On Friedrichs constant and Horgan-Payne angle for LBB condition

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condition

Monique Dauge, Christine Bernardi, Martin Costabel, Vivette Girault

To cite this version:

Monique Dauge, Christine Bernardi, Martin Costabel, Vivette Girault. On Friedrichs constant and

Horgan-Payne angle for LBB condition. Monografias Matematicas Garcia de Galdeano, pp.222, 2014,

Twelfth International Conference Zaragoza-Pau on Mathematics, ISBN: 978-84-16028-35-1. �hal-

00797642�

O N F RIEDRICHS CONSTANT AND

H ^ORGAN -P ^{AYNE ANGLE}

FOR LBB CONDITION

Monique Dauge, Christine Bernardi, Martin Costabel, and Vivette Girault

Abstract. In dimension 2, the Horgan-Payne angle serves to construct a lower bound for the inf-sup constant of the divergence arising in the so-called LBB condition. This lower bound is equivalent to an upper bound for the Friedrichs constant. Explicit upper bounds for the latter constant can be found using a polar parametrization of the boundary.

Revisiting carefully the original paper which establishes this strategy, we found out that some proofs need clarification, and some statements, replacement.

Keywords:LBB condition, inf-sup constant, Friedrichs constant, Horgan-Payne angle.

AMS classification:30A10, 35Q35.

§1. The inf-sup constant and some general properties

Here we only considerbounded connectedopen setsΩinR², the generic point inR² being denoted by x = (x1,x2). For such a domain Ω, the inf-sup constant of the divergence as- sociated with Dirichlet boundary conditions, also called LBB constant after Ladyzhenskaya, Babuˇska[3, 2] and Brezzi[4], is defined as

β(Ω)= inf

q∈L²◦(Ω)

sup

u∈H₀¹(Ω)²

divu,q

Ω

|u|_1,

Ωkqk

0,Ω

. (1)

Here

• L²_◦(Ω) stands for the space of square integrable scalar functionsqwith zero mean value inΩendowed with its natural normk·k_0,Ωand natural scalar producth·,·i_Ω,

• H¹₀(Ω)²is the standard Sobolev space of vector functionsu=(v1, v2) with square inte- grable gradients and zero traces on the boundary, endowed with its natural semi-norm

|u|

1,Ω=

2

X

k=1 2

X

j=1

k∂_xjv_kk²

0,Ω

^1/2 .

SinceΩis bounded, by virtue of the Poincaré inequality, the above semi-norm onH₀¹(Ω)²is equivalent to the usual norm inH¹(Ω)².

We list some elementary properties ofβ(Ω):

(a) β(Ω)≥0,

(b) β(Ω)≤1, because of the identity|u|²

1,Ω=kcurluk²_0,

Ω+kdivuk²_0,

Ωfor anyu∈H₀¹(Ω)², (c) β(Ω) is invariant by translations, dilations, symmetries and rotations by virtue of Piola

transform. Thusβ(Ω) only depends on theshapeofΩ.

The constantβ(Ω) is positive for Lipschitz domains (see [8, Chap. 1, Section 2.2], which relies on [12, Chap. 3, Lemme 7.1]), and also for domains with less regular boundary like John domains [1]. In contrast, domains with an external cusp (also called thin peak) satisfy β(Ω)=0, see [15, Chap. 15].

Finding calculable lower bounds forβ(Ω) is of great interest, since it is involved in any analysis of the Stokes and Navier-Stokes equations with no-slip boundary conditions. More- over, discrete inf-sup constants between finite dimensional subspaces ofH¹₀(Ω)² andL²_◦(Ω) are influenced by both the continuous inf-sup constantβ(Ω) and the type of chosen (mixed) discrete spaces, see [13] and also [6].

In reference [10], Horgan & Payne design an efficient strategy for calculating lower bounds ofβ(Ω) in domainsΩwhose boundary can be described in polar coordinates (r, θ) by a relationr= f(θ) with a Lipschitz-continuous function f:

◦ First, state a relation betweenβ(Ω) and the Friedrichs constantΓ(Ω),

◦ Second, find bounds forΓ(Ω) using f and its first derivative f⁰.

