Pascal Belan
email: pascal.belan@sc-eco.univ-nantes.fr.

Len C3e

Philippe Eurequa

Greqam & Michel

Bertrand Eurequa

Eurequa Wigniolle
email: wignioll@univ-paris1.fr.

Philippe Michel

Bertrand Wigniolle

Does imperfect competition foster capital accumulation in a developing economy ?

Keywords: JEL classification: D43, D9, D64. Mots-clés : concurrence imparfaite, accumulation du capital, altruisme JEL classification: D43, D9, D64 imperfect competition, capital accumulation, altruism

Nous analysons la relation entre la concurrence imparfaite et l'accumulation du capital dans une économie duale, composée d'un secteur traditionnel et d'un secteur moderne, et peuplée par deux types d'agents, des travailleurs et des capitalistes. Les travailleurs allouent leur temps disponible entre les deux secteurs productifs. Les capitalistes accumulent de la richesse dans le secteur moderne. L'économie est ouverte aux mouvements de capitaux, mais l'endettement des capitalistes est contraint. Le comportement non concurrentiel des capitalistes leur procure une rente supplémentaire, qui est obtenue aux dépends des travailleurs et diminue l'emploi dans le secteur moderne. A long terme, si l'endettement des capitalistes n'est pas contraint, la concurrence imparfaite favorise l'accumulation de capital et la croissance, alors qu'elle est néfaste dans le cas contraire. De plus, lorsque les capitalistes ne sont pas contraints, l'emploi et les salaires sont plus élevés. Cette propriété peut suggérer l'introduction d'une subvention des héritages, qui permet à l'économie d'atteindre le régime sans contrainte d'endettement, et qui accroît le bien-être des travailleurs.

Introduction

Does imperfect competition foster capital accumulation in a developing economy ? This paper is an attempt to answer this question. We explore the relationship between imperfect competition and capital accumulation, in a dual economy. Considering an economy with two sectors, a traditional agricultural sector and a modern capitalistic one, we assume a non-competitive behavior of capitalists in the modern sector. Capitalists take into account the effect of their labor demand on the equilibrium wage. This behavior results in an extra rent for them, but introduces a distortion in factor prices. We aim at evaluating its effect on capital accumulation.

Since [START_REF] Lewis | Economic Development with Unlimited Supplies of Labour[END_REF] contribution (developed through a formal model in [START_REF] Ranis | A Theory of Economic Development[END_REF]), numerous articles have explored issues relating to the migration of workers from a traditional sector to a modern one. Development is viewed as an unbalanced growth process, in which labor shifts toward the manufacturing sector. Capital accumulation in this sector increases labor productivity and results in higher wages, attracting more and more workers. The article by [START_REF] Kongsamut | Beyond Balanced Growth[END_REF] can be viewed as a modern reformulation of the argument, which deals with three sectors of activity (agriculture, manufacturing and services), and shows how such models are consistent with historical stylized facts.

As it is apparent in Lewis' model, a direct consequence of the dual production structure is that the two sectors have conflicting interests. Capitalists in the modern sector can hire more workers at lower costs if the labor productivity in the traditional sector is low enough. The mechanism would be reinforced if the capitalists had a non-competitive behavior taking account of the effect of their decisions on the traditional sector. Worsening the situation in the traditional sector by such a behavior could favor a quicker development of the modern sector. Typically, by strategically reducing their labor demand, capitalists can maintain lower wages.

In order to formulate this idea, we consider a framework that retains the main ideas of [START_REF] Lewis | Economic Development with Unlimited Supplies of Labour[END_REF] and [START_REF] Ranis | A Theory of Economic Development[END_REF] models. We use a model with successive generations of workers and capitalists. Workers are identical within a generation and consume their whole revenue at each period. Capitalists are altruistic and leave as a bequest a stock of wealth to their descendants. Thus, our model is close to [START_REF] Mankiw | The savers-spenders theory of fiscal policy[END_REF]'s model with savers and non-savers agents.

We consider a dual economy with two production sectors, a traditional agricultural sector and a modern capitalist one. The traditional sector employs labor as the only input, when the modern one uses capital and labor.

Workers must allocate their time endowment between the two activities. The optimal allocation of workers time determines labor supply for the modern sector.

We assume that capitalists behave non-competitively. They maximize their profit taking into account the impact of their labor demand on the equilibrium wage. Under this assumption, they tend to demand less labor than in the competitive case, in order to decrease the equilibrium wage. The equilibrium concept is in the line of the Cournot-Walras equilibrium (cf. [START_REF] Gabszewicz | Oligopoly à la Cournot in General Equilibrium Analysis[END_REF], [START_REF] Codognato | Cournot-Walras Equilibria in Markets with a Continuum of Traders[END_REF] and [START_REF] Gabszewicz | Oligopoly Equilibrium in Exchange Economies[END_REF]) : some agents, called strategic agents, take into account the influence of their choice on the walrassian equilibrium and play between them a game of the Cournot-Nash type. In our framework, strategic agents are the capitalists, who consider the effect of their labor demand on the walrassian equilibrium wage.

The economy is small and open to capital flows and the international interest rate is exogenous and constant. Capital markets are imperfect in the sense that capitalists cannot borrow more than a given fraction of their wealth. Such kind of borrowing limit can result from an enforcement problem, as in [START_REF] Matsuyama | Endogenous Inequality[END_REF].

Whether borrowing constraints are binding or not will be crucial for the consequences of imperfect competition. If the borrowing constraint is not binding, capital return is determined by the international interest rate. Thus, the non-competitive behavior of capitalists cannot affect the marginal return of capital, and the economy works as a small open economy. On the contrary, if the borrowing constraint is binding, the total capital stock of each capitalist is proportional to his own wealth, and the economy works as a closed economy. The capital return now is endogenous and depends on the capitallabor ratio. Therefore, the non-competitive behavior of capitalists tends to decrease the marginal return of capital as it leads to a smaller labor demand. To sum up, in both cases, the non-competitive behavior of capitalists provides them an extra rent due to a lower wage. In the case of a non-binding constraint, the marginal return of capital is not affected. But in the case of a binding constraint, the marginal return of capital decreases.

We study the long run equilibrium of this economy, and we analyze how the non-competitive behavior of capitalists affects capital accumulation. Two long run growth rates can be define. One can be called the international growth rate, driven by the international rate of return of capital. The other one is the natural growth rate corresponding to the growth rate in the population of workers. If the international growth rate is greater than the natural one, the borrowing constraint is not binding in the long run. In this case, the rentability of wealth is exogenously given by the international rate of return. Capitalists benefit from an extra rent induced by their non-competitive behavior and the marginal return of their wealth is not affected. Therefore, capitalists non-competitive behavior favors capital accumulation.

On the contrary, if the international growth rate is smaller than the natural one, the borrowing constraint is binding in the long run. Capitalists benefit from an extra rent provided by their non-competitive behavior but the marginal return of their wealth decreases. In this case, the economy works as a closed economy. Capitalists non-competitive behavior leads to smaller employment and capital stock in the modern sector. Thus, imperfect competition involves very different consequences on capital accumulation according to the borrowing constraint is binding or not. Capital accumulation is fostered in the unconstrained regime and is weaken in the constrained one.

Our work can be viewed as a contribution to the literature about inequality and economic growth, following [START_REF] Stiglitz | Distribution of income and wealth among individuals[END_REF] and [START_REF] Bourguignon | Pareto superiority of unegalitarian equilibria in Stiglitz' model of wealth distribution with convex saving function[END_REF]. If the saving function is convex, inequality is growth-enhancing. In our model, savers are the capitalists and their non-competitive behavior that impoverishes workers may have a positive or negative effect on growth. This paper is also related to an emerging literature that focuses on the interaction between long-run capital accumulation and imperfect competition. Laitner (1982) is one of the first attempt to study the consequences on long-run capital accumulation of the existence of oligopolies on commodity market. More recently, [START_REF] Sorger | On the long-run distribution of capital in the Ramsey model[END_REF] assumes, in the Ramsey model, that households exercise market power on the capital market: they take into account the impact of their own investment on the capital return. Still, [START_REF] Becker | Stationary Strategic Ramsey Equilibrium[END_REF] uses the same framework and studies the existence of stationary equilibria in a two players game. In these two contributions, the strategic variable is the investment in capital. Belan, Michel andWigniolle (2002, 2004) have introduced the Cournot-Walras equilibrium in a growth model to represent the long-run effect of pension funds.

Section 2 presents the model. Section 3 studies the short run equilibrium, for given levels of wealth for capitalists. It analyzes the impact of imperfect competition on labor demand in the modern sector and on the marginal return of wealth for capitalists. Section 4 considers the long run equilibrium and the impact of non-competitive behavior on capital accumulation. Section 5 concludes. Section 6 is a final technical appendix.

The model

Throughout the paper, we consider a small economy open to capital flows, with an international constant gross interest rate R. In each period, population is divided between workers and capitalists. The workers live for one period and consume their income (non-savers in [START_REF] Mankiw | The savers-spenders theory of fiscal policy[END_REF]). Capitalists are altruistic dynasties à la [START_REF] Barro | Are government bonds net wealth[END_REF] who consume, save and accumulate wealth.

The workers

In each period, a new generation of identical workers is born and lives for one period. The number of workers, N t in period t, grows at rate n: N t = (1 + n) N t-1 , where the growth factor (1 + n) is less that R. Each worker possesses a time endowment of one unit. He spends l units of time at work in modern firms for a wage w t , and the remainder in the traditional activity which provides him a revenue φ (1l). Labor supply l t of a worker living in period t maximizes

w t l + φ (1 -l) , l ∈ [0, 1] (1)
where w t is the real wage. Alternatively (1) can be interpreted as a utility function of consumption and leisure which is linear with respect to consumption c t = w t l t .

We make the following assumption on φ.

