Limit behaviour of thin insulating layers around multiconnected domains

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LIMIT BEHAVIOUR OF THIN INSULATING LAYERS AROUND MULTICONNECTED DOMAINS

Mohamed BOUTKRIDA^a, Nathalie GRENON^b, Jacqueline MOSSINO^a, Gonoko MOUSSA^c

aEcole Normale Supérieure de Cachan, C.M.L.A. – U.M.R. 8536, 61, Avenue du Président Wilson, 94235 Cachan cedex, France

bFaculté des Sciences, Rue Gaston Gerger, 18000 Bourges, France

cEcole Normale Supérieure de Cachan and Université Henri Poincaré, Nancy I, Institut Elie Cartan, BP 239, 54506 Vandoeuvre les Nancy cedex, France

Received 17 July 2000, revised 11 January 2001

ABSTRACT. – Let be a bounded domain of R^N, with boundary ∂. Let ₀ and be connected components of∂. We assume thatis surrounded along₀andby thin insulating layers₀^εand^εof varying respective thicknessesh^ε₀(s)andh^ε(s),sbeing the generic point of ₀and. We denote by₀^εand^εthe parts of∂₀^εand∂^εwhich do not meet. We consider a class of quasilinear elliptic problems with different exponents (pin,q₀in^ε₀,q in^ε) and with the following boundary conditions:

•on^ε₀,u^ε=0,

•on^ε, the total flux is prescribed andu^εis constant, but unprescribed,

•on0and, the natural transmission conditions.

The restricted equations in₀^ε and^εhave nonconstant coefficients,µ^ε₀andµ^ε, in the form µ^ε₀(x)=µ^ε₀(σ0(x))(respectivelyµ^ε(x)=µ^ε(σ (x))),σ0andσ being the respective projections on0and. We predict the asymptotic behaviour of this problem ash^ε₀andµ^ε₀(respectivelyh^ε andµ^ε) tend to zero in a suitable sense, provided they are related in a convenient way.2002 Éditions scientifiques et médicales Elsevier SAS

1991 MSC: 35B40; 35J60

Keywords: Reinforcement; Boundary layer;-convergence; Torsional rigidity; Quasilinear elliptic problem

RÉSUMÉ. – Soitun domaine borné deR^N, de frontière∂. Soient₀etdes composantes connexes de∂. Nous supposons queest entouré le long de₀et par de fines couches isolantes^ε₀et^ε d’épaisseurs variables respectivesh^ε₀(s)eth^ε(s),s étant le point générique de0ou. Nous désignons par₀^εet^εles parties de∂^ε₀et∂^εqui ne bordent pas. Nous considérons une classe de problèmes elliptiques quasilinéaires, avec des exposants de Sobolev différents (pdans,q0dans^ε₀,qdans^ε) et avec les conditions au bord suivantes :

•sur^ε₀,u^ε=0,

E-mail address: [email protected] (J. Mossino).

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•sur^ε, le flux total est prescrit etu^εest constant, mais indéterminé,

•sur₀et, les conditions de transmission naturelles.

Les équations restreintes à^ε₀et^εont des coefficients non-constants,µ^ε₀etµ^ε, de la forme µ^ε₀(x)=µ^ε₀(σ0(x))(resp.µ^ε(x)=µ^ε(σ (x))),σ0etσ étant les projections respectives sur0

et. Nous prédisons le comportement asymptotique de ce problème, lorsqueh^ε₀etµ^ε₀(resp.h^ε etµ^ε) tendent vers zéro simultanément, tout en vérifiant une relation de corrélation convenable.

