ Z γ r−2Lc(θ;r ∂r, ∂θ) rλAc[λ]−1Gc[λ] dλ = 1 2iπ Z γ r−2rλ_L c(θ; λ, ∂θ) Ac[λ]−1Gc[λ]dλ = 1 2iπ Z γ r−2rλAc[λ] Ac[λ]−1Gc[λ]dλ = 0 45/58

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Smooth coefficients

In the case of smooth coefficients, and with possible lower order terms, the results are similar if we

Define the corner Mellin symbol Acafter freezing the coeffs of a atc according to

aαβc (x ) =

aαβ(c) if |α| + |β| = 2 0 if |α| + |β| ≤ 1

Modify the singular functions Φcby completing them by theirshadowsthat have

higher degrees in rc: Replace Φcby Ψc

Ψc= Φc+ XP p≥1 XQ q=1r p+λ c logqrc ψc,λ,p,q(θc)

With this definition of the corner Mellin symbol Ac, the regularity theorem is the same

in homog./non-homog. and constant/variable cases. Theorem G.2

Let Ω be a plane polygon with corners setC 3 c. Let a be a V -coercive form. Assume that X (Ω) = H1(Ω)d.

Let m ≥ 2 and β < −1. Set η = −β − 1. Assume

∀c ∈C , the strip Re λ = [0, η] is disjoint from the Mellin spectrum σ(Ac) Let u ∈ V such that Au = f with f ∈ K_β+2m−2(Ω)d_.

Then u ∈ K_βm(Ω)d

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And Maxwell in all that?

1 _{Theorem S.3}_{is still valid when Ω is a}_{convex polygon}_.

The reason for this is the global H2_(Ω)_{regularity of the solutions of Dirichlet}

or Neumann Laplacians with right hand sides f ∈ L2_(Ω).

This means that the variational spaces XN(Ω)or XT(Ω)for electric or

magnetic cases are subspaces of H1_(Ω)2_.

As a consequenceTheorem G.1applies.

The spectrum of the Mellin symbols Acfor electric or magnetic cases can be

made explicit: σ(Ac) = (_ωπ

cZ∗− 1) ∪ (

π ωcZ∗+1).

2 _{If the polygon Ω has non-convex corners, H}_N_(Ω)_{is a strict, closed, subspace of}

XN(Ω)and HT(Ω)is a strict, closed, subspace of XT(Ω). Theorem G.1does not apply as is.

But, by the same method of proof, we can prove a similar theorem in the Maxwell case:The integration contour has to be enlargedand (G.1) replaced with (G.1?₎ _{u = u} reg+ X c∈C χc(x ) X λ∈σ(Ac),Re λ∈(−1,η) Φc,λ

with, still, σ(Ac) = (_ωπ

cZ∗− 1) ∪ (

π ωcZ∗+1).

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Outline

3D corners

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3D corners

3D cones

Definition

1 _{A subset Γ of R}3_{is called a cone if ∀x ∈ Γ, ∀ρ > 0, ρx ∈ Γ.} 2 _{Denote for x 6= 0:}

r = |x | and ˆx =x r

3 _{Define the}_sectionˆ_Γ_{of a cone Γ}

ˆ

Γ = Γ ∩ S2 Definition

Let Γ be a cone in R3_{such that ˆ}_Γ_{is a connected open subset of S}2_.

1 _Γ_{is called a}_{regular cone}_{if ˆ}_Γ_{is a smooth submanifold with boundary of S}2_. 2 _Γ_{is called a}_wedge_{if after a possible rotation, Γ = Γ}0_{× R with a sector Γ}0_. 3 _Γ_{is called a}_{polyhedral cone}_{if ∂Γ is a finite union of plane faces. Then the}

boundary of ˆΓis a finite union of great-circle arcs.

4 _Γ_{is called a}_{cone with polygonal section}_{if ˆ}_Γ_{is a polygonal subset of S}2_.

Note that 2 _{is a particular case of} 3_{, which is a particular case of} 4_.

Here we understand that a wedge has an edge, and that a cone has a vertex.

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3D corners

3D domains with corners

Definition

Let Ω be an open bounded and connected subset of R3. If for any x ∈ ∂Ω, Ω is locally diffeomorphic

1 _{to R}2_{× R}₊_{or to a regular cone, Ω is called a}_{domain with conical points}_. 2 _{to R}2_{× R}₊_{or to a wedge, Ω is called a}_{domain with edges}_.

3 _{to R}2_{× R}₊_{, to a wedge or to a polyhedral cone, Ω is called a}_{polyhedral domain}_. 4 _{to R}2_{× R}₊_{, to a wedge or to a cone with polygonal section, Ω is called a}_domain

with corners.

