A GENERALIZED PENALTY METHOD FOR DIFFERENTIAL VARIATIONAL-HEMIVARIATIONAL INEQUALITIES*

Liang LU (盧亮)

Guangxi Key Laboratory of Cross-border E-commerce Intelligent Information Processing，Guangxi University of Finance and Economics，Nanning 530003，China E-mail:gxluliang@163.com

Lijie LI (李麗潔)

Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing，Yulin Normal University，Yulin 537000，China E-mail:lilijiemaths@126.com

Mircea SOFONEA?

Laboratoire de Mathématiques et Physique，Université de Perpignan Via Domitia，52 Avenue Paul Alduy，66 860 Perpignan，F(xiàn)rance E-mail:sofonea@univ-perp.fr

Abstract We consider a differential variational-hemivariational inequality with constraints，in the framework of reflexive Banach spaces.The existence of a unique mild solution of the inequality，together with its stability，was proved in[1].Here，we complete these results with existence，uniqueness and convergence results for an associated penalty-type method.To this end，we construct a sequence of perturbed differential variational-hemivariational inequalities governed by perturbed sets of constraints and penalty coefficients.We prove the unique solvability of each perturbed inequality as well as the convergence of its solution to the solution of the original inequality.Then，we consider a mathematical model which describes the equilibrium of a viscoelastic rod in unilateral contact.The weak formulation of the model is in a form of a differential variational-hemivariational inequality in which the unknowns are the displacement field and the history of the deformation.We apply our abstract penalty method in the study of this inequality and provide the corresponding mechanical interpretations.

Key words differential variational-hemivariational inequality;generalized penalty method;Mosco convergence;viscoelastic rod;unilateral constraint

1 Introduction

Differential variational inequalities are systems which couple a differential or partial differential equation with a time-dependent variational inequality.Such systems arise in Physics，Mechanics and Engineering Sciences and，in particular，Contact Mechanics，where，they represent the weak formulation of mathematical models which describe the contact of a deformable body with an obstacle.The notion of differential variational inequalities (DVI，for short) was introduced in[2].A systematic study of DVIs in the framework of Euclidean spaces was carried out in[3]and developed in[4-8].Recently，the study of DVIs in in finite dimensional spaces has attracted increasing interest，as described by[9-16]and the references therein.

Differential hemivariational inequalities (DHVIs) and，more general，differential variationalhemivariational inequalities (DVHVIs)，represent an important extension of DVIs，since they couple a differential or partial differential equation with a hemivariational inequality and a variational-hemivariational inequality，respectively.References in the field are[1，17]，where existence and uniqueness results for various classes of DVHVIs have been obtained.Nevertheless，to the best of our knowledge，there are very few results concerning the convergence of the solutions of DVHVIs and，in particular，the convergence of the penalty methods associated to this kind of problem.

As is well known，penalty methods represent a mathematical tool for dealing a large variety of problems with constraints.In the classical penalty method，the constraints are removed by introducing an additional term governed by a penalty parameter.Then the unique solution of the original problem can be approached by the unique solution of the penalty problem as the penalty parameter converges to zero.Penalty methods can be used to prove the solvability of constrained problems，and are also very effective in the numerical solution of constrained problems.They have been intensively used in the study variational inequalities，mainly for numerical purposes.Details can be found in[18]and the references therein.Recent references in the field of penalty methods for various inequality problems are[19-22].

In this paper，we extend part of the results obtained in[21，22].We consider a DVHVI denoted by P，which involves a set of constraints K.The existence and uniqueness of the solution to the problem P was proved in[1].We proceed by introducing an an approximating sequence of DVHVIs，denoted by{Pn}，such that，for each n∈N，Problem Pnis a DVHVI governed by a set of constraints Kn?K and a penalty parameter ρn＞0.Under appropriate assumptions on the data，we prove the existence of a unique mild solution to Problem Pn，as well as its convergence to the unique solution of Problem P，as n→∞.To this end，we use arguments of compactness，pesudomonotonicity and Mosco convergence，among others.Note that，in contrast with[21，22]，in the method described above the constraints imposed to the solution are not completely removed (since Knis not supposed to be the whole space)，but are relaxed (since Kn?K).For this reason，we refer to this method as a generalized penalty method.The analysis of this method，together with its applications，constitutes the novelty of the current paper.

