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Localization and multiplicity in the homogenization of nonlinear problems

  • Renata Bunoiu and Radu Precup EMAIL logo
Published/Copyright: May 22, 2019
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Advances in Nonlinear Analysis
From the journal Advances in Nonlinear Analysis Volume 9 Issue 1

Abstract

We propose a method for the localization of solutions for a class of nonlinear problems arising in the homogenization theory. The method combines concepts and results from the linear theory of PDEs, linear periodic homogenization theory, and nonlinear functional analysis. Particularly, we use the Moser-Harnack inequality, arguments of fixed point theory and Ekeland's variational principle. A significant gain in the homogenization theory of nonlinear problems is that our method makes possible the emergence of finitely or infinitely many solutions.

Keywords: Nonlinear elliptic problem; homogenization; localization; positive solution; multiple solutions
MSC 2010: 35B27; 35J25

1 Introduction

The aim of homogenization theory is that, starting from an initial problem depending on a small positive parameter ε related to the heterogeneous character of the problem, to find an approximation of its solution which solves a problem independent of ε, called the homogenized problem. From a mathematical point of view this is obtained by passing to the limit with ε ⟶ 0 in the initial problem.

One of the main features in classical homogenization problems is the uniqueness of the solution of the problem under study. This is crucial in order to justify the convergence of the whole sequence of solutions of the microscopic problem to the solution of the homogenized problem. In the nonlinear case, there is a wide class of problems for which the uniqueness of the solution fails. When several solutions of a problem exist, it is important to be able to carry on the analysis on one particular solution and this needs its localization. A lot of results for nonlinear problems in homogenization theory have been obtained in the last years (see for instance [1], [3], [7], [8] [10], [12], [13], [18]). In most cases, in order to pass to the limit, it was necessary to obtain upper estimates of the solutions. However, up to our knowledge, none of these works had to deal with estimates from below of the solutions.

The aim of this paper is to develop a method of localization in bounded sets for the solutions of nonlinear homogenization problems, which allows the emergence of multiple solutions for a given problem. The localization which will be given on the initial problem, for each ε small enough, will be preserved after passage to the limit with ε ⟶ 0. We will illustrate this method on a simple model, namely the homogenization of an elliptic nonlinear Dirichlet problem stated in a fixed domain, with periodic fast oscillating coefficients depending on the small parameter ε, which expresses the heterogeneous properties of the medium. We emphasize that our method applies to more involved problems in nonlinear homogenization theory.

The results are then extended to systems, for which we are able to localize each component of the solution, independently. Moreover, the solution appears as a Nash equilibrium with respect to the energy functionals associated to the equations of the system. In this case again, for a given nonlinear system, by the localization technique, we are able to prove the existence of finitely or infinitely many local Nash equilibrium solutions.

The starting point of our analysis is a linear microscopic pivot problem with a right-hand side, g ϵ L∞(Ω) which allows to estimate the solutions uε both from below and from above, independently of ε, via two constants mg and M g respectively. At this point, we underline the role of the Moser-Harnack inequality in establishing the lower bound mg (see Lemma 3.1), which, up to our knowledge, is new in the frame of homogenization theory.

The question we answer in this paper is how the source term g can be modified by a nonlinear source factor f(uε) with a feedback action, such that the solution uε of the new problem remains L∞-bounded independently of ε, between two given bounds r, R. The same question is also addressed for nonlinear systems.

Our study is motivated by real-world applications in physics, engineering and biology, from where at least two physical requirements emerge. In these cases the solution corresponds to a density rate.

First requirement: Find a suitable nonlinear state-dependent source factor f(uε), in order to guarantee that the density rate uε stays bounded between two a priori given bounds r, R, 0 < r < R. Such a requirement is natural in problems arising from medicine, where for instance the concentration rate of a drug in a tissue with a nonhomogeneous structure must remain between some prescribed limits. The same question can be addressed in problems of thermodynamics, when the heat diffusion in strongly heterogeneous media has to be limited.

Second requirement: The nonlinear state-dependent source factor f being given, find the bounds of the corresponding density rate uε.

Our approach makes use of methods and results from the linear theory of PDEs (weak and strong maximum principle, Poincaré and Moser-Harnack inequalities, compact embedding theorems), linear periodic homogenization theory, and concepts and techniques of nonlinear functional analysis (Banach's contraction principle and its vector version owed to Perov, Ekeland variational principle, Nash equilibrium).

The paper is organized as follows: In Section 2 we give the statement of the problem and we recall some basic results of periodic linear homogenization theory. In Section 3 we state and prove the main existence and localization result giving a variational characterization to the solution and, as a consequence, we obtain a multiplicity result for nonlinearities with a repeated suitable behavior. These solutions are local minima of the associated energy functionals. Section 4 is devoted to the study of a nonlinear system, for which we are able to localize componentwise the solution.