Assumption 1. The function φ is three times continuously differentiable on the interval (0, 1). It satisfies 1 ∀z ∈ (0, 1) , φ 0 (z) > 0, φ 00 (z) < 0 and φ 000 (z) ≥ 0 lim z-→0 φ 0 (z) = +∞ Individual labor supply l t satisfies φ 0 (1l t ) = w t , if l t > 0. The total labor supply is L t = N t l t . Let us define the inverse function of aggregate labor supply

W t (L) ≡ φ 0 µ 1 - L N t ¶ (2)
for 0 < L < N t . Under assumption 1, this function is increasing from φ 0 (1) to +∞ and convex. 2 These properties have two consequences. First, if 1 The assumption φ 000 ≥ 0 implies that the inverse function of aggregate labor supply is convex.

W 0 t (L) = -1 N t φ 00 ³ 1 -L N t ´> 0 W 00 t (L) = 1 N 2 t φ 000 ³ 1 -L N t ´≥ 0    w t ≤ φ 0 (1)
, labor supply is zero and home production is the only resource of the economy. In order to hire some workers, firms must pay a wage higher than φ 0 (1) ≡ W min , that we call minimum wage in the following. Second, whatever the actual wage is, people will spend part of their time endowment in traditional production.

The capitalists

There are I dynasties of capitalists, I ≥ 1. In period t, capitalist i allocates his wealth X it between capital stock in the familial firm or investment on capital markets. All the firms produce the same commodity with capital K it and labor L it according to a constant-return-to-scale technology F (K it , L it ) which includes capital after depreciation. Marginal products are positive and decreasing.

The international gross interest rate R applies for borrowing and lending. Let Z it = K it -X it , the amount borrowed by family i in period t. A negative Z it means that a fraction of family wealth is invested on international capital markets. Capitalists face a borrowing limit in the sense that Z it must be less than a fraction µ of the wealth of family i3

Z it ≤ µX it , µ > 0. ( 3 )
Capitalists have a Nash behavior on labor market. They make their decisions in factor demands by taking into account the effect of their own labor demand on the equilibrium wage w t . At equilibrium of the labor market, labor supply L t is equal to the sum of labor demands L it + L -it , where L -it = P j6 =i L jt is labor demand of other capitalists. From equation ( 2), real wage paid by all firms is then w t = W t (L it + L -it ). Thus, income of capitalist i in period t writes

Π it = F (X it + Z it , L it ) -RZ it -W t (L it + L -it ) L it (4)
and is allocated between consumption C it and savings

Π it = C it + X it+1 (5)
Capitalist i behaves like an infinite-lived agent and maximizes the discounted sum

∞ X t=0 γ t C 1-1 σ it 1 -1 σ , 0 < γ < 1, σ > 0 (6)
subject to equations ( 3), ( 4) and ( 5), for a given initial wealth X i0 and taking labor demands of other capitalists (L -it ) t≥0 as given. The discount factor γ stands for the degree of altruism.

Intertemporal equilibrium

Given the initial wealth of capitalists, X i0 (i = 1, . . . , I), an intertemporal equilibrium consists of sequences of

• wages (w t ) t≥0 , • decisions of capitalists (L it , Z it , C it , , X it+1 ) t≥0 , i = 1, . . . , I,
• decisions of workers (l t ) t≥0 ,

• aggregate labor (L t ) t≥0 , which, in addition to all the individual optimality conditions, satisfy the equilibrium conditions

L t = N t l t = I X i=1
L it , and w t = W t (L t ) .

The competitive economy without restriction on borrowing

We first study the benchmark case of an economy where capitalists have a competitive behavior and face no borrowing constraint. It will allow us to measure the impact of imperfect competition and imperfect capital market and to introduce some basic assumptions that will be maintained throughout the paper.

In the competitive economy without borrowing restriction, the marginal productivities of capital and labor are respectively equal to the gross interest rate R and the wage w t in period t

F 0 K (K t , L t ) = R and F 0 L (K t , L t ) = w t
The competitive capital-labor ratio K t /L t = k c is the solution of F 0 K (k c , 1) = R. The equilibrium of the labor market implies

φ 0 µ 1 - L t N t ¶ = w t = F 0 L (k c , 1) ≡ W c
Let l c = L t /N t be the competitive labor supply of each worker, solution of

φ 0 (1 -l c ) = F 0 L (k c , 1). It is positive if and only if W c is larger than the minimum wage W min = φ 0 (1). Assumption 2. W c = F 0 L (k c , 1) > W min = φ 0 (1).
The dynamics of wealth of the competitive capitalist i are then

X it+1 = RX it -C it
and consumptions satisfy the Euler equation

C it+1 C it = (γR) σ .
As a consequence of Lemma A1 in the appendix, the intertemporal budget constraint of capitalist i is satisfied with equality (lim t-→+∞ R -t X it = 0) if and only if the growth factor of consumption is smaller than the gross interest rate R. This property is rather intuitive. Indeed, if (γR) σ ≥ R, capitalist consumption and wealth would grow at a higher rate than his income, which is unfeasible. The assumption (γR) σ < R is equivalent to

Assumption 3. γ < γ ≡ R 1 σ -1 .
Assumptions 2 and 3 made in the benchmark case of a competitive economy will be maintained for analyzing imperfect competition. Under these assumptions, the properties of the competitive equilibrium can be further analyzed. It is straightforward to show that wealth evolves according to:

X it = (γR) σt X i0 (7)
and consumption is such that:

C it = (γR) σt X i0 ¡ 1 -γ σ R σ-1 ¢

Short run equilibrium

The problem of a capitalist is broken up into a static problem and a dynamic problem. The former consists in characterizing labor demand and capital demand in period t of a capitalist, given his current wealth and labor demands of the other firms. The resulting optimal income function of period t is studied. Then, we analyze the labor market equilibrium and we state conditions for the borrowing constraints to be binding or not in each period.

Capital and labor demands of capitalists

In period t, given wealth X it and labor demand of other firms L -it , capitalist i maximizes his income Π it subject to the constraint Z it ≤ µX it . We first treat the case where the borrowing constraint is not binding and state the condition ensuring that capitalist i will be unconstrained. Then, we will turn to the binding case.

The borrowing constraint is not binding. The problem of capitalist i is equivalent to choose capital stock

K it = X it + Z it and labor L it , which maximize F (K i , L i ) -R (K i -X it ) -W t (L i + L -it ) L i
with respect to K i and L i . The solution is independent of X it . The capitallabor ratio has the same value as in the competitive economy

K it L it = k c . ( 8
)
Moreover, optimal labor demand L it satisfies

W t (L it + L -it ) + W 0 t (L it + L -it ) L it = W c . ( 9
)
Since the LHS in (9) increases from W t (L -it ) to +∞ when L it increases from 0 to N t -L -it , there is a unique positive solution

L it = ψ t (L -it ) of (9) if and only if W t (L -it ) < W c .
Note that the borrowing constraint is not binding if and only if

Z it = K it -X it ≤ µX it at the unconstrained optimum. Since K it = k c ψ t (L -it ), this inequality rewrites X it ≥ k c ψ t (L -it ) 1 + µ . ( 1 0 )
The borrowing constraint is binding. From the above analysis, we deduce that, if X it < k c ψ t (L -it ) / (1 + µ), capitalist i cannot reach his unconstrained optimal income. In this case, labor demand L it maximizes

F ((1 + µ) X it , L i ) -RµX it -W t (L i + L -it ) L i with respect to L i . Since F 0 L (1, 0) > F 0 L ¡ 1, 1 k c ¢ = W c , the assumption W c > W t (L -it ) implies that there exists a unique positive solution in L it of W t (L it + L -it ) + W 0 t (L it + L -it ) L it = F 0 L ((1 + µ) X it , L it ) (11)
In both cases, the effect of non-competitive behavior appears in equations [START_REF] Kongsamut | Beyond Balanced Growth[END_REF] and [START_REF] Lewis | Economic Development with Unlimited Supplies of Labour[END_REF] with the term W 0 t (L it + L -it ) L it . It renders the fact that capitalists take into account the impact of their labor demands on wages.

Optimal income function of a capitalist

Let us define Π * it (X it , L -it ) as the maximum profit of capitalist i, that we call optimal income function of capitalist i. We have stressed that the characterization of the optimal factor demands of a capitalist depends on his level of wealth, but the differentiability of the optimal income function with respect to wealth is needed for obtaining the usual Euler equation. We prove this property in the following proposition.

Proposition 1. Assume that L -it satisfies W t (L -it ) < W c . Then, the op- timal income function Π * it (X it , L -it ) of capitalist i is continuously dif- ferentiable with respect to X it . (i) If the borrowing constraint is not binding ( X it ≥ k c ψ t (L -it ) / (1 + µ)), then (∂Π * it /∂X it ) (X it , L -it ) = R. (ii) If it is binding ( X it < k c ψ t (L -it ) / (1 + µ)), then (∂Π * it /∂X it ) (X it , L -it ) = (1 + µ) F 0 K ((1 + µ) X it , L it ) -Rµ > R.
Proof. The derivatives in (i) and (ii) are respectively obtained for

X it < k c ψ t (L -it ) / (1 + µ) and X it > k c ψ t (L -it ) / (1 + µ), by the envelope theorem.
For

X it = k c ψ t (L -it ) / (1 + µ), we have F 0 K ((1 + µ) X it , L it ) = R
, which implies that, at this point, left-hand derivative and right-hand derivative are equal. Thus, Π * it is continuously differentiable with respect to X it . Finally, if the borrowing constraint is binding, we have

(1 + µ) X it /L it < k c which implies (1 + µ) F 0 K ((1 + µ) X it , L it ) -Rµ > R.
This concludes the proof. ¥ This proposition shows that, when the borrowing constraint is not binding, the marginal return of wealth of some capitalist remains the same as in the benchmark competitive case, i.e. equal to R. On the contrary, when the borrowing constraint is binding, the return is endogenous. It depends on the non-competitive levels of the labor demands L it , and thus, will be affected by imperfect competition.