2002 Éditions scientifiques et médicales Elsevier SAS

1. Introduction

Let be a bounded domain of R^N and let 0 and be connected components of the boundary ∂ of . We assume that is surrounded along 0 and by thin reinforcements

₀^ε=s+tn(s), s∈0,0< t < h^ε₀(s)(ε), (1.1) ^ε=s+tn(s), s∈,0< t < h^ε(s)(ε), (1.2) where n(s) denotes the outer normal to at the point s of0 or . Then we denote by ^ε=∪₀^ε ∪^ε ∪0∪ the reinforced domain and we define ₀^ε and ^ε as the parts of ∂^ε₀ and ∂^ε which do not meet . We study the limit behaviour (when ε tends to zero) of some quasilinear problems with three (possibly different) exponents p, q0, q∈(1,∞)of the type:











−div|∇u^ε|^p⁻²∇u^ε=f^ε in,

−divµ^ε₀|∇u^ε|^q⁰⁻²∇u^ε=g₀^ε in₀^ε,

−divµ^ε|∇u^ε|^q⁻²∇u^ε=g^ε in^ε,

u^ε=0 on^ε₀=∂₀^ε\0, u^εis constant (undetermined) on^ε=(∂^ε)\,

^ε

µ^ε|∇u^ε|^q⁻²∂u^ε

∂n ds=I^ε (given),

∂u^ε

∂n =0 on∂\(^ε₀∪^ε), +transmission conditions on0and,

(E)

where we write “transmission conditions” for:

u^ε_|= u^ε_|^ε

0 on0,

u^ε_|ε on,

∇u^ε_|^p⁻²∂(u^ε_|)

∂n =











µ^ε∇u^ε_|ε 0

^q⁰⁻²∂(u^ε_|ε 0)

∂n on0, µ^ε∇u^ε_|ε^q⁻²∂(u^ε_|ε)

∂n on,

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and wherendenotes the outer normal to∂or∂^ε. In an electric (or heat propagation) setting, if ^ε is doubly connected, with ^ε as inner part of the boundary ∂^ε and ₀^ε as outer part, the boundary constraints mean that the potential (or temperature) has a given value on₀^ε and that^ε surrounds a perfect conductor, in which consequently the potential (or temperature) is constant, but unprescribed. The total flux, i.e. I^ε, is then given in terms of the integral of the source term over the perfectly conducting region.

The assumption thatµ^ε₀and µ^ε are small means that ₀^ε and ^ε modelize (unperfect) insulating layers. The torsional rigidity problem for a cable of multiconnected cross section also is of the same form.

To be more explicit, consider D^ε such that ∂D^ε =₀^ε and C^ε such that ∂C^ε=^ε (D^ε=^ε∪C^ε); the model problem consists in minimizing the energy

E^ε(v)=

D^ε

c^ε|∇v|²dx−

D^ε

f^εvdx,

over the subset of functions v inH₀¹(D^ε)such that ∇v=0 in C^ε, with a conductivity coefficient c^ε having value 1 in, µ^ε in^ε, µ^ε₀ in₀^ε. This problem is equivalent to minimizing

E^ε(v)=

^ε

c^ε|∇v|²dx−

^ε

f^εvdx−v_|^ε C^ε

f^εdx,

over the subset of functions vinH¹(^ε)such thatv=0 on₀^ε and v is constant (but unprescribed) on^ε. Now the Euler equation of the above minimization problem is











−u^ε=f^ε in,

−divµ^ε₀∇u^ε=f^ε in₀^ε,

−divµ^ε∇u^ε=f^ε in^ε,

u^ε=0 on^ε₀,

u^εis constant (undetermined) on^ε,

^ε

µ^ε∂u^ε

∂n ds=

C^ε

f^εdx, u^ε_|=u^ε_|^ε

0 and ∂u^ε_|

∂n =µ^ε₀∂(u^ε_|ε)

∂n on0, u^ε_|=u^ε_|ε and ∂u^ε_|

∂n =µ^ε∂(u^ε_|ε)

∂n on, which is the model equation for(E).

Similar problems (not involving ^ε) were considered by Boutkrida, Mossino and Moussa in [4] and [5]. Previous works on reinforcement problems were done by Sanchez-Palencia [13,14], Acerbi and Buttazzo [1], Brezis, Caffarelli and Friedman [6], Buttazzo and Kohn [7] and by Buttazzo, Dal Maso and Mosco [8]. To our knowledge, the condition on the given flux and the undetermined constant boundary value appeared only very recently in this context of boundary layers [11].