In the present paper, we revisit these two steps, with more emphasis on the second one.

§2. The Friedrichs constant

In dimension 2, the coordinates (x₁,x₂) are identified with the complex number x₁+ix₂. Two real valued functionsh and g are said to be harmonic conjugate if they are the real and imaginary parts of a holomorphic functionh+ig. The functionshandgare harmonic conjugate if and only if they satisfy the relations

∆h=0, ∆g=0, and gradh=curlg in Ω.

LetF(Ω) denote the space of complex valuedL²(Ω) holomorphic functions and letF_◦(Ω) be its subspace of functions with mean value 0.

Definition 1. TheFriedrichs constant (named after [7]) denoted byΓ(Ω), is the smallest constantΓ∈R∪ {∞}such that for allh+ig∈F_◦(Ω)

khk²

L²(Ω)≤Γkgk²

L²(Ω).

Theorem 1([10], [5]). LetΩbe any bounded connected domain inR². The LBB constant β(Ω)is positive if and only if Γ(Ω)is finite and

Γ(Ω)+1= 1 β(Ω)² .

This relation betweenβ(Ω) andΓ(Ω) was proved in [10] under additional regularity prop- erties on the domain. A new proof is provided in [5], in which no regularity assumption is needed.

§3. An upper bound for the Friedrichs constant

LetΩbestrictly star-shaped, which means that there is an open ballB ⊂Ωsuch that any segment with one end inBand the other inΩ, is contained inΩ. LetObe the center ofBand (r, θ) be polar coordinates centered atO. Letθ 7→ r = f(θ) be the polar parametrization of the boundary∂Ω, defined on the torusT=R/2πZ.

Lemma 2([11, Lemma 1.1.8]). LetΩbe a bounded strictly star-shaped domain, and f be a polar parametrization of its boundary as described above. Then f belongs to W^1,∞(T).

SinceΓ(Ω) is invariant by dilation, we may assume without restriction that maxθ∈T

f(θ)=1 (2)

Following the approach in [10], we are prompted to introduce the following notation.

Notation3. Under condition (2), letP=P(α, θ) be the function defined onR+×Tas P(α, θ)= 1

αf(θ)²

1+ f⁰(θ)² f(θ)²−αf(θ)⁴

. (3)

LetM(Ω) andm(Ω) be the following two positive numbers M(Ω)= inf

α∈(0,1)

( sup

θ∈T P(α, θ) )

and m(Ω)=sup

θ∈T

( inf

α∈0, ¹

f(θ)2

P(α, θ) )

. (4)

Remark1. Let us chooseθ∈T. Calculating the second derivative of the functionP_θ :α7→

P(α, θ) defined on the interval 0, _f(θ)¹₂, we find thatPθis strictly convex. The functionPθ

tends to+∞asα→0, and if f⁰(θ),0, asα→ ¹

f(θ)². In any case, there exists a uniqueα(θ) in 0, _f(θ)¹2

such that

P(α(θ), θ)= inf

α∈0,_f(θ)2¹ P(α, θ).

So,

m(Ω)=sup

θ∈T

P(α(θ), θ). (5)

Since, in particular, for allα∈(0,1) andθ∈T,P(α(θ), θ)≤P(α, θ), we find that

M(Ω)≥m(Ω). (6)

The quantitym(Ω) is the original bound introduced by Horgan-Payne in [10] andM(Ω) is our modified Horgan-Payne like bound.

Theorem 4(Estimate (6.24) in [10]). LetΩbe a bounded strictly star-shaped domain. Its Friedrichs constant satisfies the bound

Γ(Ω)≤M(Ω). (7)

Proof. We assume for simplicity that the originOof polar coordinates coincides with the origin0of Cartesian coordinates. Letg ∈D(Ω) be an harmonic function and leth ∈D(Ω) be its harmonic conjugate such thath(0)=0. If we bound theL²(Ω) norm ofh, we bound a fortiori theL²(Ω) norm ofh−_|¹

Ω|

R

Ωhwhich is the harmonic conjugate ofginL²_◦(Ω), hence with minimalL²(Ω) norm. The extension of the estimate to all pairs of harmonic conjugate functions inL²(Ω) follows from a density argument.