2 Statement of the problem and preliminaries results

We are concerned with the following nonlinear problem with homogeneous Dirichlet boundary conditions, and strongly periodic oscillating coefficients:

(2.1) − d i v A ε ( x ) ∇ u ε = g ( x ) f u ε in  Ω u ε = 0 on  ∂ Ω .

Here Ω⊂ℝn is an open domain with sufficiently smooth boundary ∂Ω. We make the following assumptions on the data:

Let Y = (0, 1) n. For α , β ϵ R , with 0 α ≤ β , let M α , β , Y be the set of all the matrices A ϵ (L∞(Y))n×n such that for any ξϵℝn,

(2.2) α| ξ |2≤(A(y)ξ,ξ)≤β| ξ |2,                         

almost everywhere in Y. For a Y-periodic symmetric matrix A ∈ M(α, β, Y), we set

Aε(χ)=A({ χε })   a.e. in Ω.

The function g belongs to the space L∞(Ω) and the function f:ℝ→ℝ is continuous.

Throughout this paper Ω′ is a fixed subdomain of Ω,Ω′   ⊂⊂  Ω. We first look for sufficient conditions on f and g such that, for given numbers 0 < r < R, problem (2.1) has a solution uε such that the following boundedness conditions

(2.3) 0≤uε(χ)≤R   for  a.a. x ϵ Ω,                                      
(2.4) r≤uε(χ)   for a.a.  x ϵ Ω′,                           

are satisfied for each ε>0 small enough.

Then, we prove that, after passage to the limit in (2.1) with ε→0, the limit u of the sequence uε solves uniquely the problem

(2.5) − d i v A 0 ∇ u = g ( x ) f u in  Ω u = 0 on  ∂ Ω ,

which is therefore the homogenized problem of (2.1). The solution u of the homogenized problem, satisfies the same boundedness conditions. Here the entries of the constant matrix A0 are defined in (2.8)-(2.9) below. This localization is useful for the numerical computation of the solutions, especially when an initial approximation is required.

In addition, this localization result, applied to distinct pairs of numbers (rk, R k) yields finitely or infinitely many solutions for problem (2.1), provided that either

0<rk<Rk<rk+1<Rk+1

or

0<rk+1<Rk+1<rk<Rk.

Analogously, in Section 4, we discuss the localization of solutions for nonlinear systems. In this case the localization can be obtained componentwise, which is a significant gain for the treatment of systems in comparison with the classical approach.

2.1 The linear homogenization problem

We start by recalling a classical result from the homogenization of linear elliptic problems. With the domain Ω and the coefficients Aε defined as before, we consider the linear problem (see [2, Chapter 1]):

(2.6) − d i v A ε ( x ) ∇ u ε = h ( x ) in  Ω u ε = 0 on  ∂ Ω .

If h ϵ L2 (Ω) is given, then one can prove (see for instance [5, Chapter 6] and references therein) that problem (2.6) admits a unique solution uεϵ H01(Ω) which converges to some u weakly in H01(Ω), and strongly in L2 (Ω). The limit u is the unique solution of the homogenized problem

(2.7) { −div(A0∇u)=h(x)  in Ωu=0                                    on ∂Ω                                                   

with the constant homogenized coefficients given by

(2.8) A i j 0 = ∫ Y a i j − ∑ k = 1 n a i k ∂ χ j ∂ y k d y ,

where, for i, j = 1,..., n, aij are the entries of the matrix A. The j-th components of the vectorial function , χjϵ Hper1(Y)(j=1,…,n) are the unique weak solutions of the following cell problems:

(2.9) − d i v y ( A ( y ) ( ∇ y χ j − e j ) ) = 0 in  Y , M Y ( χ j ) = 0.

Here Hper1(Y) denotes the space of H1 functions which are Y-periodic and My(v) represents the mean value on the set Y of a function v.

When we make additional assumptions on the right-hand side h, one has more information on the corresponding solution of problem (2.6), as illustrated by the two remarks below.

Remark 2.1

If h(x)≥0  for a.e. x ϵ Ω, then the weak maximum principle (see [9, Theorem 8.11) guarantees that for every ε > 0, one has uε(x) ≥ 0 a.e. in Ω.

Remark 2.2

h ϵ L∞(Ω) then, according to [9, Theorem 8.15], the solution uε belongs to L ∞(Ω) and there is a constant Γh, independent of ε, such that

‖ uε ‖L∞(Ω)≤Γh.

In view of Remark 2.2, we define for each ε ∈ (0, 1), the solution operator

Sε:L∞(Ω)→V:=H01(Ω)∩L∞(Ω),                     h↦Sε(h):=uhε,

where uhε is the corresponding solution of problem (2.6).

The operator Sε is linear and due to Remark 2.1, it preserves the positivity and the order, more exactly

if  0≤h1 ≤h2,   then   0≤Sε(h1)≤Sε(h2).