The differentiability of the optimal income function in period t + 1 allows to write the marginal optimality condition for the program of capitalist i (defined by equations ( 3) to ( 6)). The Euler equation of capitalist i is

C it+1 C it = µ γ ∂Π * it+1 ∂X it+1 ¶ σ . ( 12
)

Equilibrium of the labor market

In order to analyze the equilibrium of the labor market in period t, we introduce the following deflated variables

x it = X it N t , l it = L it N t , and
l t = L t N t = I X i=1 l it .
Using equations ( 2), the marginal cost of labor of the firm held by capitalist i, W t (L t ) + W 0 t (L t ) L it , can be written as a function of aggregate labor per worker l t and labor demand per worker l it m (l t , l it ) ≡ φ 0 (1l t )φ 00 (1l t ) l it [START_REF] Matsuyama | Endogenous Inequality[END_REF] where m increases with respect to l t and l it .

The borrowing constraint is not binding for capitalist i. Then, using equation ( 9), the marginal cost of labor is

m (l t , l it ) = W c (14)
which implies

l it = g (l t ) ≡ W c -φ 0 (1 -l t ) -φ 00 (1 -l t ) .
Under Assumptions 1 and 2, the function g is positive and decreasing with l t .

The borrowing constraint is binding for capitalist i. By equation [START_REF] Lewis | Economic Development with Unlimited Supplies of Labour[END_REF], labor demand l it satisfies

m (l t , l it ) -F 0 L ((1 + µ) x it , l it ) = 0. ( 1 5 )
We study this equation in l it . The LHS is increasing with l it , and goes to +∞ (since m (l t , +∞) = +∞ and F 0 L ((1 + µ) x it , +∞) = F 0 L (0, 1) is finite). When l it goes to zero, we have

m (l t , 0) -F 0 L ((1 + µ) x it , 0) < W c -F 0 L (1, 0)
, and the LHS is negative since W c < F 0 L (1, 0). Thus, there is a unique solution to equation ( 15)

l it = h (x it , l t )
where h is increasing with respect to x it and decreasing with respect to l t .

Remark. It is possible to make a comparison with the case where capitalist i has a competitive behavior and a binding borrowing constraint. In this case, labor demand l it satisfies:

φ 0 (1 -l t ) -F 0 L ((1 + µ) x it , l it ) = 0. ( 1 6 )
This equation has a unique solution that can be written:

l it = H (x it , l t )
where H is increasing with respect to x it and decreasing with respect to l t . Moreover, for given values of x it and l t , imperfect competition reduces labor demand : h (x it , l t ) < H (x it , l t ) .

Characterization of the labor market equilibrium. The condition for capitalist i to be constrained is (1 + µ) x it /l it < k c , or equivalently

F 0 L ((1 + µ) x it , l it ) < W c = F 0 L (k c , 1
) . From equation ( 14) and ( 15), we deduce that capitalist i is constrained if and only if h (x it , l t ) < g (l t ) .

Therefore, labor demand of capitalist i can be written as a function λ of wealth x it and aggregate labor l t ,

l it = λ (x it , l t ) ≡ min {h (x it , l t ) , g (l t )}
where λ is non-decreasing with respect to x it and decreasing with respect to l t . An equilibrium of the labor market l t is a solution of

l t = I X i=1 λ (x it , l t ) .
Proposition 2. Under Assumptions 1 and 2, and given wealths of capitalists (X it ) i=1,...,I , there is a unique positive equilibrium l * t of the labor market in period t. Labor per worker l * t is lower than its competitive level l c (without borrowing constraint).

Proof. The LHS of the equilibrium condition l t -P I i=1 λ (x it , l t ) = 0 is a continuous increasing function of l t . Its limit when l t goes to 0 is negative, since both g (0) and h (x it , 0) are positive. Indeed,

g (0) = W c -φ 0 (1) -φ 00 (1) > 0 and h (x it , 0) is such that φ 0 (1) -φ 00 (1) h (x it , 0) -F 0 L ((1 + µ) x it , h (x it , 0)) = 0.
Moreover, since g (l c ) = 0, we have λ (x it , l c ) = 0. Thus, when l t goes to l c , the limit of l t -P I i=1 λ (x it , l t ) is positive. By consequence, there is a unique l * t in the interval (0, l c ) satisfying the equilibrium condition of the labor market. ¥ At equilibrium of the labor market, labor demand of capitalist i is l * it = λ (x it , l * t ). The borrowing constraint of capitalist i is binding if and only if h (x it , l * t ) < g (l * t ).

Comparison of competitive and non-competitive equilibria

What is the impact of capitalists non-competitive behavior on the labor demand, wages and profits ? To answer this question, we first compare the resulting equilibrium with the benchmark competitive case where there is no restriction on borrowing. Then, we analyze how imperfect competition modifies the equilibrium when the borrowing constraint is binding.

The borrowing constraint is not binding. If all capitalists have enough wealth to be unconstrained, total labor is L * t = N t l * (I), where l * (I) is the solution of l-Ig (l) = 0. In this case, all capitalists employ the same quantity of labor given by: N t l * (I) I From the properties of g, we deduce that, if the number of capitalists I augments, total labor increases and tends to its competitive level N t l c when I goes to infinity. The difference l cl * (I) is a consequence of the Nash behavior of capitalists on labor market, and reflects the effect of imperfect competition.

The optimal income function Π * it (X it , L -it ) of capitalist i can be written at equilibrium:

RX it + N t l * (I) I • -φ 00 (1 -l * (I)) l * (I) I
Ţhe term RX it is the income of the capitalist in the case of a competitive behavior. The supplementary term, which is positive, corresponds to the extra rent of the capitalist, due to its non-competitive behavior. Thus, when the borrowing constraint is not binding, the marginal return of capital is unchanged by the non-competitive behavior, and the level of profit is increased by the extra rent due to the non-competitive behavior.

The borrowing constraint is binding for at least one capitalist i. Then, his labor demand is

l * it = h (x it , l * t ) < g (l * t ) where l * t = P I j=1 l * jt < Ig (l * t )
. Thus, if some capitalist is constrained, equilibrium labor l * t is smaller than l * (I). The difference l * (I)l * t is a consequence of an additional effect that results from the borrowing constraint.

To obtain more precise results about the role of imperfect competition, we must focus on the special case of a symmetrical equilibrium with a binding borrowing constraint for all agents.

The symmetrical equilibrium with a binding borrowing constraint. We assume that x it and l * it are equal in all firms and we note the common value of the capital stock x t . In this case, labor demand in each firm is equal to l * t /I, with l * t solution to the equation:

l * t /I = h (x t , l * t )
Let us compare the equilibrium labor demand l * t with the equilibrium labor demand obtained when capitalists have a competitive behavior and a binding borrowing constraint. The latter denoting by l * c t is solution of:

l * c t /I = H (x t , l * c t )
It is straightforward to prove that l * t < l * c t . Indeed, we proceed by contradiction, assuming l * t > l * c t . Since H is a decreasing function of l and, using the property that, for all (x, l), h (x, l) < H (x, l), we deduce

l * t I = h (x t , l * t ) < H (x t , l * t ) < H (x t , l * c t ) = l * c t I
which contradicts the initial statement. Thus, for a given capital stock, we conclude that the non-competitive behavior of the capitalists results in a smaller value for equilibrium employment. Indeed, capitalists demand less labor in order to decrease the level of wages. The optimal income function Π * it (X it , L -it ) of capitalist i can be written at equilibrium:

X it [(1 + µ) F 0 K ((1 + µ) x t , l * t /I) -Rµ] + N t l * t /I [-φ 00 (1 -l * t ) l * t /I]
The return on capital

[(1 + µ) F 0 K ((1 + µ) x t , l * t /I) -Rµ]
appears in the first term, while the second term consists of the extra rent due to the noncompetitive behavior. If capitalists behave competitively, their optimal profit becomes X it [(1 + µ) F 0 K ((1 + µ) x t , l * c t /I) -Rµ] and the rent vanishes.

By contrast with the non-binding case, imperfect competition changes the capital return. When the borrowing constraint is not binding, the capital return is equal to the international return R, whatever the behavior of capitalist is (competitive or non-competitive). When the borrowing constraint is binding, imperfect competition decreases equilibrium employment and consequently reduces capital return. Therefore, non-competitive behavior increases the income of capitalists by a rent extracted from workers, but it decreases the marginal productivity of capital at equilibrium. As we shall see, this second effect will always result in lower capital accumulation in the long-run.

Long run equilibrium

We prove the existence of two types of long-run balanced growth path. The first one is associated with the natural growth rate and arises when borrowing constraints are binding for all capitalists. When capitalists are unconstrained, consumption and wealth of every capitalist grow at a constant rate driven by the international gross interest rate. Finally, we study the possibility of a transition from the constrained regime toward the unconstrained regime.

Stationary growth path at rate n

Let assume that all variables grow at the same rate as N t , i.e. ratios

C it N t = b c i , X it N t = b x i , L it N t = b l i , Π * it N t = b π i ,
are constant. Along such a path, labor per worker L t /N t is also constant and equal to b l, solution of l -

P I i=1 λ (b x i , l) = 0. Consequently, labor demand of capitalist i is such that b l i = λ ³ b x i , b l
´.

From the Euler equation ( 12), the marginal revenue ∂Π * it+1 /∂X it+1 must be constant and equal to (1+n) 1/σ γ . Furthermore, we know from Proposition 1 that

ρ i ≡ ∂Π * it+1 ∂X it+1 = ( R, if i is not constrained (1 + µ) F 0 K ³ (1 + µ) b x i , b l i ´-Rµ > R, if i is constrained
(17) Thus, we have

ρ i = ρ ≡ (1 + n) 1/σ γ ≥ R (18)
and this inequality is strict if i is constrained.

Proposition 3. If γ < (1+n) 1/σ R
≡ γ, then there exists a unique stationary equilibrium growth path at the natural rate. Along this path, all dynasties of capitalists have equal wealth and their borrowing constraints are binding.