As in [4] and [5] the shape of the reinforcement may depend onε, that is we consider

“general” functions h^ε₀ and h^ε, and the insulating (or reinforcing) material may be

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inhomogeneous along 0 and , but µ^ε₀ and µ^ε are constant along each normal to 0 and : in other words µ^ε₀(x)=µ^ε₀(σ0(x)) (respectively µ^ε(x)=µ^ε(σ (x))), where σ0(x) (respectively σ (x)) denotes the projection of x ∈₀^ε (respectively ^ε) on 0

(respectively ). We definea^ε₀:0→R(respectively a^ε:→R) by a^ε₀=µ^ε₀(h^ε₀)¹⁻^q⁰ (respectively a^ε=µ^ε(h^ε)¹⁻^q). We assume essentially that a^ε₀ are positive functions in L^∞(0)(and the same fora^ε ∈L^∞()), with uniformly bounded inverses, and that0

(respectively) is divided into two parts:

•0(respectively) such that either0(respectively) is empty or_a¹ε

0 tends to zero inL^q⁰⁻¹(0)(respectively _a¹ε →0 inL^q⁻¹()),q₀ andqdenoting the conjugates ofq0

andq (e.g. ¹_q+_q¹ =1),

• 0 (respectively ) such that either 0 (respectively ) is empty or a₀^ε tends to a0 in weak −L^∞(0) (respectively a^ε→a in weak −L^∞()) and such that h^ε₀ (respectivelyh^ε) does not oscillate too much on0(respectively).

We prove that the limit problem has the form











−div|∇u|^p⁻²∇u=f in,

u=0 on0,

|∇u|^p⁻²∂u

∂n+a0|u|^q⁰⁻²u=0 on0, u=k(undetermined constant) on,

|∇u|^p⁻²∂u

∂n+a|u−k|^q⁻²(u−k)=0 on,

|∇u|^p⁻²∂u

∂nds=I,

∂u

∂n=0 on∂\(0∪),

(L)

where f and I are respective limits off^ε and I^ε. As Acerbi and Buttazzo did in [1], we use the -convergence theory introduced by De Giorgi [10] (see also Attouch [2]

and Dal Maso [9]) and we actually are able to predict the explicit limit of more general minimization problems than those associated with (1.1), (1.2) and(E):

•One can introduce additional constraints onu^ε_|;

•The reinforcements (1.1) and (1.2) can be generalized to ₀^ε=s+tnH0(s), s∈0,0< t < h^ε₀(s)(ε),

^ε=s+tnH(s), s∈,0< t < h^ε(s)(ε),

where nH0(s) (respectively nH(s)) is supported by the line of points having H0

(respectively H)-projection s on 0 (respectively ) and where H0and H are general norms (nH=n, the unit normal vector, ifH is the euclidian norm);

•In the energy functional, the term 1

p

|∇v|^pdx+ 1 q0

₀^ε

µ^ε₀◦σ0|∇v|^q⁰dx+ 1 q^ε

µ^ε◦σ|∇v|^qdx

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can be generalized to the anisotropic one F(v_|)+

^ε₀

a₀^ε◦σ0

h^ε₀◦σ0

G0

h^ε₀◦σ0H₀^o(∇v)dx+

^ε

a^ε◦σ

h^ε◦σGh^ε◦σ H^o(∇v)dx, where (e.g.)(µ^ε◦σ )(x)=µ^ε(σ (x)),σ (x)(respectivelyσ0(x)) is the projection ofxfor theH-norm (respectivelyH0-norm) and where (e.g.)H^o denotes the dual norm ofH, defined as

H^o(ξ^o)=sup

ξ=0

ξ^o. ξ H (ξ ).

We give the precise assumptions onF , G, G0, H, H0in the following.

We would like to emphasize the connection between the geometry of the reinforcements ^ε, ₀^ε (defined byH and H0) and the energy functional, whose integrands on ^εand₀^ε are defined in terms of the dual normsH^oandH₀^o.

Finally, let us mention that the result can be easily generalized to reinforcements along a finite number of components of∂.