Searching now a weak solution uε ∈ V of problem (2.1) is thus equivalent to solving the nonlinear equation

(2.10) uε=Sε(gf(uε)).                                           

For each ε, denoting

(2.11) Nε(v):=Sε(gf(v)),  v ϵ V,                     

any solution of problem (2.10) is a fixed point of the operator Nε. The existence and the uniqueness of such a fixed point will be obtained by using Banach's contraction principle. Alternatively, the existence and the uniqueness of the solution of (2.1) can be obtained using the variational approach in connection with the energy functional associated to (2.1), namely

Jε : V→ℝ,Jε(v)=12∫ΩAε(χ)∇v⋅∇v−∫Ωg(χ)∫0v(χ)f(τ)dτ.

Similarly, we associate to problem (2.5) the energy functional

J : V→ℝ,J(v)=12∫ΩA0∇v⋅∇v−∫Ωg(χ)∫0v(χ)f(τ) dτ.

3 Main results for equations

3.1 A lower bound lemma

Our aim in this paper being the localization of solutions of problem (2.1), as shown in (2.3)–(2.4), we need bounds from above and from below. For the upper bounds, we use classical results from the literature, as mentioned in Remark 2.2. Getting lower bounds is a more difficult task and, up to our knowledge, it is new in the frame of the homogenization theory. To obtain such lower bound estimates, we make use of an auxiliary result, Lemma 3.1, whose proof is based on Moser-Harnack's inequality. The key auxiliary result is the following.

Lemma 3.1

For each h ∈ L∞ (Ω) with h ≥ 0 in Ω and h > 0 on a subset of nonzero measure of Ω, there exist εh, > 0 and yh > 0 such that uε (x) ≥ yh for a.a. x in the fixed subdomain Ω′ ⊂⊂Ω and every ε < εh.

Proof. The proof will be done by contradiction. Assume the contrary. Then for every integer ≥ ≥ 1, there exists εk<1/k with

(3.1) infΩ′uεk<1k.       

Let p ϵ ℝ be fixed such that 1≤p<n/(n−2) if n≥3  and p=2 if n<3. Using Moser-Harnack's inequality (see [11, Theorem 12.1.2]) and covering arguments as in [15, Theorem 1.3], we have

(3.2) infΩ′uεk≥C‖ uεk ‖LP(Ω),                               

where the constant C > 0 only depends on p, Ω, Ω′, and the ellipticity constants α and β in (2.2). From (3.1) and (3.2) we deduce that

(3.3) ‖ uεk ‖LP(Ω′)→0 as k→∞.                        

Since p<n/(n−2)<2n/(n−2)=2∗, the injection of H01(Ω) in Lp(Ω) is compact, and consequently up to a subsequence we nave uεk→u in Lp(Ω), Where u is the solution of the homogenized problem (2.7). Then from (3.3), ‖ u ‖Lp(Ω′)=0, that is u = 0 in Ω which is impossible in view of the strong maximum principle applied to the homogenized problem (2.7) in which h ≥ 0 and h is not the null function.

Coming back to our initial question, namely the localization of the solution for problem (2.1), let us define, for the given numbers 0 < r < R, the set

DrR={ v∈V:0≤v(x)≤R a.e. in Ω, r≤v(x)a.e. in Ω′ }.

Note that the set DrR is a closed subset of H01(Ω). Indeed, if vk∈DrR  and vk→v in H01(Ω), then vk→v in L2(Ω) and there is a subsequence (vkj) of the sequence (vk) such that vki(x)→v(x) for a.a. x ∈ Ω (see [4, Theorem 4.9]). Then, from the definition of DrR, since

0≤vkj(x)R a.e in Ω, and r≤vkj(x)a.e. in Ω′,

passing to the limit we obtain

0≤v(x)≤R a.e in Ω, and r≤v(x) a.e. in Ω′,

that is v ∈ DrR.

3.2 Invariance of DrR through the operator Nε

The first step for applying Banach's contraction principle to the nonlinear operator (2.11) is to guarantee the invariance condition. To this aim we state our first hypothesis.

(h1) g(x)≥0 a.e in Ω, g(x)=0 for x∈Ω/Ω′, and g > 0 on a subset of nonzero measure; f ( τ ) ≥0 for every τ≥0.

Let v ∈ DrR. Since r≤v(x)≤R a.e. in Ω′ and f is continuous, we have

mf:=minτ∈[ r,R ]f(τ)≤f(v(x))≤Mf:=maxτ∈[ r,R ]f(τ),a.e. in Ω′.

Clearly 0≤mf≤Mf. This, together with the property of g of being nonnegative in Ω and to be zero outside Ω′, gives

g(x)mf≤g(x)f(v(x))≤g(x)Mf, a.e. in Ω.

Then using the positivity and monotonicity properties of Sε, Remark 2.2, by applying Nε to the previous inequalities we obtain

0≤Nε(v)(x)=Sε(gf(v))(x)≤Sε(gMf)(x)    = MfSε(g)(x)≤MfΓg, for a.a. x∈Ω.