The rest of the manuscript is structured as follows:in Section 2，we introduce some preliminaries on nonsmooth analysis，then we state the problem and recall the unique solvability result proved in[1].In Section 3，we introduce the generalized penalty problems，prove their unique solvability，and state and prove our main convergence result，Theorem 3.1.Next，in Section 4，we introduce a mathematical model which describes the equilibrium of a viscoelastic rod in contact with a rigid obstacle covered by a rigid-elastic layer.We list the assumptions on the data and derive its variational formulation.Finally，in Section 5，we apply Theorem 3.1 in the study of this model and provide some mechanical interpretations.

2 Problem Statement and Preliminaries

Everywhere in this paper，unless stated otherwise，(E，‖·‖E) is a real Banach space，(V，‖·‖V) is a real reflexive Banach space，0Eand 0Vrepresent the zero elements of E and V，respectively，V*denotes the dual of V，and〈·，·〉represents the duality paring mapping.Moreover，E×V represents the product of spaces E and V endowed with the canonical topology，and L (V，E) denotes the space of bounded linear operators from V to E，endowed with the operator norm‖·‖L (E).All the limits，upper limits and lower limits are considered as n→∞，even if we do not mention it explicitly.

Let T＞0 and let I be the interval of time I=[0，T].We denote by C (I，E) and C (I，V) the space of continuous functions on I with values in E and V，respectively，endowed with the norm of the uniform convergence.We assume that A:D (A)?E→E is the infinitesimal generator of a C0-semigroup{T (t)}t≥0of linear continuous operators on E and we refer the reader to[23]for more details on this topic.Moreover，we assume that f:I×E→E，g:I×E→L (V，E) and x0∈E.We also consider a set K?V，the operators B:V→V*and h:I×E→V*，and the functions φ:V×V→R and j:V→R.We assume that φ is convex with respect to the second argument，that the function j is locally Lipschitz，and we denote by j0its generalized (Clarke) directional derivative.With these notations，we now consider the following problem:

ProblemPFind a pair of functions (x，u) with x:I→E and u:I→V such that x (0)=x0and，for each t∈I，the following hold:

We note that Problem P is a system which couples a differential equation with a variationalhemivariational inequality，associated to an initial condition.Therefore，following the terminology introduced above，we refer to Problem P as a differential hemivariational inequality.The solution of problem P is understood in the following sense:

Definition 2.1A pair of functions (x，u) is said to be a mild solution of Problem P if，with x∈C (I，E)，u∈C (I，V)，(2.2) holds for all t∈I and，moreover，

For the solvability of Problem P we consider the following assumptions on the data:

We now recall the following existence and uniqueness result:

Theorem 2.2([1，Theorem 3.6]) Assume that (2.4)-(2.13) hold.Then，there exists a unique mild solution (x，u)∈C (I，E)×C (I，V) to Problem P.

We end this section by a collection of definitions and preliminary results which are useful throughout the rest of the paper.More details on the material that follows can be found in[24，25，27，28].

Definition 2.3Let{Kn}be a sequence of nonempty subsets of V anda nonempty subset of V.The sequence{Kn}converges toin the sense of Mosco if the following conditions hold:

(b) For each sequence{vn}such that vn∈Knfor each n∈N，if vn→v weakly in V，we have v∈.

Recall that the notion of the Mosco convergence was introduced in[26];in this paper we shall denote it by KnK in V.

Definition 2.4An operator G:V→V*is said to be

(a) bounded，if G maps bounded sets of V into bounded sets of V*;

(b) monotone，if〈Gu-Gv，u-v〉≥0 for all u，v∈V;

(c) pseudomonotone，if G is bounded and for every sequence{un}?V with un→u weakly in V，such that limsup〈Gun，un-u〉≤0，we have〈Gu，u-v〉≤liminf〈Gun，un-v〉for all v∈V;

(d) demicontinuous，if un→u in V implies Gun→Gu weakly in V*;

(e) hemicontinuous，if the function λ〈G (u+λv)，w〉is continuous on[0，1]for all u，v，w∈V.

Definition 2.5An operator P:V→V*is said to be a penalty operator of K if P is bounded，demicontinuous and monotone and K={u∈V:Pu=0V*}.