In fact 3 _(and 4_{) are more general than the}_{“manifold with corners”}_{for which the}

cones have 3 plane faces. We are going to

1 _{Give hints on model problems (homogeneous with constant coefficients) in regular}

cones,

2 _{Tackle the problem of unweighted Sobolev spaces.} 3 _{Chat on polyhedral cones.}

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3D corners

3D domains with corners: Examples

1 _{Domains with conical points}

2 _{Domains with edges}

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3D corners

3D domains with corners: Examples

3 _{A polyhedral domain (Fichera corner) — on the left} 4 _{A domain with corners (non polyhedral) — on the right}

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Regular cones in Rn and unweighted spaces

Regular cones in R

n

The variable ˆx ∈ ˆΓ ⊂ Sn−1plays the same role as the angle θ ∈ (0, ω) = I: ∆ becomes e−2t ∂t2+ (n − 2)∂t− Ln−1

with Ln−1the (positive) Laplace-Beltrami operator on Sn−1. Let (µ`, φ`)be its Dirichlet

eigenpairs on ˆΓ. The Mellin symbol of the Dirichlet Laplacian on Γ is A[λ] :U 7−→ λ2+ (n − 2)λ − Ln−1

(H2∩ H1

0)(ˆΓ) −→L2(ˆΓ)

The spectrum σ(A) = {λ`, ` ∈ Z∗} with

λ±`=1 −n₂± q µ`+ (n2− 1)2, ` ∈ N∗ The equivalence u ∈ Km β(Γ) ⇐⇒e−ηt˘u ∈ H m_{(R × ˆ}_Γ)_{holds with η = −β −}n 2. Theorem L.5 Theorem L.3

Let Γ be a regular cone in Rn_{. Let m ≥ 2 and β < −1. Set η = −β −}n

2. Assume

The line Re λ = η is disjoint from the Mellin spectrum σ(A) Let u ∈ H1

0(Γ)with compact support such that ∆u ∈ K m−2 β+2(Γ) Then (L.5) u = uη+ X `∈Z∗,1−n₂<λ`<η γ`rλ`φ`(ˆx ) with uη∈ Kβm(Γ) 52/58

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Unweighted spaces in 3D (odd nD) and Taylor expansions

The question of a right hand side f posed in a standard unweighted Sobolev space Hm−2(Ω)is more or less still pending.

1 _{If f ∈ L}2_{(Ω) =}_K0 0(Ω), OK with β = −2 and m = 2. Regular part in K2 −2(Ω) ⊂H 2_(Ω) 2 _{If f ∈ H}1_{(Ω), then f ∈ K}1

−1(Ω)because in dimension n = 3, the two spaces

coincide! OK with β = −3 and m = 3. Regular part in K₋₃3 (Ω) ⊂H3(Ω)

3 _{If f ∈ H}2_{(Ω), then f ∈}_C12_(Ω)_{and, moreover}

f −X

c∈C

χc(x ) f (c) ∈ K−22 (Ω)

4 _{General case in nD, n odd: If f ∈ H}k_{(Ω), f ∈}Ck −n2(Ω)and

[Taylor ] f −X c∈C χc(x ) X |α|<k −n₂ (x −c)α α! ∂ α xf (c) ∈ K−kk (Ω)

Proofs useHardy’s inequality The curse of even dimensions

If n = 2, or more generally if n is even, we are in limit cases of Sobolev embedding theorem for integer regularity exponents. [Taylor ] can be extended to odd dimensions with |α| < k −n₂but not to even dimensions.

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Unweighted spaces in 3D (odd nD) domains with conical points

Theorem G.3 Theorem G.1 Theorem G.4

For odd n ≥ 3, let Ω ⊂ Rnbe a domain with set of conical pointsC 3 c [here for simplicity we assume that local diffeomorphisms at corners are affine functions]. Let a be a V -coercive form homogeneous with constant coefficients. Assume that X (Ω) = H1_(Ω)d_{. Recall that A is the op. associated with a.}

Let m ≥ 2. Set η = m −n

2. We assume that

∀c ∈C , the line Re λ = η is disjoint from the Mellin spectrum σ(Ac) Let u ∈ V such that Au = f , f ∈ Hm−2_(Ω)d_.