Proof. From (17) and (18), along a stationary growth path, all capitallabor ratios (1 + µ) xi / li are equal to the solution k of

(1 + µ) F 0 K (k, 1) -Rµ = ρ > R. ( 1 9 )
Moreover, the marginal cost of labor m( l, li ) is equal to its marginal product F 0 L ( k, 1). This implies that all labor demands are equal to l/I. This in turn leads to the equality of wealth xi = k l/ [(1 + µ) I].

Conversely, these variables satisfy all the equilibrium conditions when all capitalists are constrained. Indeed, from (19), we have

F 0 K ( k, 1) > R = F 0 K (k (R) , 1
) and we deduce that (1 + µ) xi / li = k < k c , which implies that all the borrowing constraints are binding. ¥

We are now in a position to discuss the effect of imperfect competition on labor and capital income in the long-run when all capitalists are constrained on their borrowing. Indeed, the long-run capital-labor ratio k does not depend on the competition environment. It would be the same in a competitive economy with binding borrowing constraints. In the short run, for a given level of wealth X i , we have seen in Section 3.4 that non-competitive behavior leads capitalists to reduce their labor demand. The resulting marginal product of capital is smaller than in the competitive case. Now, in the long-run, capitalists wealths being endogenous, marginal product of capital remains unchanged whatever the competition environment is. Consequently, since labor demand is lower in the non-competitive case, wealth must also be lower. Imperfect competition leads to lower capital accumulation.

Increasing the number of families I allows to modify the intensity of competition, an infinite number of families corresponding to perfect competition. Note that the capital-labor ratio does not depend on I. Total labor per worker l satisfies m( l, l / I) = F 0 L ( k, 1) and is increasing with respect to the number of capitalists I. When I tends to +∞, it converges to l∞ solution of m( l∞ , 0) = F 0 L ( k, 1) < W c . Since l is increasing with I, total wealth per worker x = P I i=1 xi = k l/ (1 + µ) is also increasing with I. As a consequence, non-competitive behavior of capitalists results in lower capital accumulation, lower employment and lower income for workers.

Remark. If capitalists cannot borrow (µ = 0) and if the population growth rate n is zero, the equilibrium capital-labor ratio corresponds to the Modified Golden-Rule level, solution of γF 0 K (k, 1) = R. When n > 0 and µ > 0, two additional modifications affect the stationary capital-labor ratio. First, along a stationary growth path, a positive growth rate of worker population leads to a positive growth rate of the consumption of any capitalist. This augments long-run marginal return of wealth ρ. Second, with µ > 0, capitalists can borrow and use higher capital stock than with µ = 0. The capital-labor ratio k increases with respect to µ.

Dynamics with unconstrained capitalists

If no capitalist is constrained in period t, then all capital stocks K * it and labor demands L * it are identical among capitalists, and we have L * it = L * t /I = N t l * , where l * = l * (I) is the solution of Ig (l)-l = 0. Optimal income of capitalist i is

Π * it = F (K * it , L * it ) -R (K * it -X it ) -W t (L * t ) L * it = RX it + [W c -W t (L * t )] L * it
where the second equality is obtained appealing to linear homogeneity of the production function F and to the equality

F 0 K (K * i , L * i ) = R, which also implies F 0 L (K * i , L * i ) = W c
. Thus, we can rewrite the optimal income as

Π * it = RX it + b * N t (20)
where

b * = [W c -φ 0 (1 -l * )] l * I .
is the rent per worker of each capitalist. Non competitive behavior allows the capitalist to levy the rent b * on worker income. Differentiating the aggregate rent per worker Ib * with respect to I leads to

[W c -φ 0 (1 -l * ) + φ 00 (1 -l * ) l * ] dl * dI = • φ 00 (1 -l * ) µ l * - l * I ¶¸d l * dI
which is negative since l * is increasing with I. Therefore, the rent per worker for each capitalist b * is also decreasing with I and tends to zero as I becomes infinite. As I represents the intensity of competition, stronger competition tends to diminish the rent.

From equations ( 12) and ( 17), we obtain

C * it+1 C * it = (γR) σ ≡ G.
From the budget constraint Π * it = C * it + X * it+1 and (20), we have

X * it+1 = RX * it + b * N t -C * it (21)
Dynamics of wealth and consumption of capitalist i are characterized by the two-dimensional system that consists of these two last equations, where initial wealth X i0 is given.

Proposition 4. Let Assumptions 1, 2 and 3 be satisfied. If

γ ≥ (1 + n) 1/σ R ≡ γ and X i0 ≥ k c N 0 1 + µ , for i = 1, . . . , I,
then there exists a unique growth path where consumption and wealth of any capitalist asymptotically increase at factor G. Along this path, no capitalist is constrained. For the given initial wealth X i0 of family i, wealth X it is given by

X it = X i0 G t + b * N 0 R -(1 + n) £ G t -(1 + n) t ¤ (22)
and consumption C it grows at factor G.

Proof. In the constrained case, we have

C it + X it+1 = Π * it = F ((1 + µ) X it , L it ) -RµX it -W t (L t ) L it Since X it+1 ≥ 0 and 0 ≤ L it ≤ N t , C it ≤ C it + X it+1 ≤ F ((1 + µ) X it , L it ) ≤ F ((1 + µ) X it , N t )
and we deduce that

(1 + n) -t C it ≤ F ¡ (1 + µ) (1 + n) -t X it , N 0 ¢ . Now, the condition γ > (1 + n) 1/σ /R is equivalent to G > 1 + n and implies lim t→+∞ (1 + n) -t X it = +∞.
A similar argument applies in the unconstrained case. Indeed, we have

C it ≤ C it + X it+1 = Π * it < RX it + ω (R) L * it < RX it + ω (R) N t which implies (1 + n) -t C it < R (1 + n) -t X it + ω (R) N 0
and the result follows from the assumption G > 1 + n. Thus, at some date, the condition X it ≥ k(R)Nt 1+µ is satisfied, for all i and all capitalists are unconstrained. ¥ Proposition 3, 4 and 5 show that the long-run growth path depends crucially on the respective values of the natural growth factor 1 + n and the internationally driven growth rate G = (γR) σ . When 1 + n is higher (lower) than G, capitalists are constrained (unconstrained) in the long-run. Now, in the unconstrained case, workers benefit from higher wages and higher employment in the modern sector. These results suggest an interesting policy issue. In an economy with 1 + n > G, an appropriate economic policy can shift the economy towards the growth path where capitalists are unconstrained.

To this aim, the government may subsidy bequests and finance such payments with a lump-sum tax. The budget constraint of a capitalist writes

C it + (1 -τ ) X it+1 = Π it -θ t
where τ is the share of the bequests financed by the government through a lump-sum tax θ t . The corresponding Euler equation is

C it+1 C it = µ γ 1 -τ ∂Π * it+1 ∂X it+1 ¶ σ .
The condition for capitalists to be unconstrained in the long-run rewrites µ γR 1τ ¶ σ > 1 + n.

Thus, the unconstrained regime will be reached if the subsidy rate τ is high enough.

This result shows that subsidizing capitalists bequests improve the welfare of workers. Nevertheless, this policy may increase inequality among capitalists since the subsidy is proportional whereas the lump-sum tax is uniform.

Conclusion

We have explored the relationship between imperfect competition and capital accumulation in a two-sector economy with a borrowing constraint. Noncompetitive behavior of capitalists results in a rent which is extracted from the workers and lower employment in the modern sector.

In the long-run, the borrowing constraints remain binding if the internationally driven growth rate is lower than the natural one. In this case, imperfect competition decreases capital accumulation and is detrimental for growth. In the converse case, the borrowing constraint is not binding and imperfect competition is beneficial for capital accumulation and growth.

Finally, subsidy on bequests, that incites capitalists to accumulate wealth, allows the economy to take off and reach the unconstrained regime. Since employment and wages are higher in this case, such a policy is welfare-enhancing for workers.

Appendix : Equilibrium without borrowing constraint

If all capitalists are unconstrained, dynamics of consumptions and wealths are such that

X it+1 = RX it + bN t -C it C it+1 /C it = (γR) σ ≡ G
where X i0 is given. We are looking for initial consumption level C i0 such that the intertemporal budget constraint is satisfied with equality lim t-→+∞ R -t X it = 0.

Lemma A1. Assume R > N > 0, B > 0 and G > 0. For a given X 0 , there exist C 0 > 0 such that the difference equation

X t+1 = RX t + BN t -C 0 G t (23)
has a solution which satisfies lim t-→+∞ R -t X t = 0 if and only if G < R. This value of C 0 is unique and equal to

C 0 = (R -G) µ X 0 + B R -N ¶ .
Proof. We distinguish three cases depending on the sign of R -G.

• G > R. The general solution of ( 23) is written as

X t = λ 1 R t + λ 2 N t + λ 3 G t ( 24
)
where λ 1 , λ 2 and λ 3 have to be determined. The intertemporal budget constraint lim t-→+∞ R -t X t = 0 implies λ 1 = λ 3 = 0. Thus X t = X 0 N t . Substituting in (23), one obtains

C 0 µ G N ¶ t = (R -N) X 0 + B which contradicts C 0 > 0.
The growth rate of consumption cannot be higher than the interest rate.

• G = R. The general solution of (23) is written

X t = (λ 0 t + λ 1 ) R t + λ 2 N t
and the intertemporal budget constraint implies λ 0 = λ 1 = 0. Substituting X t = X 0 N t in (23), one still obtains than G = R is not consistent with C 0 > 0.

• G < R. The general solution of ( 23) is given by (24). The intertemporal budget constraint implies λ 1 = 0 and X t = λ 2 N t + λ 3 G t . By substitution in (23), we obtain

λ 2 N t+1 + λ 3 G t+1 = R ¡ λ 2 N t + λ 3 G t ¢ + BN t -C 0 G t .
For N 6 = G, we deduce that λ 2 = -B/ (R -N) and λ 3 = C 0 / (R -G).