Definition 2.6A function φ:V→R is said to be lower semicontinuous (l.s.c.) if liminf φ(un)≥φ(u) for any sequence{un}?V with un→u in V.

Definition 2.7The Clarke generalized directional derivative of a locally Lipschitz function j:V→R at x in the direction v，denoted by j0(x;v)，is defined by

The generalized Clarke subdifferential of j at x is a subset of V*given by

?j (x)={x*∈V*|j0(x;v)≥〈x*，v〉for all v∈V}.

Lemma 2.8([27，Lemma 3，p.113]) If A:V→V*is a bounded，hemicontinuous and monotone operator，then it is pseudomonotone.Moreover，if A，B:V→V*are pseudomonotone operators，then A+B:V→V*is pseudomonotone，too.

Lemma 2.9([28，Proposition 2.1.2]) Let j:V→R be a locally Lipschitz function.Then the following statements hold:

(1) j0(x;v)=max{〈ξ，v〉|ξ∈?j (x)}for all x，v∈V.

(2) For each x∈V，the function U?vj0(x;v)∈R is positively homogeneous and subadditive，i.e.，j0(x;λv)=λj0(x;v) for all λ≥0，v∈U，and j0(x;v1+v2)≤j0(x;v1)+j0(x;v2) for all v1，v2∈V，respectively.

3 Problem Statement and Main Result

In this section we introduce the generalized penalty method in the study of Problem P.It consists of defining a sequence of penalty problems{Pn}in order to prove their unique solvability and also the convergence of the sequence of their solutions to the unique solution of P，obtained in Theorem 2.2.In this section we keep assumptions (2.4)-(2.13)，and consider an operator P:V→V*.Moreover，for each n∈N，we introduce a DVHVI in which Kn?V and ρn＞0.

ProblemPnFind a pair of functions (xn，un) with xn:I→E and un:I→V such that xn(0)=x0and，for each t∈I，it holds that:

Following Definition 2.1 we recall that the pair of functions (xn，un) is said to be a mild solution to Problem Pnif xn∈C (I，E)，un∈C (I，V)，and (3.2) holds for all t∈I and，moreover，

In the study of Problem Pnwe consider the following hypotheses on the data:

Our main result in this section is

Theorem 3.1Assume that (2.4)-(2.13)，(3.4)-(3.10) hold.Then，

(1) for each n∈N，there exists a unique mild solution (xn，un)∈C (I，E)×C (I，V) to Problem Pn;

(2) the mild solution (xn，un) of Problem Pnconverges to the mild solution (x，u) of Problem P，i.e.，

(xn(t)，un(t))→(x (t)，u (t)) in E×Vasn→∞，for all t∈I.

Proof(1) For each n∈N，define an operator Bn:V→V*by equality

From hypotheses (3.5)，(3.6) and Lemma 2.8，we find that Bnis pseudomonotone and strongly monotone with the constant mB.Therefore，using Theorem 2.2 with Knand Bninstead of K and B，respectively，we deduce that there exists a unique mild solution (xn，un)∈C (I，E)×C (I，V) to Problem Pn.

(2) For each n∈N，we consider the auxiliary problem of finding a function∈C (I，V) such that

for all t∈I.Using standard arguments ([1，Lemma 3.3]，for instance)，it is easy see that problem (3.11) has a unique solution∈C (I，V).

The rest of the proof is divided into five steps，based on arguments similar to those used in[22，27]，in the case when Kn=K=V，and P is a penalty operator of K.Because of the similarity to[22，27]，we skip some details in the proof and concentrate only on the differences which arise from the fact that here we are working in the case where KnK and P satisfies (3.7)(c)，(d).

Step (i)We prove that for each t∈I，there existsand a subsequence of，again denoted by，such thatweakly in V as n→∞.

Let n∈N，t∈I and u0∈K.We take v=u0in (3.11) to deduce that

and，using assumptions (3.7)(c) and (3.9)，we find that

Next，we use assumption (2.10) and Lemma 2.9(1) to see that

Moreover，using conditions (2.8)(2)，(3.9)，inequalities (3.12)，(3.13) and the strong monotonicity of operator B，it follows that

This inequality，combined with condition (2.11)，implies that

where M0:=?+cφ(u (t))+κ0+κ1‖u0‖V+‖Bu0‖V*.Note that M0depends on t but does not depend on n.Thus，the sequenceis bounded in V.Therefore，the reflexivity of V implies that there exists an element∈V such that，passing to a subsequence if necessary，we have thatweakly in V as n→∞.Since∈Kn，assumption (3.7)(b) combined with Definition 2.3(b) shows that.