Sinceh+igis holomorphic, its square is holomorphic too and we deduce that the function H :=h²−g²is harmonic conjugate ofG :=2gh. Hence equation gradH =curlGleads to the relation in polar coordinates

∂_ρHe=1 ρ∂_θGe

whereH(r, θ)e = H(x) andG(r, θ)e =G(x) for x =(rcosθ,rsinθ). Thus for anyθ ∈Tand r∈(0,f(θ)) we have

H(r, θ)e −H(0)=Z r 0

∂ρH(ρ, θ) dρe =Z r 0

1

ρ∂θG(ρ, θ) dρ .e We divide by f(θ)²and integrate forθ∈Tandr∈(0,f(θ)):

Z

T

Z f(θ) 0

H(r, θ)e −H(0)

f(θ)² rdrdθ=Z

T

Z f(θ) 0

1 f(θ)²

( Z r

0

1

ρ∂_θG(ρ, θ) dρe ) rdrdθ

=Z

T

Z f(θ) 0

1 f(θ)²

1

ρ∂θG(ρ, θ)e ( Z f(θ)

ρ rdr

) dρdθ

= 1 2

Z

T

Z f(θ) 0

f(θ)²−ρ²

ρ²f(θ)² ∂θG(ρ, θ)e ρdρdθ . Since the function f(θ)²−ρ²is 0 on the boundary, integration by parts yields

Z

T

Z f(θ) 0

H(r, θ)e −H(0)

f(θ)² rdrdθ=− Z

T

Z f(θ) 0

f⁰(θ)

f(θ)³G(ρ, θ)e ρdρdθ . We set for anyθ∈T

t(θ)= f⁰(θ) f(θ) . Coming back tohandgwe find:

Z

Ω

h(x)² f(θ)²dx=Z

Ω

g(x)²−g(0)² f(θ)² dx−2

Z

Ω

t(θ)h(x)g(x)

f(θ)² dx. (8)

In order to take the best advantage of the previous identity we introduce a parameter α∈(0,1)

and write for anyθ∈T(here we use condition (2) which ensures that 1−αf(θ)²>0) 2

t(θ)h(x)g(x) ≤n

1−αf(θ)²o

h(x)²+ t(θ)²

1−αf(θ)²g(x)²

and deduce from (8) that (note that the sameαis used for allθ) αZ

Ω

h(x)²dx≤ Z

Ω

g(x)²

f(θ)² + t(θ)² 1−αf(θ)²

g(x)² f(θ)²dx. Thus, for anyα∈(0,1)

Z

Ωh(x)²dx≤sup

θ∈T

1 αf(θ)²

1+ t(θ)² 1−αf(θ)²

Z

Ωg(x)²dx. Optimizing onα∈(0,1) and coming back to the definition oftandP, we find

Z

Ωh(x)²dx≤ inf

α∈(0,1)

sup

θ∈TP(α, θ) Z

Ωg(x)²dx,

which is nothing else thankhk²_0,Ω≤M(Ω)kgk²_0,Ω, whence the theorem.

Remark2. The proof above is due to Horgan and Payne in [10, § 6]. Unfortunately, instead of simply concluding that M(Ω) is an upper bound forΓ(Ω), they try to show that M(Ω) coincides withm(Ω) and this part of their argument is flawed. In the rest of our paper we discuss cases where equality or non-equality holds between these two quantities.

§4. The Horgan-Payne angle

Stoyanin [14] propose an interesting geometrical interpretation of the lower bound onβ(Ω) under the condition thatm(Ω) is an upper bound forΓ(Ω).

Notation5. Forθ∈T, letxbe the point (f(θ) cosθ,f(θ) sinθ) in∂Ω, letγ(θ)∈[0,^π₂) denote the (non-oriented) angle between the line [0,x] and the outward normal vector to∂Ωat x.