For the lower estimation, we make use of Lemma 3.1, applied for g, and we obtain:

(3.4) Nε(v)(x)≥Sε(gmf)(x)=mfSε(g)(x)≥mfyg, for a.a. x∈Ω′.   

We are now in position to derive a sufficient condition for Nε (v) ∈ DrR to hold, and thus for the invariance condition Nε(DrR)⊂DrR to be verified, namely

(h2) MfΓg≤R and r≤mfyg.

Remark 3.1

If in addition the function f is nondecreasing on the interval [r, R] , then mf = f (r), M f = f (R) and condition (h2) becomes

(h2)' f(R)Γg≤R and r≤f(r)yg.

This gives an answer to the second requirement stated in the Introduction: f being given, such bounds r, R always exist if the following asymptotic conditions are satisfied:

limτ→+∞f(τ)τ=0, limτ→0f(τ)τ=+∞.

3.3 The contraction condition

Assume that f is Lipschitz continuous on the interval [r, R] , more precisely that the following condition holds:

(h3) There exists a constant 0≤l<αλ1/‖ g ‖L∞(Ω) such that

(3.5) | f(τ)−fτ¯ |≤| τ−τ¯ | for all τ,τ¯∈[ r,R ].      

Here λ1 is the first eigenvalue of the Dirichlet problem for −Δ.

We recall that imposing on f the Lipschitz continuity condition is natural for getting uniqueness in nonlinear problems. Nevertheless, in our case, this condition is assumed only locally, as is often the case in real-world applications, and mathematically it makes the possibility to study by localization problems having multiple solutions.

For any fixed ε > 0, we prove that, under hypothesis (h3), the operator Nε satisfies the contraction property on DrR. To this aim, let v1, v2 ∈ DrR and denote wi = Nε (vi), i = 1, 2. Then

− d i v A ε ∇ w i = g f v i in  Ω w i = 0 on  ∂ Ω

in the weak sense, for i = 1, 2. These give

− d i v A ε ∇ w 1 − w 2 = g f v 1 − f v 2 in  Ω w 1 − w 2 = 0 on  ∂ Ω

in the weak sense. Thus, multiplying by w1 – w2 and integrating over Ω yield

(3.6) ∫ Ω A ε ∇ w 1 − w 2 ⋅ ∇ w 1 − w 2 = ∫ Ω g f v 1 − f v 2 w 1 − w 2 .

For the left-hand side integral in (3.6), by the ellipticity condition (2.2) on Aε, one has

(3.7) α w 1 − w 2 H 0 1 Ω 2 ≤ ∫ Ω A ε ∇ w 1 − w 2 ⋅ ∇ w 1 − w 2 .

For the right-hand side integral in (3.6), since g is zero outside and by using (h3), we have

∫Ωg[ f(v1)−f(v2) ](w1−w2)=∫Ωg[ f(v1)−f(v2) ](w1−w2)                                                                 ≤l∫Ω′g| v1−v2 || w1−w2 |                                                                ≤l‖ g ‖L∞(Ω)‖ v1−v2 ‖L2(Ω)‖ w1−w2 ‖L2(Ω).

Two times application of Poincaré's inequality with the sharp constant 1/λ1 yields

(3.8) ∫Ωg[ f(v1)−f(v2) ](w1−w2)≤l‖ g ‖L∞(Ω)λ1‖ v1−v2 ‖H01(Ω)‖ w1−w2 ‖H01(Ω).   

Now (3.6)-(3.8) imply

(3.9) ‖ w1−w2 ‖H01(Ω)≤l‖ g ‖L∞(Ω)αλ1‖ v1−v2 ‖H01(Ω),       

which in view of (h3) shows that N ε is a contraction on DrR, with the contraction constant

L:=l‖ g ‖L∞(Ω)αλ1<1.

3.4 Existence and localization result

We are now in position to state the existence and localization result for the nonlinear problem (2.1).

Theorem 3.2

Assume that conditions (h1)-(h3) hold. Then there exists ε0 > 0 such that

  1. for any ε < ε0, problem (2.1) has a unique solution uε in DrR, which is in DrR the unique minimum point of the energy functional JΕ.

  2. uε→u as ε→0, weakly in H01(Ω) and strongly in L2 (Ω) , where u is the unique function in DrR which solves the homogenized problem (2.5). Also u is in DrR the unique minimum point of the energy functional J, and

Jε(uε)→J(u)  as  ε→0.