Step (ii)We prove that∈K for all t∈I.

Thus，since{vn}，are bounded sequences and B is a bounded operator，we deduce that there exists a constant＞0 which does not depend on n such that

and，therefore，

and，therefore，(3.15) yields

On the other hand，assumption (3.6) combined with Lemma 2.8(b) guarantees that P is a pseudomonotone operator.Thus，inequality (3.17)，together with the pseudomonotonicity of P，implies that

for all v∈V and，therefore，(3.16) yields

We now combine inequality (3.19) with assumption (3.7)(c) to find that

then we use assumption (3.7)(d) to obtain the regularity

Step (iii)We now prove that?u (t) in V，for all t∈I.

Let n∈N，t∈I and v∈K.We use inequality (3.11) and inclusion K?Knto see that

and，therefore，(3.7)(c) implies that

We now pass to the upper limit in the previous inequality，then we use the lower semicontinuity of φ with respect to the second argument and the hypothesis (3.10) to find that

This inequality，together with the pseudomonotonicity of operator B，implies that

We now combine inequalities (3.24) and (3.22) to see that

and，therefore，

Then，we use assumptions (2.7) and (2.10)(2) to find that

This inequality combined with the smallness assumption (2.11) implies that=u (t).

In the meantime，a careful examination reveals that any weakly convergent subsequence of the sequenceconverges weakly to u (t) as n→∞.Moreover，since the sequenceis bounded，it follows that the whole sequenceconverges weakly to u (t)(see[27，Theorem 71，p.12]).

Step (iv)We now prove that→u (t) in V，for all t∈I.

which concludes the proof of this step.

Step (v)Finally，we prove that (xn(t)，un(t))→(x (t)，u (t)) in E×V，for all t∈I.

Let t∈I and n∈N.We write (2.2) with v=un(t).Then we take (3.2) with v=and add the resulting inequalities to see that

Therefore，assumptions (2.8)-(2.10) and the monotonicity of the operator P yield

then use inequality (3.26) and assumption (2.11) to obtain that

which implies that there exist two constants，D0＞0 and D1＞0 such that

In the meantime，using (2.3)，(3.3)，assumptions (2.5) and (2.6)，and inequality (3.27)，after some elementary calculus，we obtain that there exist two constants，＞0 and＞0，such that

see[1，Lemma 3.4]for details.Therefore，using Gronwall’s inequality，it follows that there exists a constant D＞0 such that

This inequality，the convergence→u (s) in V，valid for each s∈[0，T]，and the Lebesguedominated convergence theorem (see[24，Theorem 1.65]) imply that

We conclude here that xn(t)→x (t) in E.On the other hand，using this convergence，the convergence→u in V proven in Step (iv)，and inequality (3.27)，we deduce that un(t)→u (t) in V，which completes the proof of the theorem. □

4 A Contact Model for a Viscoelastic Rod

The abstract results in Sections 2 and 3 are useful in the study of various initial and boundary value problems.A wide class of examples can be considered，some of them arising from Contact Mechanics.Below，we restrict ourselves to providing a one-dimensional example which describes the contact of a viscoelastic rod with an obstacle made of a rigid material covered with an rigid-elastic material.Its novelty arises in the analysis of the model，based on a non standard variational formulation.

The physical setting is depicted in Figure 1 and is described below:a viscoelastic rod occupies，in the reference con figuration，the interval[0，L]on the Oz axis.The rod is fixed in z=0，is acted upon by body time-dependent forces of density fbalong Oz，and its extremity z=L is in contact with an obstacle made of a rigid body covered by a rigid-elastic layer of thickness k＞0.The time interval of interest is I=[0，T]with T＞0.We denote by a prime the derivative with respect to the time variable t∈I and by the subscript z the derivative with respect to the spatial variable z∈[0，L]，i.e.，x′=and uz=.With these preliminaries，the contact problem we consider is formulated as follows:

Figure 1 Physical setting

ProblemQ Find a displacement fieldu:[0，T]×[0，L]→R and a stress field σ:[0，T]×[0，L]→R such that

We now present a brief description of the equations and boundary conditions in this problem.