Since X 0 = λ 2 + λ 3 , we have

C 0 = (R -G) • X 0 + B R -N ¸.
The same formula applies in the case N = G.¥ Under the assumption G < R (assumption 3 in the text), the unique solution of (23) writes

X t = X 0 G t + B R -N ¡ G t -N t ¢

2

 2

This kind of borrowing limit can result, for instance, from the enforcement problem[START_REF] Matsuyama | Endogenous Inequality[END_REF]). Specifically, the capitalist would refuse to honour its payment obligation RZ it if it is greater than the cost of default, which is taken to be a fraction of family wealth λX it . Thus, the lender would allow the capitalist to borrow only up to µX it , with µ = λ/R.

Proof. If no capitalist is constrained, then consumption C * it of capitalist i grows at factor G and, from (21), dynamics of his wealth are

Using Lemma A1 in the appendix, there exists a unique solution to this difference equation, given by ( 22). The assumption γ ≥ γ is equivalent to G ≥ 1 + n. Thus,

If all capitalists are unconstrained in period 0 and G ≥ 1 + n, then they are still unconstrained in any future period. ¥ Equation ( 22) can be compared to [START_REF] Gabszewicz | Oligopoly Equilibrium in Exchange Economies[END_REF] obtained in the case of perfect competition when all capitalists are unconstrained : X it = X i0 G t . The extra rent due to imperfect competition enhances wealth accumulation. In contrast to the case with binding borrowing constraints, the imperfect competition provides extra rent without modifying the marginal productivity of capital.

Considering two families i and j with different initial wealths, X i0 > X j0 , the inequality among capitalists will also hold in the long-run since

, which is larger than 1. But, it is also smaller than X i0 /X j0 that would be the limit of the ratio in the competitive case. Since the extra rent is the same for all capitalists, whatever their wealths, imperfect competition tends to reduce inequality among capitalists.

Transition

We have shown that, if γ > γ and if capitalists are not constrained initially, they accumulate enough wealth in each period for remaining unconstrained in all future periods. In the long-run, the economy follows a growth path with an asymptotic growth factor equal to G. Now, even if γ > γ, capitalist could be constrained initially because of insufficient wealth. The following proposition proves that they will become unconstrained in the long-run.

Proposition 5. If γ > (1 + n) 1/σ /R ≡ γ, whatever the initial wealth of capitalists, all capitalists becomes unconstrained.


Sylvain Guerber
email: sylvain.guerber@st.com

Carlos Alonso-Ramos

Daniel Benedikovic

Elena Duran-Valdeiglesias

Xavier Le Roux

Nathalie Vulliet

Eric Cassan

Delphine Marris-Morini

Charles Baudot

Frédéric Boeuf

E Durán-Valdeiglesias

Laurent Vivien

Broadband Polarization Beam Splitter on a Silicon Nitride Platform for O-Band Operation

Keywords: Optical polarization, Photonic integrated circuits, Silicon photonics, Silicon Nitride, Polarization beam splitter, Wavelength division multiplexing

come   L'archive ouverte pluridisciplinaire

I. INTRODUCTION

N recent years, emerging technologies from Cloud services (computing, working, streaming, gaming) together with Big Data are driving Internet traffic towards unprecedented density and speed levels [START_REF]Global Cloud Index 2015-2020[END_REF]. A key point to this ramp up is that more than 75% of the global data traffic occurs inside datacenters [START_REF]Global Cloud Index 2015-2020: Forecast and Methodology[END_REF], i.e. in a short distance range. Within this context, optical data communication links in the original telecom band (O-band) are spreading widely due to standardization. Silicon photonics, based on CMOS industrial manufacturing infrastructure, can fulfill those new requirements in terms of bandwidth and costs. However, silicon photonics circuits mostly rely on the high-indexcontrast (∆n≈2) silicon-on-insulator (SOI) platform, that suffers from high temperature sensitivity, tight fabrication tolerances, strong polarization dependence and large waveguide dispersion which in turn significantly constraints the devices performance. These limitations may be overcome by co-integration of silicon nitride (SiN) [START_REF] Baudot | Developments in 300mm silicon photonics using traditional CMOS fabrication methods and materials[END_REF], a CMOS compatible material. Indeed, SiN has a seven-fold lower thermo-optical coefficient compared with silicon [START_REF] Zhang | A Silicon Nitride Platform by PVD for Dense Wavelength Division Multiplexing on Chip[END_REF]. Moreover, at the cost of a slightly larger footprint compared to SOI, SiN reduced index contrast (∆n≈0.5) leads to relaxed fabrication tolerances, efficient chip-fiber coupling as well as reduced wavelength-dependence of devices operation. Taking benefit of those properties, various applications have been demonstrated such as integrated spectrometers [START_REF] Nie | CMOScompatible broadband co-propagative stationary Fourier transform spectrometer integrated on a silicon nitride photonics platform[END_REF], quantum photonic circuits [START_REF] Poot | Design and characterization of integrated components for SiN photonic quantum circuits[END_REF] and low power Kerr-comb generation [START_REF] Xuan | High-Q silicon nitride microresonators exhibiting low-power frequency comb initiation[END_REF].

Light polarization management has been recognized as a critical functionality in photonic chips over the years. Polarization splitting grating couplers (PSGC) [START_REF] Bogaerts | A polarization-diversity wavelength duplexer circuit in silicon-on-insulator photonic wires[END_REF], smartly implement this function together with chip-fiber coupling. Yet, their narrow bandwidth is incompatible with wavelength division multiplexing (WDM) applications. Another promising solution consists in using a broadband and polarization independent fiber-chip interface based on edge coupling [START_REF] Kasaya | A simple laterally tapered waveguide for lowloss coupling to single-mode fibers[END_REF][START_REF] Barwicz | An o-band metamaterial converter interfacing standard optical fibers to silicon nanophotonic waveguides[END_REF] together with a wideband polarization beam splitter (PBS). Compact and wideband PBS have been demonstrated in the SOI platform, based on highly birefringent bent directional couplers (DC) [START_REF] Dai | Novel ultra-short and ultrabroadband polarization beam splitter based on a bent directional coupler[END_REF]. However, this approach is not well suited for the lower index contrast SiN platform. Wideband SOI PBS have also been shown [START_REF] Lu | Wideband silicon photonic polarization beamsplitter based on point-symmetric cascaded broadband couplers[END_REF], based on phase-controlled DC [START_REF] Lu | Broadband silicon photonic directional coupler using asymmetric-waveguide based phase control[END_REF]. In this case, the large birefringence of SOI precludes simultaneously achieving both, proper coupling ratios for orthogonal polarizations and wideband operation. To circumvent this limitation, two wideband half splitters need to be cascaded, complicating device implementation. In this work, we propose and experimentally demonstrate a novel and flexible strategy that leverages the low birefringence of the SiN to implement wideband PBS with a single phase-controlled DC. The device is optimized to operate in the O-band, targeting WDM systems for datacom applications.

II. DESIGN AND SIMULATION

The PBS proposed here exploits the phase-controlled DC Sylvain Guerber , Carlos Alonso-Ramos, Daniel Benedikovic, Elena Durán-Valdeiglesias, Xavier Le Roux, Nathalie Vulliet, Eric Cassan, Delphine Marris-Morini, Charles Baudot, Frédéric Boeuf and Laurent Vivien

Broadband polarization beam splitter on a silicon nitride platform for O-band operation I concept to yield wideband operation [START_REF] Lu | Broadband silicon photonic directional coupler using asymmetric-waveguide based phase control[END_REF]. As shown on Fig. 1, the PBS comprises a phase-controlled DC, which can be divided into three parts: two symmetric coupling sections at each end, and a phase control region in between. According to this configuration, the wavelength-dependent behavior of the first symmetrical coupler section is compensated introducing a small phase shift prior to the second section.

The thickness of the SiN layer is 600 nm, its refractive index is 1.92 at 1.3µm wavelength. In order to ensure single mode operation simultaneously for both orthogonal transverse electric (TE) and transverse magnetic (TM) polarizations, waveguide width (W std ) is fixed to 700 nm. Coupling section gap (G) is set to 600 nm as a tradeoff between footprint and fabrication sensitivity.

Waveguide widths in the phase control section are respectively set to 850 nm for top arm (W w ), limiting high order mode excitation, and 550 nm for bottom arm (W n ), avoiding additional losses. Transition tapers of 6 µm length (L t ) are used to provide smooth mode transformation (low losses) while maintaining a compact footprint. S-bends are based on 40 µm radius circular curves.

First, the device is analyzed using a semi-analytical transfer matrix modeling (TMM) approach [START_REF] Lu | Broadband silicon photonic directional coupler using asymmetric-waveguide based phase control[END_REF]. Here, each particular section of the PBS is depicted by a transfer matrix as follows:

(1) Input and output electric fields, E 1 to 4 , are as described in Fig. 1. Matrices C stand for symmetric coupler sections, whereas the transition tapers and the phase control region are described by Pt and P, respectively. Transfer matrices account for waveguide properties such as modal effective indices, losses and section length. Since no coupling is considered within the transition tapers and phase control sections, P t and P stand as diagonal matrices. In this modeling approach, the coupling occurring in the S-bend sections is not taken into account. Modal effective indices of each particular sections are calculated using a full-vectorial finite difference eigenmode (FDE) solver [14].