2. Statement of the problem and of the result LetH:R^N→R⁺be a norm. In particular

∀t∈R, ∀ξ∈R^N, H (tξ )= |t|H (ξ ), (2.1)

∃δ1, δ2>0,∀ξ∈R^N, δ1|ξ|H (ξ )δ2|ξ|, (2.2)

∀ξ1, ξ2∈R^N, H (ξ1+ξ2)H (ξ1)+H (ξ2) (2.3) and from (2.1), (2.3),H is convex. The dual function ofH, defined as

Hô(ξô)=sup ξô.ξ

H (ξ ), ξ∈R^N, ξ=0

=supξ^o.ξ , ξ∈R^N,0< H (ξ )1 (2.4) is also a norm, with

∀ξ^o∈R^N, 1 δ2

ξôHô(ξô) 1 δ1

ξ^o; (2.5)

HandH^oare dual to each other and satisfy

∀ξ∈R^N,∀ξô∈R^N, ξô. ξH (ξ )Hôξô; (P .1) In this paper we consider two such norms H and H0; we assume that they are differentiable at any point but zero and thatHôandH₀ôare strictly convex.

Now consider a bounded regular domain in R^N and let and 0 be connected components of ∂. We introduce the distances (related toH andH0) fromx toand 0, defined by

t (x)=min

s∈ H (x−s), t0(x)=min

s∈₀H0(x−s).

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Let us remark that the above minima are achieved. Moreover if, for example, σ minimizesH (x−s)fors∈and ify is on the segment[σ, x], thenσ also minimizes H (y−s)fors∈. (Actually assume σ is on withH (y−σ) < H (y−σ ). Then H (x−σ)H (x−y)+H (y−σ) < H (x−y)+H (y−σ )=H (x−σ )and we get a contradiction.) It follows that the set of points having H-projection σ on is a line emanating fromσ.

Let⊃be the domain having boundary (∂\(∪0))∪∪₀, withand ₀ at given small distancet(relative toHandH0respectively) fromand0. We have \=∪₀ and we assume thatand₀ areC¹-diffeomorphic to×(0, t)and 0×(0, t)by the mappingsDandD0:

D:x∈→σ (x), t (x)∈×(0, t),

witht (x)=mins∈H (x−s)as above andσ (x)=arg mins∈H (x−s). (D0is defined similarly). We notice that (e.g.) D⁻¹(σ, t)=σ +tnH(σ ) with H (nH(σ ))=1 (since t=H (tnH(σ ))).

Let ε < t be a small parameter (hereafter ε will describe a sequence of positive numbers tending to zero) and leth^ε:→R⁺\ {0}be a positiveC¹-function such that

∀σ∈, h^ε(σ )ε; (2.6)

h^εdefines the reinforcement^εofalong:

^ε=x∈,0< t (x) < h^εσ (x)=σ+tnH(σ ), σ∈,0< t < h^ε(σ ). We set

^ε=σ+h^ε(σ )nH(σ ), σ∈.

We consider also a similar function h^ε₀ defined on 0 and we associate with it the reinforcement ₀^ε (of along 0) and the part ₀^ε of its boundary which does not meet. We denote by ^ε=∪^ε ∪₀^ε∪∪0the reinforced domain. Note that ⊂^ε⊂=∪∪₀ ∪∪0.

With the above geometrical data and given p, q, q0 in (1,∞), we consider the functional space

V^ε=v:^ε→R, v_|∈W^1,p(), v_|^ε ∈W^1,q(^ε), v_|₀^ε∈W^1,q⁰(^ε₀),

v_|=v_|^ε on, v_|=v_|₀^ε on0, v_|^ε=(undetermined) constant, v_|₀^ε=0. We are given dataI^ε,f^ε,g^ε,g₀^ε,F,K,G,G0,a^ε,a₀^εsuch that

• I^ε ∈ R, f^ε ∈ L^p(), g^ε ∈ L^q(^ε), g₀^ε ∈ L^q⁰(₀^ε), with p, q and q0 the conjugates ofp,q andq0,

• F:W^1,p()→Ris a continuous strictly convex functional such that

∃λ >0,∃λ>0,∀v∈W^1,p(), F(v)λ∇v^p_Lp()^N−λvW^1,p(), (2.7)

• K is a nonempty closed convex subset of W^1,p(), corresponding to conditions that concern neithernor0,

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• GandG0:R⁺→R⁺are increasing, strictly convex functions and

∃µ1, µ2>0,∀η∈R⁺, µ1η^qG(η)µ2η^q (2.8) (respectively µ1η^q⁰G0(η)µ2η^q⁰),

• a^ε ∈L^∞(), a^ε >0 a.e. and _a¹ε ∈L^∞(), a^ε₀∈ L^∞(0), a^ε₀>0 a.e. and _a¹ε

0 ∈

L^∞(0).