First，equation (4.1) represents the viscoelastic constitutive law in which β＞0 is the Young modulus of the material andare constitutive functions which will be described below.Here，for simplicity，we restrict ourselves to the homogenuous case.Using the notation ε=uz，equation (4.1) reads as

This equality shows that at each moment t the stress field σ(t) can be split in two parts:an elastic part σE(t)=βε(t)(which depends on the current value of the deformation) and an anelastic part σAN(t)=(which depends on the history of the deformation).Concrete examples of elastic constitutive laws of the form (4.5) can be found in[27]which provides the analysis of various contact models for materials of this form under various assumptions on the constitutive functions.

Equation (4.2) is the equilibrium equation in which fbdenotes the density of body forces acting on the rod.Condition (4.3) represents the displacement condition;we use it here since the rod is assumed to be fixed at z=0.Conditions (4.4) represents the boundary conditions which model the contact of the point z=L of the rod with a rigid body covered by a layer made of rigid-elastic material，say，a crust.Here k denotes the thickness of this layer，F(xiàn) represents its yield limit，and peis a real-valued function which describes its elastic properties.More details regarding such boundary conditions，including their extension in the three-dimensional case，can be found in[27].

In the study of Problem Q we assume the following conditions on the data.

Note that condition (4.8)(d) is imposed for mechanical reasons.This shows that when there is contact between the rod and the obstacle，the reaction of the obstacle is towards the rod，and when there is separation，it vanishes.

We now denote by E and V the spaces

Both E and V are real Hilbert spaces endowed with the inner products

and the associated norms‖·‖Eand‖·‖V，respectively.Moreover，from the Sobolev trace theorem it follows that

We denote by V*and〈·，·〉the dual of V and the duality pairing between V*and V，respectively.In addition，we use the notation r+for the positive part of r.

Next，de fine the set K，the operators A:E→E，B:V→V*and the functions φ:V×V→R，q:R→R，j:V→R，f:I×E→E，g:I×E→L (E，V)，h:I×E→V*by equalities

Note that the assumptions (4.6)-(4.9) above guarantee that these definitions make sense.For instance，we specify that the definition (4.19) has to be understood in the following sense:for all t∈I and x∈E，g (t，x) represents the operator which associates to each function u∈V，the function g (t，x) u:(0，L)→R defined by

It is easy to see that this function belongs to E and that the operator ug (t，x) u:V→E is linear and continuous，i.e.，it belongs to L (V，E).Also，note that the operator B and the function h are defined using Riesz’s representation theorem.The function q could be nonconvex.Nevertheless，it is a regular function in the sense of Clarke.Moreover，it satisfies the equality

where q0(s;θ) denotes the generalized directional derivative of q at the point s in the direction θ.On the other hand，using a standard argument (Theorem 3.47 in[24]or Lemma 8(vi) in[27]，for instance) we have that

where j0(u;v) denotes the generalized directional derivative of j at the point u in the direction v.

Assume now that (u，σ) is a regular solution to Problem Q，and consider the history of the deformation field x:I×[0，L]→R defined by

We have that

and，using (4.1)，we deduce that

Moreover，performing interation by parts and using (4.2)-(4.4)，it follows that

Therefore，using equalities (4.26)，(4.21) and (4.22)，we find that

Finally，using (4.24)-(4.27) and notations (4.12)-(4.20)，we obtain the following variational formulation of the contact problem Q:

ProblemQVFind a displacement field u:I→V and a history of deformation field x:I→E such that x (0)=0Eand，for all t∈I，it holds that

Note that Problem QVrepresents a differential variational-hemivariational inequality.In the next section we shall apply the abstract results provided by Theorems 2.2 and 3.1 in the study of this problem.

5 Existence，Uniqueness and Convergence Results

To present the generalized penalty method in the study of Problem QV，we consider a function p，two sequences{kn}，{ρn}，and a positive numberwhich satisfy the following properties:

Then，we define the operator P:V→V*and the sets Knby equalities

Moreover，for each n∈N we introduce the following perturbation of Problem QV:

ProblemFind a displacement field un:I→V and a history of deformation field xn:I→E such that xn(0)=0Eand，for each t∈I，it holds that:

Note that the concept of a mild solution，for both Problem Pnand Problem P，is understood in the sense of Definition 2.1.