From this semi-analytical approach, two important parameters [START_REF] Lu | Broadband silicon photonic directional coupler using asymmetric-waveguide based phase control[END_REF] are extracted:

(2)

(3) First one, as stated in (2), is the cross-coupling coefficient, η, which is defined as the normalized cross-power ratio between the two output arms. Second parameter, defined in (3), is the cross-coupling coefficient variation over 100 nm bandwidth, ∆η, accounting for the device wavelength sensitivity. Both parameters have been computed for TE (η TE , ∆η TE ) and TM (η TM , ∆η TM ) polarizations. To perform the optimization of the splitting efficiency and the bandwidth of the PBS, we have defined two figures of merit, and , that relate aforementioned parameters as:

(4)

, as stated in (4), represents the PBS splitting efficiency, with the best splitting performance obtained for . Whereas , defined in ( 5), stands for the PBS wavelength sensitivity; the lower , the broader bandwidth. Coupler characteristics, η PBS and ∆η PBS , are then tuned by adjusting the length of symmetrical couplers L c , and the length of phase control section L pc . The results are respectively reported in Fig. 2 Fig. 2(a) reveals two (L c, L pc ) couples, denoted by C1 and C2, that achieve efficient polarization splitting. Those specific configurations occur thanks to the slightly birefringent response of the SiN waveguides. Indeed, different phase shift and coupling strength are introduced for TE and TM polarizations allowing the phase-controlled DC to act as a PBS. As shown, for L c = 36.2 µm and L pc = 27.8 µm (C1), η PBS = 0.96 while for L c = 36.6 µm and L pc = 18.4 µm (C2), η PBS = 0.97. Fig. 2(b) shows that ∆η PBS is slightly higher for C1 (∆η PBS ≈ 7.2%) compared to C2 (∆η PBS ≈ 5.5%). Consequently, with a higher splitting efficiency and a lower wavelength sensitivity, the second (L c, L pc ) couple (C2) is selected for the final device design.

The selected design is fine-tuned using three-dimensional Finite Difference Time Domain (3D-FDTD) [START_REF] Lu | Broadband silicon photonic directional coupler using asymmetric-waveguide based phase control[END_REF] simulations. Considering the S-bend contributions, the optimal values for the coupling and phase control regions are determined to be L c = 30 µm and L pc = 18 µm. Fig. 3 shows the propagation of both TE and TM fields for the optimized design at a nominal wavelength of 1300 nm.

III. FABRICATION AND CHARACTERIZATION

Devices have been fabricated on the 300 mm silicon photonics R&D platform DAPHNE in ST Crolles (France) [START_REF] Baudot | Developments in 300mm silicon photonics using traditional CMOS fabrication methods and materials[END_REF][START_REF] Guerber | Integrated SiN on SOI dual photonic devices for advanced datacom solutions[END_REF]. We used silicon bulk wafers with 1.4 µm-thick silicon dioxide (SiO 2 ) layer as a buried oxide. A 600 nm-thick SiN layer is then deposited at low temperature via plasma enhanced chemical vapor deposition (PECVD). SiN patterning is performed using 248 nm deep-ultraviolet (deep-UV) optical photolithography, followed by dry etching process. An encapsulation layer of a 1.5 µm thick silicon dioxide is finally deposited to complete the fabrication process.

Individual dies have been singled out to perform optical characterization in an edge coupling scheme. High precision stealth dicing has been carried out to achieve a good lateral surface uniformity. Consequently, reliable edge coupling is obtained leading to repeatable coupling efficiency for all waveguides included within the sample.

Devices optical testing has been performed over a wavelength range from 1260 nm to 1355 nm. Light is injected in the bottom arm (E1) and collected for both outputs (E3 and E4) and for both TE and TM polarizations. The control of light polarization state has been performed using standard fiber paddle polarization controller together with calcite-made polarizers in order to proper reject the unwanted polarization states on a large wavelength range. The measured spectral responses are normalized with a straight waveguide transmission in the respective polarization state. Device performance is then evaluated from the insertion losses (IL) and the extinction ratio (ER), which is defined by the power difference between both polarizations in a given output arm.

Fig. 4(a) shows the experimental spectral response of a single PBS. Insertion losses lower than 0.6 dB and ER above 10 dB are achieved for both TE and TM polarizations, over the 95 nm wavelength range. This confirms the simulated broadband behavior of our devices.

However, some applications like coherent communication systems [START_REF] Savory | Digital coherent optical receivers: algorithms and subsystems[END_REF] typically require comparatively higher ER levels. Therefore, a cascaded structure comprising three PBS has been realized in order to achieve stronger polarization filtering. This circuit configuration is shown in the inset of Fig. 4(b). As a result, two times improvement in respect to ER is achieved, reaching more than 20 dB for both polarizations over 95 nm bandwidth while keeping the IL below 1.3 dB. An ER level of 24 dB is even achieved over 85 nm bandwidth with IL lower than 1.1 dB.

Comparison with state of the art is presented in Table I. The reported devices exhibit comparable performances with C and L-band SOI-based PBS in terms of insertion losses, extinction ratio, and bandwidth. It is also worth noting that the wideband PBS's realized in this work are the first experimental demonstration of such photonic components on a SiN platform, completely fabricated in an industrial environment. These results pave the way towards implementation of dualpolarization applications, such as polarization diversity or coherent communications, in a CMOS-compatible SiN material system, taking advantage of the platform stability related to environment and fabrication.

IV. CONCLUSION

In conclusion, a novel architecture of broadband polarization beam splitter is proposed and realized using a 300 mm SiN platform. The devices were seamlessly fabricated in a single-etch step process by conventional 248-nm deep-UV optical lithography, typically used by industrially-driven photonic applications. Optical characterizations demonstrated insertion loss lower than 0.6 dB and extinction ratio larger than 10 dB for both TE and TM polarizations, covering the full O-band. Furthermore, a cascaded architecture achieves insertion losses below 1.3 dB, with an extinction ratio larger than 20 dB on the same wavelength range.

  (a) and 2(b).

Fig. 2 .

 2 Fig. 2. Filled contour maps of PBS splitting efficiency, ηPBS (a), and wavelength sensitivity, ΔηPBS (b), as function of Lc and Lpc. Red crosses denoted C1 and C2 shows two interesting (Lc, Lpc) couples for PBS design.

Fig. 1 .

 1 Fig. 1. In-plane (top) schematic view of the proposed polarization beam splitter.

Fig. 4 .

 4 Fig. 4. Simulated and experimental spectral response for (a) single PBS and (b) cascaded PBS. Insets shows top view of devices from optical microscope with polarization response.

Fig. 3 .

 3 Fig. 3. Optical field distribution in the PBS from 3D-FDTD for (a) TE and (b) TM polarizations at λ = 1.3 µm.

  This work, design of record; WK2: This work, cascaded version; WK3: This work, cascaded version, high ER.

				TABLE I
		COMPARISON WITH STATE OF THE ART
	Ref.	BW (nm) IL (dB)/	(dB) ER	(nm) Wavelength	(L×l) µm Footprint	Mat.
	[12]	>NA/125 >20 1477-1602	97.4×5.5	SOI
	[17]	>1/35	>17 1530-1565	4.8×1.6	SOI/
						SiN
	[18]	>~0.8/95	>25 1500-1595	20×6.9	SOI
	[19]	>1.3/100 >9.9 1470-1570	5.8×NA	SOI
	[20]	>2.2 /55	>10 1525-1580 132.6×4.2 SOI
	[21]	>3.4/150	>10 1350-1500 1400×1.8 SOI
	WK1	>0.6/95	>10 1260-1355	136×8.7	SiN
	WK2	>1.3/95	>20 1260-1355	-	SiN
	WK3	>1.1/85	>24 1270-1355	-	SiN
	IL = Insertion Loss, BW = Bandwidth, ER = Extinction Ratio.
	WK1:

This work was supported in part by the European Research Council (ERC POPSTAR) under Grant 647342, in part by the European Commission H2020-ICT-27-2017 (COSMICC) under Grant 688516 and in part by the French Industry Ministry Nano2017 program. S. Guerber is with the Centre de Nanoscience et Nanotechnologies (C2N),


Jean Bouin

Christian Dolbeault

Schmeiser

Partial differential equations -A variational proof of Nash's inequality, by EMERIC

Keywords: Nash inequality, interpolation, semi-linear elliptic equations, compactness, compact support, Neumann homogeneous boundary conditions, Laplacian, radial symmetry 35J20, 26D10, Secondary: 37B55, 49K30

This paper is intended to give a characterization of the optimality case in Nash's inequality, based on methods of nonlinear analysis for elliptic equations and techniques of the calculus of variations. By embedding the problem into a family of Gagliardo-Nirenberg inequalities, this approach reveals why optimal functions have compact support and also why optimal constants are determined by a simple spectral problem.

INTRODUCTION AND MAIN RESULT

Nash's inequality [START_REF] Nash | Continuity of solutions of parabolic and elliptic equations[END_REF] states that, for any u ∈ H 1

(R d ), d ≥ 1, u 2+ 4 d 2 ≤ C Nash u 4 d 1 ∇u 2 2 , (1)
where we use the notation v q = R d |v| q d x 1/q for any q ≥ 1. The optimal constant C Nash in (1) has been determined by E. Carlen and M. Loss. To state their result, let us introduce ω d , the volume of the unit ball B 1 in R d , λ 1 , the principal eigenvalue of the Laplacian with homogeneous Neumann boundary conditions, and x → ϕ 1 (|x|) an eigenfunction associated with λ 1 , normalized by ϕ 1 (1) = 1.

THEOREM 1.1 ( [9]). Inequality (1) holds with optimal constant

C Nash = (d + 2) 1+ 2 d d λ 1 (2 ω d ) 2 d (2)
Moreover, there is equality in [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF] if and only if, up to translation and scaling of x as well as multiplication of u by a constant, u(x) := 1 -ϕ 1 (|x|) for |x| ≤ 1 , 0 for |x| > 1 .

The compactness of the support of the optimizers in (1) can be understood by deriving Nash's inequality as a limiting case of a family of Gagliardo-Nirenberg inequalities [START_REF] Gagliardo | Proprietà di alcune classi di funzioni in più variabili[END_REF][START_REF] Nirenberg | On elliptic partial differential equations[END_REF] Nash's inequality corresponds to the limit case as p → 1.

The inequality (3) is equivalent to the minimization of u → ∇u 2 2 + u 2 p under the constraint that u 2 2 is a given positive number: any minimizer solves, up to a scaling and a multiplication by a constant, the Euler-Lagrange equation

-∆u = u -|u| p-2 u . ( 4
)
In our main result, which follows, we use the notation of Theorem 1.1.