  1. Proof. (a) The existence and uniqueness in DrR of uε follows from Banach's contraction principle, in view of the previous considerations. In order to prove the convergence of uε to u, note that, due to the boundedness in L∞(Ω) of the set { gf(uε) }, the set { uε } is bounded in H01(Ω), and so relatively compact in L2 (Ω). Thus there is a subsequence (uεk) with εk→0 which is weakly convergent in H01(Ω) to some u∈Η01(Ω), and strongly convergent in L2 (Ω) to u. Obviously, the limit u belongs to DrR. Now we are in position to pass to the limit in problem (2.1). The limit of the left-hand side of (2.1) is found as in the linear case (see [5, Chapter 9]), while for the right-hand side of (2.1) we use the uniform boundedness of (uεk) in L∞ (Ω), its strong convergence in L2 (Ω), and the continuity of Nemytskii's operator associated to f from L2 (Ω) to itself. Thus, the limit u satisfies problem (2.5). The uniqueness in DrR of the solution of the homogenized problem, can be proved with similar arguments as above, by means of Banach's contraction principle, and implies that the entire sequence (uε) converges to u, weakly in H01(Ω), and strongly in L2 (Ω).

  2. (b) To show that for each fixed ε, uε minimizes Jε in DrR, let us first note that Jε is a C1 functional on V and under the standard identification of H−1(Ω) to H01(Ω), one has

(3.10) (Jε)′(v)=v−Nε(v).       

Hence the fixed points of Nε are the critical points of Jε. It is easy to see that Jε is bounded from below on DrR. Then, using Ekeland's variational principle (see, e.g., [6], [17]), there is a minimizing sequence (vk) of elements from DrR such that

(3.11) Jε(vk)≤infDrRJε+1k        

and

(3.12) Jε(vk)≤Jε(v)+1k‖ v−vk ‖H01(Ω)             

for all v ∈ DrR. For any fixed index k, choose

vt=vk−(Jε)′(vk), 0<t<1.

Using (3.10), one has

vt=(1−t)vk+tNε(vk).

Here, one has v k ∈ DrR and by the invariance property in Section 3.2, N ε (vk) also belongs to DrR. Since DrR is convex, it follows that vt ∈ DrR for every t ∈ (0, 1). Replacing v by vt into (3.12) and then dividing by t, yields

t−1(Jε(vk)−Jε(vk−t(Jε)′(vk)))≤1k‖ (Jε)′(vk) ‖H01(Ω),

whence letting t go to zero, one finds

J ε ′ v k , J ε ′ v k ≤ 1 k J ε ′ v k H 0 1 Ω ,

that is

J ε ′ v k H 0 1 Ω ≤ 1 k .

Hence

(3.13) w k := J ε ′ v k → 0 in  H 0 1 Ω .

Using (3.10), we have wk = vk – Nε (vk) and by the contraction property of Nε, we find

‖ vk+p−vk ‖H01(Ω)≤‖ Nε(vk+p)−Nε(vk) ‖H01(Ω)+‖ wk+p−wk ‖H01(Ω)                                ≤L‖ vk+p−vk ‖H01(Ω)+‖ wk+p−wk ‖H01(Ω),

whence

‖ vk+p−vk ‖H01(Ω)≤11−L‖ wk+p−wk ‖H01(Ω).

This ensures that (vk) is a Cauchy sequence, so it converges in to some vε. Clearly vε ∈ DrR and from (3.11) and (3.13), passing to the limit with k→∞ we obtain that vε minimizes Jε in DrR and (Jε)' (vε) = 0. Hence vε is a fixed point in DrR of Nε. The uniqueness of the fixed point, guaranteed by Banach's contraction principle, implies that vε = uε as desired. The proof of the fact that the limit u, solution of the homogenized problem, minimizes J in DrR is analogous.

Finally the convergence Jε(uε)→J(u) follows from a standard result in homogenization theory (see [5, Section 8.2]) and the properties of Nemytskii's operator.

Remark 3.2

(Multiplicity) If the conditions (h1)-(h3) are satisfied for two pairs of numbers (r, R), let them be (r1,R1) and (r2, R2), with 0 < r1 < R1 < r2 < R2, then the sets Dr1R1 and Dr2R2 are disjoint, and according to Theorem 3.2, problem (2.1), as well as the corresponding homogenized problem (2.5), have two distinct solutions, one in Dr1R1 and the other in Dr2R2. This argument, extended to a finite or infinite sequence of pairs (r, R) , can be used in order to obtain multiple (finitely or infinitely many) solutions of problem (2.1) and of the corresponding homogenized problem (2.5).

Remark 3.3

Under the assumption that g vanishes in Ω/Ω′, we were able to localize the solutions assuming the Lipschitz continuity condition for f only locally, in the interval [r, R]. It is precisely this local Lipschitz condition which allows us to obtain, via Theorem 3.2, multiple solutions. If the assumption that g vanishes in Ω\Ω' does not hold, one can however localize the solutions in the set DrR, but under the stronger condition that f is Lipschitz continous in [0, R] instead of [r, R]. Correspondingly, in this case M f is

Mf=maxτ∈[ 0,R ]f(τ)

and the estimation from below (3.4) should be driven as follows:

Nε(v)(x)≥Sε(1Ω′gf(v))(x)≥Sε(1Ω′gmf)(x)                       = mfSε(1Ω′g)(x)≥mfy1Ω′g.