We set

γ(Ω)=sup

θ∈T

γ(θ) and ω(Ω)=π

2−γ(Ω). (9)

The angleω(Ω) is referred as the Horgan-Payne angle in [14].

Lemma 6. We have the identities m(Ω)=1+sinγ(Ω)

1−sinγ(Ω) and 1

√

m(Ω)+1 =sinω(Ω)

2 . (10)

Proof. Let us recall the formulas cosγ(θ)= f(θ)

pf(θ)²+f⁰(θ)², sinγ(θ)= f⁰(θ)

pf(θ)²+f⁰(θ)², tanγ(θ)= f⁰(θ) f(θ) . Hence we have

P(α, θ)= 1 αf(θ)²

1+ tan²γ(θ) 1−αf(θ)²

. (11)

Letθ be chosen. To determine the valueα(θ) which realizes the minimum of P(α, θ) for α∈(0,1/f(θ)²], cf. (5), we calculate

∂_αP(α, θ)=− 1 α²f(θ)²

1+ tan²γ(θ) 1−αf(θ)²

+ 1 αf(θ)²

tan²γ(θ)f(θ)² (1−αf(θ)²)² . Settingζ=αf(θ)², we see that∂_αP(α, θ)=0 if and only if

ζ²−2(1+tan²γ(θ))ζ+1+tan²γ(θ)=0. (12) We look forζ∈(0,1]. The convenient root of equation (12) is

α(θ)f(θ)²=ζ=1+tan²γ(θ)−tanγ(θ)q

1+tan²γ(θ)

= 1

1+sinγ(θ) . (13)

Hence we find

P(α(θ), θ)=1+sinγ(θ)

1−sinγ(θ), (14)

whose supremum is attained for the supremumγ(Ω) ofγ(θ), whence the first formula in (10).

The second formula is obtained using sin^ω(₂^Ω) =sin(^π₄−^γ(₂^Ω))= ^√1₂(cos^γ(₂^Ω)−sin^γ(₂^Ω)).

As a straightforward consequence of Theorem 1 and Lemma 6, we obtain the following.

Corollary 7. For any domain such thatΓ(Ω)≤m(Ω), the inf-sup constantβ(Ω)satisfies β(Ω)≥sinω(Ω)

2 . (15)

Remark3. The estimate (15) is stated in [14] for any strictly star-shaped domain. The reality is that (15) is true if and only ifΓ(Ω)≤m(Ω). The latter estimate is true for some categories of domains as we will see in the next section. We will also exhibit domains for whichm(Ω) is distinct fromM(Ω). In [5] it is proved that, in fact, there exists strictly star-shaped domains such thatΓ(Ω)>m(Ω) (equivalently,β(Ω)<sin^ω(₂^Ω)).

§5. Examples

In this section, we consider some particular shapes of domains, namely ellipses, polygons, and limaçons.

5.1. Disks and ellipses

The equation of an ellipse can always be written in suitable Cartesian coordinates as x²

a² +y² b² =1

with positive coefficientsa≤b. The constantΓ(Ω) is analytically known, cf. [7], namely Γ(Ω)= b²

a² and β(Ω)= a

√ a²+b²

. (16)

In polar coordinates, the parametrization of the ellipse is f(θ)=ab

b²cos²θ+a²sin²θ−1/2

. (17)

i) Let us calculatem(Ω). We have tanγ(θ)= f⁰(θ)

f(θ) =sinθcosθ(b²−a²)

b²cos²θ+a²sin²θ =tanθ(b²−a²)

b²+a²tan²θ . (18) The maximal value tanγ(Ω) of tanγ(θ) is obtained for

tanθ= b a , hence

tanγ(Ω)= b²−a² 2ab from which we deduce

sinγ(Ω)= b²−a² b²+a² . Formula (10) then yields

m(Ω)= b² a² .

ii) Let us calculateM(Ω). In order to comply with the condition max_θ∈T f(θ) = 1, we set

˜

a=b/aand ˜b=1, and consider f given by (17) witha,breplaced by ˜a,b. We use formula˜ (11) forPto write:

P(α, θ)= 1 α

1+tan²γ(θ)f(θ)⁻²−α 1−αf(θ)² . From (17) and (18) we deduce

1+tan²γ(θ)f(θ)⁻²=a˜⁻². Therefore

P(α, θ)= 1 α

˜ a⁻²−α 1−αf(θ)².