Now we are able to defineJ^ε:V^ε→Rby J^ε(v)=F(v_|)+

^ε

a^ε◦σ

h^ε◦σGh^ε◦σ H^o(∇v)dx+

₀^ε

a₀^ε◦σ0

h^ε₀◦σ0

G0

h^ε₀◦σ0H₀^o(∇v)dx

−

f^εvdx−

^ε

g^εvdx−

₀^ε

g₀^εvdx−I^εv_|^ε,

where (e.g.)(a^ε◦σ )(x)=a^ε(σ (x))and the first integral over^εis meaningful since by (2.5), (2.6), (2.8) the nonnegative integrand is bounded byµ2ε^q⁻¹δ₁⁻^q||a^ε||L^∞()|∇v|^q.

Our aim is to study the limit, as ε tends to zero, of the sequence of minimization problems

InfJ^ε(v), v∈V^ε, v_|∈K. (P^ε) PROPOSITION 1. –(P^ε)has a unique solutionu^ε.

Proof. – In this proof, as well as in the whole paper, C denotes various constants and we write C^ε for constants depending on ε. It follows from (2.5), (2.8) and the Lebesgue dominated convergence theorem that v→G(h^ε ◦σ H^o(∇v)) is continuous fromW^1,q(^ε)toL¹(^ε), so that the first integral over^εis a continuous function on W^1,q(^ε). The same is true for₀^ε.

Let us prove that

vV^ε= ∇vL^p()^N+ ∇vL^q(^ε)^N+ ∇vL^q0(^ε₀)^N

is a norm onV^εwhich is equivalent to the usual one induced byW^1,p()×W^1,q(^ε)× W^1,q⁰(₀^ε). Actually by Poincaré inequality

vW^1,q⁰(^ε₀)C^ε∇vL^q⁰(^ε₀)^N

and

vL^p()C∇vL^p()^N+ v_|₀L¹(0)

, vL^q(^ε)C^ε∇vL^q(^ε)^N+ v_|L¹())

. Moreover

v_|₀L¹(₀)Cv_|₀L^q⁰(₀)C^εvW^1,q⁰(^ε₀), v_|L¹()Cv_|L^p()CvW^1,p().

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It follows

vL^p()C∇vL^p()^N+C^εvW^1,q0(^ε₀)

C∇vL^p()^N+C^ε∇vL^q0(₀^ε)^N

, vL^q(^ε)C^ε∇vL^q(^ε)^N+ vW^1,p()

C^ε∇vL^q(^ε)^N+ ∇vL^p()^N+ ∇vL^q0(₀^ε)^N

. This gives the equivalence of the two norms under consideration.

ClearlyJ^ε is a continuous strictly convex functional onV^ε. Moreover J^ε is coercive since by (2.5), (2.7), (2.8) and the above equivalence, one has with α^ε such that 0< α^εmin(infσ∈a^ε(σ )h^ε(σ )^q⁻¹,infσ∈₀a^ε₀(σ )h^ε₀(σ )^q⁰⁻¹),

J^ε(v)λ∇v^p_Lp()^N −λvW^1,p()

+µ1α^εδ₂⁻^q∇v^q_Lq(^ε)^N+µ1α^εδ₂⁻^q⁰∇v^q_L⁰^q0(₀^ε)^N

− f^εL^p()vW^1,p()− g^εL^q(^ε)vW^1,q(^ε)

− g^ε_L^q₀₍ε

0)vW^1,q⁰(^ε₀)−C^εvW^1,q(^ε)

λ∇v^p_Lp()^N +µ1α^εδ⁻₂^q∇v^q_Lq(^ε)^N+δ⁻₂^q⁰∇v^q_L⁰^q0(^ε₀)^N

−C^εvW^1,p()+ vW^1,q(^ε)+ v_W^1,q0(₀^ε)