Here 11Ω' is the characteristic function of This implies that yg in (3.4) should be replaced by y1Ω′g.

Nevertheless, under this stronger Lipschitz condition on f, Theorem 3.2 cannot produce multiple solutions. Indeed, if Theorem 3.2 would apply to two pairs of numbers, let them be (r1, R1) and (r2, R2), with 0 < r1 < R1 < r2 < R2, then the sets Dr1R1 and Dr2R2 being disjoints, Nε would have two distinct fixed points, one in Dr1R1 and other in Dr2R2. On the other hand, f should be Lipschitz continuous on [0, R2] with the Lipschitz constant l<αλ1/‖ g ‖L∞(Ω). Then it is easy to see that the contraction condition on Nε, namely inequality (3.9), holds true for every v1,v2∈BR2:={ v∈V:0≤v(x)≤R2 and this guarantees that Nε has at most one fixed point in B R2. Since both disjoint sets Dr1R1 and Dr2R2 are included in BR2 , we get a contradiction. This justifies our claim about multiplicity.

4 Componentwise localization for systems

The technique used in Section 3 for one equation can be extended to systems, for the localization of each component of the solution, individually. For simplicity we discuss only a system of two equations in Ω, namely

(4.1) { −div(A1ε(x)∇u1ε)=g1(x)f1(u1ε,u2ε)−div(A2ε(x)∇u2ε)=g2(x)f2(u1ε,u2ε)                                   

under the Dirichlet boundary condition uiε=0 on ∂Ω, i=1,2.

We assume on the matrices Aiε, and on the functions gi, fi, similar hypotheses as in Section 2, allowing the ellipticity constants for the two equations to be different, namely αi and βi, i = 1,2. Then, solving the Dirichlet problem for system (4.1) in the weak sense is equivalent to finding in V x V the fixed point of the problem

u1ε=N1ε(u1ε,u2ε)u2ε=N2ε(u1ε,u2ε)

where Niε=Siε(gifi),i=1,2. Also, in terms of the energy functionals associated to the equations of the system, solving problem (4.1) is equivalent to solving the critical point problem

J11ε(u1ε,u2ε)=0J22ε(u1ε,u2ε)=0,

where by Jiiε(v1,v2) we have denoted the Fréchet derivative of Jiε(v1,v2) with respect to the variable vi, for i = 1,2, and the expressions of the functionals Jiε(v1,v2) are

J1ε(v1,v2)=12∫ΩA1ε(x)∇v1⋅∇v1−∫Ωg1(x)∫0v1(x)f1(τ,v2(x))dτ,J2ε(v1,v2)=12∫ΩA2ε(x)∇v2⋅∇v2−∫Ωg2(x)∫0v2(x)f2(v1,(x),τ)dτ.

We are interested to guarantee the existence and uniqueness of a solution (u1ε,u2ε) in a bounded subset of V x V, more exactly (u1ε,u1ε)∈D:=Dr1R1×Dr2R2, where 0 < r1 < R1 and 0 < r2 < R2 are given numbers.

Here are the hypotheses that duplicate the assumptions made in Section 3 for a single equation:

(H1) gi(x)≥0 a.e in Ω,gi(x)=0for x∈Ω/Ω′, and gi>0 on a subset of nonzero measure; fi(τ1,τ2)≥0 for every τ1,τ2≥0.

(H2) MiΓgi≤Ri and ri≤miygi, where

Mi=maxτ1∈[ r1,R1 ]τ2∈[ r2,R2 ]fi(τ1,τ2),  mi=minτ1∈[ r1,R1 ]τ2∈[ r2,R2 ]fi(τ1,τ2).

(H3) There exist constants 0 ≤ lij; (i,j = 1, 2) such that

| fi(τ1,τ2)−fi(τ¯1,τ¯2) |≤li1| τ1−τ¯1 |+li2| τ2−τ¯2 |                   for all τ1,τ¯1∈[ r1,R1 ]and τ2,τ¯2∈[ r2,R2 ],

and the spectral radius of the matrix L=[ Lij ]i,j=1,2, where

Lij:=lij‖ gi ‖L∞(Ω)αiλ1(i,j=1,2)

is less than one.

Theorem 4.1

Assume that conditions (H1)-(H3) hold. Then there exists ε0 > 0 such that for any ε < ε0 :

  1. (i) system (4.1) has a unique solution (u1ε,u2ε)∈D, which is in D the unique Nash equilibrium point with respect to the energy functionals J1ε,J2ε,i.e.,

J1ε(u1ε,u2ε)=minDr1R1J1ε(⋅,u2ε) and J2ε(u1ε,u2ε)=minDr2R2J2ε(u1ε,⋅).