Our main result in this section is

Theorem 5.1Assume (4.6)-(4.9)，(5.1)-(5.3) and，moreover，assume that β＞αeL.Then，the following statements hold:

(1) There exists a unique mild solution (x，u)∈C (I，E)×C (I，V) to Problem QV.

(2) For each n∈N，there exists a unique mild solution (xn，un)∈C (I，E)×C (I，V) to Problem.

(3) The mild solution (xn，un) of Problemconverges to the mild solution (x，u) of Problem QV，i.e.，

ProofThe proof of this is based on Theorems 2.2 and 3.1.For this reason，we check，in what follows，the validity of the conditions of these theorems.

First，note that the operator (4.12) is the generator of the semigroup{T (t)}t≥0defined by T (t) x=etx for each t≥0 and x∈E.Therefore，condition (2.4) is satisfied.Moreover，it is easy to see that the functions f and g，defined by (4.18) and (4.19)，satisfy condition (2.5)and (2.6)，respectively.In addition，the operator (4.14) satisfies condition (2.7) with mB=β.Finally，assumptions (4.6) and (4.9) guarantee that the function h defined by (4.20) satisfies condition (2.8).

Next，it is easy to see that the function φ defined by (4.15) satisfies condition (2.9) with αφ=0 and，using the standard arguments on subdifferential calculus，we see that the function j defined by (4.17) satisfies the condition (2.10)(1).Moreover，using (4.22)，(4.21) and (4.11)，we have that

which shows that condition (2.10)(2) holds with αj=αeL.We also note that inequality β＞αeL implies that the smallness condition (2.11) holds，too.Finally，we note that (2.12) and (2.13) are obviously satisfied.

On the other hand，it is easy to see that condition (3.4) is satisfied and that conditions (3.5)，(3.8) are recovered by assumption (5.3).Moreover，using the properties (5.1) of the function p and inequality (4.11)，it follows that the operator P defined by (5.4) is monotone and Lipschitz continuous and，therefore，that it satisfies condition (3.6).

We now turn to condition (3.7) and，to this end，we consider the set

Then，we note that the properties of the function p and inequality≥k imply that each term in (5.10) is negative，i.e.，

We conclude from here that〈Pu，v-u〉≤0 and，therefore，that condition (3.7)(c) holds.Assume now that〈Pu，v-u〉=0.Then (5.10) implies that

p (u (L)-k)(k-u (L))=-p (u (L)-k)(v (L)-k)，

so (5.11) shows that p (u (L)-k)(k-u (L)) is both positive and negative.It follows from here that p (u (L)-k)(k-u (L))=0.This equality，combined with assumption (5.1)(c)，shows that u (L)≤k.We conclude that u∈K and，therefore，condition (3.7)(d) holds.

Finally，using the compactness of the trace map it follows that conditions (3.9) and (3.10) hold，too.The proof is based on standard arguments，and therefore we skip them.

It follows from above that the assumptions of Theorems 2.2 and 3.1 are satisfied.Hence，using the existence，uniqueness and convergence results provided by these theorems，we are in a position to conclude the proof of Theorem 5.1. □

We end this section with some remarks and mechanical interpretations.First，consider the boundary condition

for all t∈I.Noting that this condition is obtained from (4.4)，we replace the bound u (t，L)≤k with the bound u (t，L)≤knand the function rpe(r) with the function r.Here，ρnis a deformability coefficient andrepresents the stiffness coefficient.Such conditions model the contact of the rod with a rigid obstacle covered by a rigid-elastic layer of thickness kn≥k.Nevertheless，when the penetration of this layer is larger then k，its reaction increases and goes to infinity when ρn→0.We conclude that this part of the layer rigidifies.Note that Problemrepresents the variational formulation of a perturbation version of Problem Q in which the contact condition (4.4) was replaced with the contact condition (5.12).Therefore，in addition to the mathematical interest of the convergence result in Theorem 5.1，it is important from mechanical point of view，since it shows that the solution of the contact problem with a rigid obstacle ROcovered by a layer Lkof thickness k can be approached by the solution of the contact problem obtained by considering an additional deformable layerof thickness kn-k，situated between Lkand RO，provided that the stiffness coefficient of the layer is large enough and the thickness kn-k is close to a given value-k.