λ∇v^p_L^p₍₎^N−C^ε∇vL^p()

+µ1α^εδ₂⁻^q∇v^q_L^q₍^ε₎^N −C^ε∇vL^q(^ε)^N

+µ1α^εδ₂⁻^q⁰∇v^q_L⁰^q0(₀^ε)^N −C^ε∇vL^q⁰(^ε₀)^N

and since, whenvV^ε → +∞, at least one of∇vL^p()^N,∇vL^q(^ε)^Nor∇vL^q⁰(^ε₀)^N

tends to infinity. ✷

We study the limit behaviour of(P^ε)under the following additional assumptions on a^ε,a₀^ε,h^ε,h^ε₀,I^ε,f^ε,g^ε,g₀^ε, valid whenεtends to zero. First we assume that{_a¹ε}_εand {_a¹^ε

0}

ε are respectively bounded inL^∞()andL^∞(0):

∃α >0, a.e.s∈, a.e.s0∈0, ∀ε, a^ε(s)α, a₀^ε(s0)α. (2.9) Moreover we assume that, up to a set of(N−1)-dimensional measure zero, there exists a partition of (respectively 0) into two open regular subsets and (respectively 0 and 0) independent ofε (one of them being possibly empty, none of them being necessarily connected), such that

either= ∅ or 1 a^ε_|

→0 inL^q⁻¹()

respectively either₀= ∅ or 1 a₀^ε_|

0

→0 inL^q⁰⁻¹(₀)

(2.10) and

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either= ∅ or ∃a∈L^∞(), a^ε_|1 a in weak-L^∞() (2.11) respectively either0= ∅ or ∃a0∈L^∞(0), a₀^ε_|

01 a0 in weak-L^∞(0),

∇h^ε→0 inL^q() respectively∇h^ε₀→0 inL^q⁰(0). (2.12) We also assume that

for anyv∈K withv_|=l(constant), v_|₀=0, v_|∈L^q(), v_|

0∈L^q⁰(0), there exists a sequence of elementsvn∈C¹()∩Ksuch that

vn|=l, v_|₀=0, vn→vinW^1,p(), v_n_|→v_| inL^q(), v_n_|

0→v_|

0 inL^q⁰(0).

(2.13) Finally we assume that

I^ε→I,g^ε_Lq(^ε)

εandg₀^ε

L^q⁰(₀^ε)

ε

are bounded and

∃f ∈L^p(), f^ε1 f weakly inL^p(). (2.14) Let us comment (2.12). It means that the oscillations ofh^εandh^ε₀onand0are small.

Of course this holds true if (e.g.) h^ε(s)≡εh(s)withhof classC¹(∂), 0< h1, but also if (e.g.) N =2, is a closed curve, ε =n⁻^r, h^ε(s)=n⁻^rH(ny(s)) for s ∈, y(s)= arc length(s),Hperiodic of periodY whereY is the length of, 0<H1,H of classC¹(R⁺)andr >1. (Actually in this case|∇h^ε| →0 inL^∞().) On the contrary it does not hold true in the periodic case of [7] or [11], where r =1, unless if, in the notations of [7],R^ε/S^ε tends to zero.

Under the above assumptions we have THEOREM 1. – Letu^εbe the solution of

InfJ^ε(v), v∈V^ε, v_|∈K, (P^ε) where K is a nonempty closed convex subset ofW^1,p(), corresponding to conditions that concern neithernor0,

V^ε= {v:^ε→R, v_|∈W^1,p(), v_|^ε ∈W^1,q(^ε), v_|₀^ε∈W^1,q⁰(₀^ε), v_|=v_|^ε on, v_|=v_|₀^ε on0, v_|^ε is constant, v_|^ε₀=0, J^ε(v)=F(v_|)+

^ε

a^ε◦σ

h^ε◦σGh^ε◦σ H^o(∇v)dx+

₀^ε

a₀^ε◦σ0

h^ε₀◦σ0

G0

h^ε₀◦σ0H₀^o(∇v)dx

−

f^εvdx−

^ε

g^εvdx−

₀^ε

g₀^εvdx−I^εv_|^ε. Let us define(P)by

InfJ (v, l);(v, l)∈W, v∈K, (P)