  1. uiε→ui as ε→0, weakly in H01(Ω) and strongly in L2 (Ω), where (u1, u2) is the unique couple of functions in D which solves the homogenized system

(4.2) { −div(A10∇u1)=g1(x)f1(u1,u2)−div(A20∇u2)=g2(x)f2(u1,u2)           

Also (u1,u2 ) is in D the unique Nash equilibrium with respect to the energy functionals J1, J2 associated to the equations of the homogenized system (4.2), and

Jiε(u1ε,u2ε)→Ji(u1,u2)  as  ε→0,  for  i=1,2.

Proof. (a) Existence, uniqueness and localization: Using similar arguments as in the case of a single equation, we obtain for any (v1,v2),(v¯,v¯2)∈D and i∈{ 1,2 }:

0≤Niε(v1,v2)(x)≤MiSiε(gi)(x)≤MiΓgi,   for a.a. x∈Ω,

and

(4.3) Niε(v1,v2)(x)≥miSiε(gi)(x)≥miygi,   for a.a. x∈Ω′,    

which in view of (H2) show that Nε(D)⊂D,  where  Nε=(N1ε,N2ε).

Also

‖ Niε(v1,v2)−Niε(v¯1,v¯2) ‖H01(Ω)≤Li1‖ v1−v¯1 ‖H01(Ω)+Li2‖ v2−v¯2 ‖H01(Ω), i=1,2

which, using the matrix L defined in (H3), can be written in the matrix form

N 1 ε v 1 , v 2 − N 1 ε v ¯ 1 , v ¯ 2 H 0 1 Ω N 2 ε v 1 , v 2 − N 2 ε v ¯ 1 , v ¯ 2 H 0 1 Ω ≤ L v 1 − v ¯ 1 H 0 1 Ω v 2 − v ¯ 2 H 0 1 Ω .

Then, from Perov's fixed point theorem (see [14, Theorem 1]), the operator N ε has in D a unique fixed point (u1ε,u2ε), solution of system (4.1).

(b) Nash equilibrium: In order to prove that the solution (u1ε,u2ε) is a Nash equilibrium, we use an iterative approximation scheme. We start with a fixed element u2,0ε in Dr2R2. At each step k(k≥1),u2, k−1ε been determined at step k – 1, first we apply Ekeland' s principle to J1ε(⋅,u2,k−1ε) and find an u1,kε∈Dr1R1 such that

(4.4) J1ε(u1,kε,u2,k−1ε)≤infDr1R1J1ε(⋅,u2,k−1ε)+1k,‖ J11ε(u1,kε,u2,k−1ε) ‖H01(Ω)≤1k.      

Next we apply Ekeland’s principle to J2ε(u1,kε,⋅) and obtain an u2,kε∈Dr2R2 with

(4.5) J2ε(u1,kε,u2,kε)≤infDr2R2J2ε(u1,k,ε⋅)+1k,‖ J22ε(u1,kε,u2,kε) ‖H01(Ω)≤1k.   

Let

vk:=J11ε(u1,kε,u2,k−1ε)   and   wk:=J22ε(u1,kε,u2,kε).

From (4.4) and (4.5), one has vk,wk→0 in H01(Ω). Also, as in the case of one equation, we have

(4.6) u 1 , k ε − N 1 ε u 1 , k ε , u 2 , k − 1 ε = v k u 2 , k ε − N 2 ε u 1 , k ε , u 2 , k ε = w k .

The first equality in (4.6) written for k and k + p, with p∈ℕ\{ 0 }, yields

(4.7) u 1 , k + p ε − u 1 , k ε H 0 1 Ω ≤ N 1 ε u 1 , k + p ε , u 2 , k + p − 1 ε − N 1 ε u 1 , k ε , u 2 , k − 1 ε H 0 1 Ω + v k + p − v k H 0 1 Ω ≤ L 11 u 1 , k + p ε − u 1 , k ε H 0 1 Ω + L 12 u 2 , k + p − 1 ε − u 2 , k − 1 ε H 0 1 Ω + v k + p − v k H 0 1 Ω ≤ L 11 u 1 , k + p ε − u 1 , k ε H 0 1 Ω + L 12 u 2 , k + p ε − u 2 , k ε H 0 1 Ω + L 12 u 2 , k + p − 1 ε − u 2 , k − 1 ε H 0 1 Ω − u 2 , k + p ε − u 2 , k ε H 0 1 Ω + v k + p − v k H 0 1 Ω .

The second equality in (4.6) written for k and k + p gives

(4.8) ‖ u2,k+pε−u2,kε ‖H01(Ω)≤L21‖ u1,k+pε−u1,kε ‖H01(Ω)+L22‖ u2,k+pε−u2,kε ‖H01(Ω)+‖ wk+p−wk ‖H01(Ω)⋅ 

Denote

ak,p=‖ u1,k+pε−u1,kε ‖H01(Ω), bk,p=‖ u2,k+pε−u2,kε ‖H01(Ω),ck,p=‖ vk+p−vk ‖H01(Ω),dk,p=‖ wk+p−wk ‖H01(Ω).