For eachα∈(0,1), the supremum inθofP(α, θ) is attained for f(θ) minimum, i.e. inθ=0 for which f(θ)=a. We deduce˜

supθ∈T

P(α, θ)= 1 α

a˜⁻²−α 1−α˜a² = 1

α 1

˜ a² = 1

α b² a²,

hence, taking the infimum overα∈(0,1):

M(Ω)= b² a² . Comparing with (16), we finally obtain

m(Ω)=M(Ω)=b²

a² = Γ(Ω). (19)

In particular, ifΩis a disk

m(Ω)=M(Ω)= Γ(Ω)=1. (20)

5.2. Star-shaped polygons

A polygonΩis characterized by the fact that its boundary is a finite union of segments. Let us first investigate the behavior of the functionPalong a segment.

For ease of computation, we consider a segment I lying on a vertical line of equation x1 =d withd > 0. Note thatd is the distance of this line to the origin. Normals toI are horizontal. We find

f(θ)= d

cosθ and γ(θ)=θ. (21)

Hence, under the global condition maxθ∈T f(θ)=1, the contribution to the functionPof such a segment is — here we use formula (11),

P(α, θ)=cos²θ αd²

1−αd² cos²θ−αd²

= 1 αd²

1−αd²

1−αf(θ)². (22)

For anyα∈(0,1), the maximal value ofPis attained for f(θ) maximal, i.e., at an end of the segmentI, and this end is the most distant from the origin. That is why we introduce:

Notation8. For any sideIj,j=1, . . . ,J, of a polygonΩ, we define its radiusrjas the distance between the origin and its most distant endpointE_j. Denoting by ˜I_jthe line containingI_j, we defined_jas the distance of ˜I_jto the origin.

The normalization (2) here takes the form max_jr_j =1. From the previous computation (22) we find the formula

M(Ω)= inf

α∈(0,1)

maxJ j=1

1 αd²_j

1−αd²_j

1−αr²_j (23)

= inf

α∈(0,1)

maxJ j=1

1 αr²_j

r²_jd⁻²_j −αr²_j

1−αr²_j . (24)

In order to find a similar formula for the quantitym(Ω), we are going to use (5) and we go back to expression (14) which can be written in function of cosγ(θ) instead of sinγ(θ):

P(α(θ), θ)=









 1 cosγ(θ)+

s 1 cos²γ(θ) −1











2

. (25)

For the anglesθcorresponding to the segmentIj, the supremum ofP(α(θ), θ) is attained for cosγ(θ) minimum, i.e. for cosγ(θ)=^d_rj^j. Therefore formula (25) yields

m(Ω)=max^J

j=1











 rj

dj

+ vt

r²_j d²_j −1













2

. (26)

The maximum is attained whenrj/djis maximal.

Proposition 9. LetΩbe a polygon, with djand rjthe distances in Notation 8. We have i) If all rjare equal, then M(Ω)=m(Ω).

ii) If all d_jare equal, then M(Ω)=m(Ω).

iii) If the largest value of rj/d_jis attained for two different indices j and k and if rj,rk, then M(Ω)>m(Ω).