THEOREM 1.2. For any p ∈ (1, 2), equality in (3), written with an optimal constant, is obtained after a possible translation and scaling of x, and multiplication by a constant, by the nonnegative radial solution u p of (4). The support of u p is a ball of radius R p > 0, such that lim p→1 + R p = R 1 := λ 1 , and u p converges, as p → 1 + , to

u 1 = u(•/R 1 ) in H 1 ∩ L 1 (R d ).
Our result is in the spirit of [START_REF]Sharp constant in Nash's inequality[END_REF]. In their paper, E. Carlen and M. Loss establish the optimality case by direct estimates as chain of inequalities which become equalities in the case of the optimal functions. Our contribution is to establish first why the problem can be reduced to a problem on a ball involving only radial functions, using the theory of nonlinear elliptic PDEs and methods of the calculus of variations. Identifying the optimal case is then an issue of spectral theory.

Let us highlight a little bit why we insist on the compactness of the support of the optimal function. The first application of Nash's inequality by Nash himself in [START_REF] Nash | Continuity of solutions of parabolic and elliptic equations[END_REF] is the computation of the decay rate for parabolic equations. We consider a solution u of the heat equation

∂u ∂t = ∆u , (5)
with initial datum

u 0 ∈ L 1 ∩ L 2 (R d ). If u 0 is nonnegative, so does u(t , •),
y ≤ -2 C -1 Nash u 0 -4 d 1 y 1+ 2 d ,
which, after integration, yields that the solution u of (5) satisfies the estimate

u(t , •) 2 ≤ u 0 -4 d 2 + 4 d C -1 Nash u 0 -4 d 1 t -d 4 , (6)
for all t ≥ 0. This estimate is optimal in the following sense: if we take u 0 to be an optimal function in (1) and differentiate (6) at t = 0, it is clear that C Nash is the best possible constant in the decay estimate [START_REF] Bendikov | Nash type inequalities for fractional powers of non-negative self-adjoint operators[END_REF].

With this result at hand, it comes a little bit as a surprise that the optimal function has nothing to do with the heat kernel

G(t , x) = 1 (4π t ) d 2 exp - |x| 2 4 t
and is even compactly supported. This can be explained by noting that ( 6) is an optimal result for small times, but it can be improved concerning the long time behavior. Estimation of the solution u(t , •) = G(t , •) * u 0 of the heat equation by Young's convolution inequality gives

u(t , •) 2 ≤ G(t , •) 2 u 0 1 = (8π t ) -d 4 u 0 1 . (7)
The sharpness of this result (take u 0 = G(ε, •) for an arbitrary small ε > 0) and a comparison with (6) imply 8π > 4

d C Nash . ( 8
)
An estimate, which is optimal for both small and large times, is now obviously obtained by taking the minimum of the right hand sides of (6) and of [START_REF] Lions | Nonlinear scalar field equations. I. Existence of a ground state[END_REF]. The reader interested in further consideration on Nash's inequality and the heat kernel is invited to refer to [START_REF] Davies | Heat kernels and spectral theory[END_REF].

This paper is organized as follows. Theorem 1.1 can be seen as a consequence of Theorem 1.2: see Section 2. Section 3 is devoted to the proof of Theorem 1.2. In Section 4 we adopt a broader perspective and sketch the proof by E. Carlen and M. Loss along with a review of several other methods for proving Nash's inequality.

THEOREM 1.2 IMPLIES THEOREM 1.1

The optimal constant C GN (p) in ( 3) is obtained by minimizing the quotient

Q p [u] := ∇u 2 a a+b 2 u 2 b a+b p u 2 2 .
Except when d = 1, no explicit expression of C GN (p) is available to our knowledge. Theorem 1.2 implies

Q 1 [u 1 ] = lim p→1 + Q p [u p ] = C Nash d d +2 .
As a consequence, the support of u 1 is the ball B R 1 and on ∂B R 1 x, u 1 (x) = 0 and x • ∇u 1 (x) = 0. We also deduce from Theorem 1.2 that u p uniformly converges to u 1 using standard elliptic results of [START_REF] Trudinger | Elliptic partial differential equations of second order[END_REF] or elementary ODE estimates. Moreover, since lim p→1 + u p-1 p = 1, the optimal function u 1 solves the Euler-Lagrange equation

-∆u 1 = u 1 -1 . This means that v 1 := 1 -u 1 (R 1 •) solves -∆v 1 = λ 1 v 1 , x ∈ B 1 , v 1 = 1 , x • ∇v 1 = 0 , x ∈ ∂B 1 , with λ 1 = R 2 1 . This implies v 1 = ϕ 1 (| • |).
We recall that λ 1 is defined by

λ 1 := inf B 1 |∇ϕ| 2 d x
where the infimum is taken on ϕ ∈ H 1 (B 1 ) :

B 1 ϕ d x = 0 and B 1 |ϕ| 2 d x = 1 .
As observed in [START_REF] Beckner | Geometric proof of Nash's inequality[END_REF], λ 1 can be computed in terms of the smallest positive zero z d /2 of the Bessel function of the first kind J d /2 using, for instance, [11, page 492, Chapter VII, Section 8]. Indeed, solving the radial eigenvalue problem

ϕ 1 + d -1 r ϕ 1 + λ 1 ϕ 1 = 0 , ϕ 1 (1) = 1 , ϕ 1 (1) = 0 , ϕ 1 (0) = 0, is equivalent to finding the function J such that ϕ 1 := r -α J α λ 1 • with α = (d -
2)/2 that solves

J α + 1 r J α + 1 - α 2 r 2 J α = 0 .
Hence J α is a Bessel function of the first kind and the boundary condition

ϕ 1 (1) = 0 is transformed into λ 1 J α λ 1 -α J α λ 1 = 0. With the property z J α (z) -α J α (z) + J α+1 (z) = 0
of Bessel functions of the first kind (see, e.g., [START_REF] Watson | A Treatise on the Theory of Bessel Functions[END_REF]), we obtain

λ 1 = z 2 α+1 = z 2 d /2 .
Looking at the Euler-Lagrange equation was precisely the way how E. Carlen and M. Loss realized that the optimizers must have compact support, [8]. This information heavily influenced the shape of the proof in [START_REF]Sharp constant in Nash's inequality[END_REF]. The observations of this section point in the very same direction. For p ∈ (1, 2), the compact support principle gives a very simple intuition of why the optimal function for (3) has compact support, and then it is natural to take the limit in the Euler-Lagrange equation ( 4). As we shall see in the next section, symmetrization can be avoided and replaced by the moving plane method, which also puts the focus on (4).

PROOF OF THEOREM 1.2

For readability, we divide the proof into a list of simple statements. Some of these statements are classical and are not fully detailed. Our goal here is to prove that the optimal function u p is supported in a ball with finite radius and investigate the limit of u p as p → 1 + .

1. The non-optimal inequality and the optimal constant. It is elementary to prove that (1) holds for some positive constant C Nash and without loss of generality, we can consider the best possible one. See Section 4 for various proofs of such a statement.

Scalings and Gagliardo-Nirenberg inequalities. Inequality (3) is equivalent to

∇u 2 2 + λ u 2 p ≥ K GN (p, λ) u 2 2 , (9)
for all u ∈ H 1 (R d ) ∩ L p (R d ). A scaling argument relates K GN (p, λ) and C GN (p). Indeed, take any u ∈ H 1 (R d ) ∩ L p (R d ), then so does u σ = u(σ•) for any σ > 0, and we have

K GN (p, λ) ≤ ∇u σ 2 2 + λ u σ 2 p u σ 2 2 = σ 2 ∇u 2 2 + λ u 2 p σ -2 a b u 2 2 .
where a = d (2p), and b = 2 p as in [START_REF] Bakry | Weighted Nash inequalities[END_REF]. From this, we may deduce two things.

First, choosing σ 2 = λ b a+b and optimising on u yields

K GN (p, λ) = K GN (p, 1) λ b a+b .
Second, an optimization on σ > 0 shows that

K GN (p, λ) ≤ a + b a a a+b b b a+b λ b a+b Q p [u]
so that optimizing on u yields

K GN (p, λ) = a + b a a a+b b b a+b λ b a+b C GN (p) . (10)
Throughout this paper we assume that C GN (p) and K GN (p, λ) are the optimal constants respectively in (3) and ( 9). by Hölder's inequality, we find that

C GN (p) ≤ C Nash a(p) a(p)+b(p) p→1 + -→ C Nash d d +2 ,
from which we conclude that

C Nash d d +2 = lim p→1 + C GN (p) .
4. Nonnegative optimal functions. We look for optimal functions in (3) by minimizing Q p . The existence of a minimizer is a classical result in the calculus of variations: see for instance [START_REF] Lions | Nonlinear scalar field equations. I. Existence of a ground state[END_REF]. Without loss of generality, we can consider only nonnegative solutions of (4) because

Q p [u] = Q p |u| .
5. Support and regularity. According to [START_REF] Cortázar | Uniqueness of positive solutions of ∆u + f (u) = 0 in R N , N ≥ 3[END_REF], as a special case of the compact support principle (see [START_REF] Pucci | A strong maximum principle and a compact support principle for singular elliptic inequalities[END_REF][START_REF]A strong maximum principle and a compact support principle for singular elliptic inequalities[END_REF]), nonnegative solutions of (4) have compact support. By convexity of the function t → t 2/p , we can consider solutions which have only one connected component in their support. For each p, we can pick one such solution and denote it by u p . The standard elliptic theory (see for instance [START_REF] Trudinger | Elliptic partial differential equations of second order[END_REF]) shows that the solution is continuous and smooth in the interior of its support, which is a connected, closed set in R d .