Then (G.3) u = uflat+ X c∈C χc(x ) X λ∈σ(Ac)∪N2,Re λ∈[1−n2,η) Ψc,λ with uflat∈ K−mm (Ω)d

Here Ψcare singular functions (or polynomials) associated with the cornerc

Ψc,λ(x ) = XQ q=1r λ c logqrc ψc,λ,p(ˆxc) 54/58

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General residues in 3D (odd nD) cones

Difference between Th. G.3 and Th. G.1: The regularity of the rhs and the set of poles: λ ∈ σ(Ac) ∪ {2, 3, . . .} with Re λ ∈ [1 −n₂, η)

The proof is still based on the residue formula issued from inverse Mellin transform: uη− u = X λ0∈σ(Ac)∪N2 Re λ0∈[1−n₂,η) Res λ=λ0 rλ_A c[λ]−1Gc[λ] = X λ∈σ(Ac)∪N2 Re λ∈[1−n 2,η) Ψc,λ

Now Gc[λ]has poles on integers λ ≥ 2 corresponding to the Taylor expansion of f atc

Res λ=kr λ_{G[λ] = r}2 X |α|=k −2 (x −c)α α! ∂ α xf (c)

Lemma See also: Residues without polynomials

( L Ψc,λ=0 if λ ∈ σ(Ac) \ N2 L Ψc,λ=P|α|=λ−2 (x −c)α α! ∂ α xf (c) if λ ∈ N2

and Ψc,λsatisfies zero boundary conditions on ∂Γc

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Injectivity modulo polynomials

Injectivity modulo polynomials

The notion of singular function isrelative. A function is considered as singular if it does not belong to some space of regular function. With a space like K_−mm , any nonzero polynomial of degree < m −n₂is singular, but with an unweighted Sobolev space Hm_,

the same polynomial is regular. Definition

Let Γ be a regular cone in Rn_{, with ˆ}_Γ_{its section. Let λ ∈ C.}

1 _{Let S}λ_{be the space of quasi-homogeneous d -component functions}

Sλ(Γ) = n Ψ =rλXQ q=0log q_{r ψ} q(ˆx ), ψq∈C∞(Γ)d o

We mention zero boundary conditions by an index: Sλ

bc(Γ), e.g. S λ 0(Γ)for

Dirichlet, and Sλ

N(Γ)for tangential bc’s (if n = d ).

2 _{Let P}λ_{be the space of homogeneous polynomials in Cartesian var. x , of deg. λ} 3 _{The operator L with boundary conditions bc is said to be}_{injective modulo}

polynomialsif

Ψ ∈Sλ_bc(Γ) and LΨ ∈ Pλ−2 =⇒ Ψ ∈Pλ. This condition can be written as

L : Sλ_bc(Γ)/P_bcλ(Γ) −→Sλ−2(Γ)/Pλ−2 injective

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Optimal theorem for domains with conical points in any dimension

Theorem G.4 The dimension is any integer n ≥ 2. Theorem G.3

Let Ω ⊂ Rn_{be a domain with set of conical points}_{C 3 c [here for simplicity we}

assume that local diffeomorphisms at corners are affine functions].

Let a be a V -coercive form homogeneous with constant coefficients. Assume that X (Ω) = H1_(Ω)d_{. Recall that A is the operator, L is pde system, and bc the boundary}

conditions associated with a.

Let m ≥ 2. Set η = m −n₂. We assume that

(∀c ∈_{C ), (∀λ, Re λ = η), L is injective mod. polynomials on S}λ bc(Γc)

Let u ∈ V such that Au = f , f ∈ Hm−2_(Ω).

Then (G.4) u = ureg+ X c∈C χc(x ) X λ∈σ?(a,c), Re λ∈[1−n₂,η) Ψc,λ with ureg∈ Hm(Ω)

and Ψcsingular functions associated with the cornerc (similar as inTh.G.3)

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Comments on the optimal theorem and conclusion

1 _{The regular part is in H}m_{and is not “flat” in general: It may have a nonzero Taylor}

expansion at corners.

2 _{When λ 6∈ N, the injectivity mod. polynomials is equivalent to (λ 6∈ σ(A}_c_)). 3 _{Importantly, we have overcome the curse of even dimensions! But at the expense}

of introducing modified parameter-dependent norms Hm_(ˆ_Γ

c; λ; ?)in order to relax Hm_(ˆ_Γ

c; λ)in a suitable way around positive integers.

4 _{2D cracks}_{: The condition of injectivity mod. polynomials is satisfied for} homogeneousDirichlet or Neumann conditions for ∆ at λ = 1 (and any positive integer) when ω = 2π.

5 _{Half-planes: The condition of injectivity mod. polynomials is satisfied for} homogeneousDirichlet or Neumann for ∆ at any λ when ω = π... This means that our theory is coherent with the standard regularity results at regular points of the boundary.

Regularity theorems on edge or polyhedral domains are in the same spirit: At each singular point is associated a condition of injectivity mod. polynomials (IMP) and regularity is obtained if the IMP condition is satisfied in certain complex strips. Expansions in regular and singular terms are much more involved. There is a competition between corners and edges. Both contribute to expansions, and they combine with each other in non-unique and non-canonical ways.