Proof. i)If allr_jare equal, the normalization max_jr_j =1 yields thatr_j =1. Formula (24) then gives that

M(Ω)= inf

α∈(0,1)

1 α

d_min⁻² −α 1−α

wheredminis the minimum value of thedj. The optimization with respect toαprovides the optimal value

α0= 1 d²_min −

s 1 d_min⁴ − 1

d_min² ∈(0,1) forα, hence the infimum

M(Ω)=









 1 dmin

+ s 1

d_min² −1











2

which coincides withm(Ω) given by (26) sincerj/djis maximal for 1/dmin. ii)If alld_jare equal, Formula (23) gives that

M(Ω)= inf

α∈(0,1)

1 αd²

1−αd² 1−αr²_max

wheredis the common value of thedjandrmaxthe maximum value of therj. Due to normal- ization max_jrj=1, this formula becomes

M(Ω)= inf

α∈(0,1)

1 αd²

1−αd² 1−α = inf

α∈(0,1)

1 α

d⁻²−α 1−α . As in the previous case we find

M(Ω)=







 1 d +

√ 1−d²

d









2

,

which coincides withm(Ω) given by (26) sincerj/d_jis maximal for 1/d.

iii)For`∈ {j,k}, letθ<sub>`</sub>be the angleθcorresponding to the endE_`. We have M(Ω)≥ min

α∈(0,1) max

θ∈{θj,θk}P(α, θ).

And letαmbe the value ofα∈(0,1] minimizing maxθ∈{θ_j,θ_k}P(α, θ). We have M(Ω)≥max{P(α_m, θ_j),P(α_m, θ_k)}.

Now, still for ` ∈ {j,k}, let α` be the value ofα ∈ (0,r<sub>`</sub><sup>−2</sup>) minimizing P(α, θ`). Since rj/dj=rk/dkmaximizes the quotientsri/di, we have by (26)

m(Ω)=











 rj

dj + vt

r²_j d²_j −1













2

=









 rk

dk+ sr²_k

d_k² −1











2

=P(α_j, θ_j)=P(α_k, θ_k). By (13),αjandαksatisfy

α_{`</sub>r<sub>`}²= 1

1+sinγ(θ<sub>`</sub>), `= j,k.

But sinγ(θj)=sinγ(θk) because cosγ(θ`)=r`/d`. Hence, sincerj ,rk, we haveαj ,αk, thereforeα_mcannot coincide withα_jandα_kat the same time. So, since the functionsα7→

P(α, θ) are strictly convex in the interval (0,f(θ)⁻²), we deduce

M(Ω)≥max{P(αm, θj),P(αm, θk)}>P(αj, θj)=P(αk, θk)=m(Ω),

and conclude thatM(Ω)>m(Ω) as announced in the proposition.

Here are examples for the three situationsi)–iii)investigated in Proposition 9.

Example 1. In each of the examples below, the center0of polar and Cartesian coordinates is chosen at thebarycenterof the domain.

i) IfΩis a regular polygon or a rectangle, then allrjare equal, thusM(Ω)=m(Ω).

ii) IfΩis a triangle or a rhombus, then alld_jare equal, thusM(Ω)=m(Ω).

iii) See Figure 1: For this hexagonal domain, the quotientsrj/djare all equal to

√ 2, but r1=1 andr2=1/√

2 (withθ1=0 andθ2= ^π₄). Thereforem(Ω)<M(Ω).

5.3. Limaçons

Limaçons of Pascal (named after Etienne Pascal, father of Blaise Pascal) are curves defined in polar coordinates by a formula of the type

f_ε(θ)=a(1+εcosθ), a>0, ε >0. (27)

ï1 ï0.5 0 0.5 1 ï0.5

0 0.5

m(1) = 5.83 M(1) = 6.86

0 0.5 1 1.5 2

0 2 4 6 8 10 12

large radius small radius

Figure 1: Example whereM(Ω)>m(Ω). DomainΩwith center of coordinates, left. Plot of α7→P(α, θj) forα∈(0, _f(θ¹

j)²), j=1,2, right.

Such a curve is simple ifεis less than 1, and so defines the boundary of a domainΩε. In [10] the case of such limaçons is considered. Here the constantΓ(Ωε) is analytically known, [9, 10], which provides an explicit formula forβ(Ωε) via Theorem 1

Γ(Ωε)=2+ε²

2−ε² and β(Ωε)=

√ 2−ε²

2 . (28)

This example can serve as a benchmark for boundsmandM. We have computed by a Matlab program the two constantsm(Ωε) andM(Ωε). It happens that as soon asεis not zero, i.e.,Ωε

is not a circle, these two constants are distinct, see the top two curves in Figure 2. The other curves are explained below.