Clearly,

(4.9) ck,p→0 and dk,p→0 as k→∞, uniformly with respect to p.     

With these notations, (4.7) and (4.8) become

ak,p≤L11ak,p+L12bk,p+L12(bk−1,p−bk,p),bk,p≤L21ak,p+L22bk,p+dk,p,

which can be expressed in the matrix form

(4.10) a k , p b k , p ≤ L a k , p b k , p + c k , p + L 12 b k − 1 , p − b k , p d k , p .

Under the assumption that the spectral radius of the matrix L is less than one, the matrix I – L (where I is the unit matrix), is invertible and its inverse contains only nonnegative elements (see [14, Lemma 2]). Thus (4.10) gives

a k , p b k , p ≤ I − L − 1 c k , p + L 12 b k − 1 , p − b k , p d k , p .

Let (I – L)–1 = [ρi j]. Then

(4.11) a k , p ≤ ρ 11 c k , p + L 12 b k − 1 , p − b k , p + ρ 12 d k , p b k , p ≤ ρ 21 c k , p + L 12 b k − 1 , p − b k , p + ρ 22 d k , p .

From the second inequality, one has

(4.12) bk,p≤ρ21L121+ρ21L12bk−1,p+ρ21ck,p+ρ22dk,p1+ρ21L12.      

Note that the sequence (bk,p)k≥1 is bounded uniformly with respect to p. Indeed, from (4.6),

u2,kε−wk=N2ε(u2,kε,u2,kε),

whence

{ −div(Aε(x)∇(u2,kε−wk))=g2(x)f2(u1,kε,u2,kε)   in Ωu2,kε−wk=0                                                                              on ∂Ω.

Since the right-hand side is bounded in L∞ (Ω) independently of k, this implies

‖ u2,kε−wk ‖H01(Ω)≤c0

for every k. As a result, the sequence (u2,kε)k≥1 is bounded in the norm of H01(Ω) by some constant c. Then, clearly bk,p≤2c, as claimed. Now we recall a lemma proved in 16, Lemma 3.2 .

Lemma 4.2

Let (xk , p ) , (yk,P) be two sequences of real numbers depending on a parameter p, such that

(xk , p) is bounded uniformly with respect to p,

and

(4.13) 0≤xk,p≤λxk−1,p+yk,p            

for all k, p and some λ∈[ 0,1 ).If yk,p→0 as k→∞ uniformly with respect to p, then xk,p→0 uniformly with respect to p.

We apply the previous lemma for

χk,p=bk,p,yk,p=ρ21ck,p+ρ22dk,p1+ρ21L12 and λ=ρ21L121+ρ21L12bk−1,p.

According to (4.12), one has (4.13), while due to (4.9), one has yk,p→0 as k→∞, uniformly with respect to p. Also, obviously, 0≤λ<1. It follows that bk,p→0 as k→∞, uniformly with respect to p, that is (u2,kε) is a Cauchy sequence. Next, the first inequality in (4.11) implies that (u1,kε) is also a Cauchy sequence. Let v1ε,v2ε be the limits of the sequences (u1,kε),(u2,kε) as k→∞, respectively. Clearly (v1ε,v2ε)∈D, and passing to the limit in (4.4), (4.5) we obtain that (v1ε,v2ε) solves system (4.1). The uniqueness of the solution in D implies that (v1ε,v2ε)=(u1ε,u2ε).

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Received: 2018-10-26
Accepted: 2018-11-24
Published Online: 2019-05-22

© 2020 R. Bunoiu and R. Precup, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 Public License.

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https://doi.org/10.1515/anona-2020-0001
Keywords for this article
Nonlinear elliptic problem; homogenization; localization; positive solution; multiple solutions
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Articles in the same Issue

  1. Frontmatter
  2. On the moving plane method for boundary blow-up solutions to semilinear elliptic equations
  3. Regularity of solutions of the parabolic normalized p-Laplace equation
  4. Cahn–Hilliard equation on the boundary with bulk condition of Allen–Cahn type
  5. Blow-up solutions for fully nonlinear equations: Existence, asymptotic estimates and uniqueness
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  18. A Picone identity for variable exponent operators and applications
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  73. Lack of smoothing for bounded solutions of a semilinear parabolic equation
  74. Gradient estimates for the fundamental solution of Lévy type operator
  75. π/4-tangentiality of solutions for one-dimensional Minkowski-curvature problems
  76. On the existence and multiplicity of solutions to fractional Lane-Emden elliptic systems involving measures
  77. Anisotropic problems with unbalanced growth
  78. On a fractional thin film equation
  79. Minimum action solutions of nonhomogeneous Schrödinger equations
  80. Global existence and blow-up of weak solutions for a class of fractional p-Laplacian evolution equations
  81. Optimal rearrangement problem and normalized obstacle problem in the fractional setting
  82. A few problems connected with invariant measures of Markov maps - verification of some claims and opinions that circulate in the literature
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