Here comes the question of the choice of polar coordinates definingm(Ω) andM(Ω). For limaçons, the first choice is to consider the polar coordinates in which the domain is defined by (27). But, considering thatΩεintersects the horizontal axis between the points−a+aε anda+aε, choosing new polar coordinates (r⁰, θ⁰) centered atO⁰ = (aε,0) appears more judicious. A new equation

r⁰= f_ε⁰(θ⁰) is associated withΩε, leading to new quantities

m⁰(Ωε) and M⁰(Ωε).

In fact these new quantities are very different from the old ones. We have observed that m⁰(Ωε) andM⁰(Ωε) do coincide, and are much smaller thanm(Ωε) andM(Ωε), see Figure 2.

Moreover, the asymptotic behavior ofm⁰(Ωε)=M⁰(Ωε) asε→0 is very good. Indeed, using formula (28) allows us to compute the differencem⁰(Ωε)−Γ(Ωε). We have found numerical evidence, see Figure 3, for the asymptotic behavior

m⁰(Ωε)−Γ(Ωε)=M⁰(Ωε)−Γ(Ωε)=O(ε³).

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0

0.5 1 1.5 2 2.5 3

Log10 m(1_¡) Log10 M(1_¡) Log10 m’(1_¡) Log10 K(1 )_¡

Figure 2: Plot ofε7→log₁₀m(Ωε),M(Ωε),m⁰(Ωε)=M⁰(Ωε),Γ(Ωε) forε=0.1 to 0.9.

5.4. Decentered disks

We end this section by a curiosity which sheds some light on the discrepancy betweenm(Ω) andM(Ω) and their dependency on the center of polar coordinates. LetΩbe a disk. We have seen in (20) that we have the optimal valuesm(Ω)=M(Ω)= Γ(Ω)=1.

Now, we consider decentered disks, moving offthe center of polar coordinates by a rel- ative amount δwith respect to the radius of the disk. We can assume that the new center lies on the horizontal axis. This defines new versions of the constants, denotedm[δ](Ω) and M[δ](Ω). It is not very hard to prove the following

(i)The maximal value of the angleγoccurs forθ₀= ^π₂, so sinγ=δ. Hence, cf (10), m[δ](Ω)=1+δ

1−δ.

This value is the same for the limaçon (27) choosingε=δ, see [10, (6.34)].

(ii) For any α ∈ (0,1), the max in θ of P(α, θ) is attained forθ = π, then the inf in α corresponds toP(1, π). Hence

M[δ](Ω)= 1

f(π)² =1+δ 1−δ 2

=m[δ](Ω)²>m[δ](Ω).

ï4 ï3.5 ï3 ï2.5 ï2 ï1.5 ï1 ï14

ï12 ï10 ï8 ï6 ï4 ï2 0 2 4

Log2 { M(1_¡) ï 1 } Log2 { m(1_¡) ï 1 } Log2 { m’(1_¡) ï 1 } Log2 { K(1 )_¡ ï 1 } Log2 { m’(1_¡) ï K(1 )_¡ }

Figure 3: Plot of log₂ε7→log₂M(Ωε)−1,m(Ωε)−1,m⁰(Ωε)−1,Γ(Ωε)−1,m⁰(Ωε)−Γ(Ωε) forε=0.0625 to 0.5.

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Monique Dauge and Martin Costabel IRMAR

CNRS et Université de Rennes 1 Campus de Beaulieu

35042 RENNES Cedex, FRANCE [email protected] [email protected]

Christine Bernardi and Vivette Girault Laboratoire Jacques-Louis Lions CNRS et Université Pierre et Marie Curie 4 place Jussieu

75252 PARIS Cedex 05, FRANCE [email protected] [email protected]