6. Symmetry by moving planes. According to [START_REF] Cortázar | Uniqueness of positive solutions of ∆u + f (u) = 0 in R N , N ≥ 3[END_REF], the solution u p is radially symmetric and supported in a ball of radius R p . The proof relies on moving plane techniques and applies to nonlinearities such as u → uu p-1 with p ∈ (1, 2). Up to a translation we can assume that the support is a centered ball. The solution is unique according to [10, Theorem 1.1] and [START_REF] Peletier | Uniqueness of nonnegative solutions of semilinear equations in R n[END_REF]Theorem 3].

7. Limit as p → 1 + . This is the point of the proof which requires some care. As p → 1 + , u p converges to a nonnegative solution u 1 of

   -∆u 1 = u 1 -1 on B R 1 , x |x| • ∇u 1 = 0 , u 1 = 0 on ∂B R 1 . ( 11
)
Let us give some details. For any p ∈ (1, 2), the solution u p of ( 4) satisfies the two identities because of [START_REF] Cortázar | Uniqueness of positive solutions of ∆u + f (u) = 0 in R N , N ≥ 3[END_REF]. As a consequence we have that lim p→1 + λ p = λ 1 . The two identities and the definition of λ p provide an explicit expression of ∇u p 2 , u p p and u p 2 in terms of λ p . This proves the strong convergence of u p to some u 1 in H 1 ∩ L 1 (R d ) and, as a consequence, this proves that equality in (1) is achieved by u 1 which solves [START_REF] Courant | Methods of mathematical physics[END_REF].

∇u p 2 2 + u p p p = u p 2 2 , d -2 2 d ∇u p 2 2 + 1 p u p p p = 1 2 u p
8. Convergence of the support. By adapting the results of [START_REF] Balabane | Nodal solutions for a sublinear elliptic equation[END_REF][START_REF] Dolbeault | Qualitative properties and existence of sign changing solutions with compact support for an equation with a p-Laplace operator[END_REF], it is possible to prove that lim p→1 + R p =: R 1 ∈ (0, ∞). Let us give an elementary argument. The dynamical system

U = V , V = 1 -U - d -1 r V
has smooth solutions such that, if initial data coincide, then U (r ) = u 1 (r ) and V = u 1 (r ) on the support of u 1 . As a limit of u p , we know that u 1 is nonnegative. Since the sequence (r n ) n∈N of the zeros of V , with the convention r 0 = 0, is such that the sequence (-1) n U (r n ) n∈N is monotone decaying, u 1 has compact support and R 1 = r 1 . This completes the proof of Theorem 1.2.

OTHER PROOFS OF NASH'S INEQUALITIES

This section is devoted to a brief review of various methods that have been used to derive Nash's inequality and estimate the optimal constant C Nash . The method of E. Carlen and M. Loss is the only one which provides the optimal value of C Nash . A comparison of the various results is summarized in Fig. 1.

Interpolation with Sobolev's inequality

In dimension d ≥ 3, it follows from Hölder's inequality that

u 2 2 ≤ u 4 d +2 1 u 2 d d +2 2 * with 2 * = 2 d d -2
, and from Sobolev's inequality that

u 2 2 * ≤ S d ∇u 2 2
where

S d = 1 d (d -2) π Γ(d ) Γ(d /2) 2 d
is the optimal constant in Sobolev's inequality [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF][START_REF] Rodemich | The Sobolev inequalities with best possible constants[END_REF][START_REF] Talenti | Best constant in Sobolev inequality[END_REF], so that C Nash ≤ S d .

Nash's inequality and the logarithmic Sobolev inequality are equivalent

In the interpolation strategy, we can replace Sobolev's inequality by the logarithmic Sobolev inequality. The equivalence of Nash's inequality and the logarithmic Sobolev inequality is a well known fact, in which optimality of the constants is not preserved: see for instance [START_REF] Bakry | On Sobolev and logarithmic Sobolev inequalities for Markov semigroups[END_REF]. In [START_REF] Beckner | Geometric proof of Nash's inequality[END_REF], W. Beckner made the following observation: Jensen's inequality applied to the convex function σ(u) := log(1/u) and the probability measure u -2

2 |u| 2 d x shows that log u 2 2 u 1 = σ u 1 u 2 2 = σ R d 1 |u| |u| 2 d x u 2 2 ≤ R d σ 1 |u| |u| 2 d x u 2 2 = R d log |u| |u| 2 d x u 2 2
and can be combined with the logarithmic Sobolev inequality in scale invariant form of [START_REF] Stam | Some inequalities satisfied by the quantities of information of Fisher and Shannon[END_REF][START_REF] Weissler | Logarithmic Sobolev inequalities for the heat-diffusion semigroup[END_REF], that is, According to [START_REF] Beckner | Geometric proof of Nash's inequality[END_REF], this estimate is asymptotically sharp as d → ∞, which can be verified by using classical results on the zeroes of Bessel functions [START_REF] Watson | A Treatise on the Theory of Bessel Functions[END_REF]. Notice that an information-theoretic proof of Nash's inequality based on Costa's method of entropy powers can be used to directly prove this inequality (see for instance [START_REF] Villani | A short proof of the "concavity of entropy power[END_REF], with application to Nash's inequality in [START_REF] Toscani | An information-theoretic proof of Nash's inequality[END_REF]Section 4]).

R d log |u| |u| 2 d x u 2 2 ≤ log u 2 + d 4 log 2 π d e ∇u 2
Reciprocally, let us consider Hölder's inequality u q ≤ u α p u 1-α s with α = p q s-q s-p , p ≤ q ≤ s, and let us take the logarithm of both sides. Then we obtain

log u q u p + (α -1) log u p u s ≤ 0 .
This inequality becomes an equality when q = p. We may differentiate it with respect to q at q = p and

R d u p log |u| u p d x ≤ s s -p u p log u s u s . ( 12
)
immediately follows.

The equivalence of Nash's inequality and the logarithmic Sobolev inequality (although with non-optimal constants) is a well known fact that has been exploited in various related problems. See for instance [START_REF] Bendikov | Nash type inequalities for fractional powers of non-negative self-adjoint operators[END_REF][START_REF] Bakry | Weighted Nash inequalities[END_REF].

The original proof by J. Nash

It is a very simple argument based on Fourier analysis, due originally to E.M. Stein according to J. Nash himself (see [17, page 935]). Let us denote by û the Fourier transform of u defined as

û(ξ) = (2π) -d 2 R d e i x ξ u(x) d x , so that by Plancherel's formula u 2 2 = R d | û(ξ)| 2 d ξ ≤ û 2 ∞ |ξ|≤R d ξ + 1 R 2 R d |ξ| 2 | û(ξ)| 2 d ξ ≤ (2π) -d ω d R d u 2 1 + 1 R 2 ∇u 2 2
for any R > 0. By optimizing the right hand side with respect to R, we obtain the estimate

C Nash ≤ 1 4 π d + 2 d 1+ 2 d Γ d 2 -2 d =: C 2 (d ) .

The method of E. Carlen and M. Loss

Without loss of generality, we can assume that the function u is nonnegative. If u denotes the spherically non-increasing rearrangement of a function u, then [START_REF] Lieb | Analysis[END_REF] u q = u q and ∇u 2 ≤ ∇u 2 , so we can consider nonnegative radial non-increasing functions without loss of generality. For any R > 0, let u R := u 1 B R .

We observe that

u -u R ≤ u(R) ≤ u R := u R 1 |B R | because u is radial non-increasing, so that u -u R 2 2 ≤ u R u -u R 1 = u R 1 R d ω d u -u R 1 . (13)
On the other hand, using

u R 2 2 = u R -u R 2 2 + u R 1 B R 2 2 ,
we deduce from the Poincaré inequality

B R |v| 2 d x ≤ R 2 λ 1 B R |∇v| 2 d x,
for all v ∈ H 1 (R d ) such that B R v d x = 0 and from the definition of u R that u R

2 2 ≤ R 2 λ 1 ∇u 2 2 + u R 2 1 R d ω d , (14)
using ∇u R 2 ≤ ∇u 2 . By definition of u R , we also know that

u 2 2 = u R 2 2 + u -u R 2 2 .
After summing ( 13) and ( 14), we arrive at

u 2 2 ≤ R 2 λ 1 ∇u 2 2 + u R 1 R d ω d u R 1 + u -u R 1 and notice that u R 1 u R 1 + u -u R 1 = u R 1 u 1 ≤ u 2 1 . (15)
Altogether, this means that

u 2 2 ≤ R 2 λ 1 ∇u 2 2 + u 2 1 R d ω d .
An optimization on R > 0 determines a unique optimal value R = R and provides the expression of C Nash . Equality is achieved by functions meeting equalities in all previous inequalities, that is functions such that u = u R and u R is an optimal

  ∇u

  u ∈ H 1 ∩ L p (R d ), where 1 < p < 2, a = a(p) = d (2p), and b = b(p) = 2 p.

3 .

 3 Comparison of the optimal constants. By taking the limit in Q p [u] as p → 1 + for an arbitrary smooth function u and arguing by density, one gets that p → C GN (p) is lower semi-continuous and

2 2 , 2 2 + λ p u p 2 p = u p 2 2 with λ p := u p p- 2 p

 222222 obtained by testing (4) respectively by u p and by x • ∇u p (Pohozaev's method). The first identity is rewritten as ∇u p . As a consequence, we have that 1 = K GN (p, λ p ) = a +

Figure 1 :

 1 Figure 1: With d ≥ 1 considered as a real parameter, plots of d → S d and d → C 1 (d ) corresponding respectively to an interpolation with Sobolev's inequality (only for d > 2) and to an interpolation with the logarithmic Sobolev inequality are shown as dashed curves. The dotted curve is the estimate d → C 2 (d ) of J. Nash in[START_REF] Nash | Continuity of solutions of parabolic and elliptic equations[END_REF]. The optimal value d → C Nash (d ) is the plain curve and it is numerically well approximated from below by d → 1/(2πd ) deduced from(8).

© 2018 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.

Acknowledgments: This work has been partially supported by the Projects EFI (E.B., J.D., ANR-17-CE40-0030) and Kibord (E.B., J.D., ANR-13-BS01-0004) of the French National Research Agency (ANR). The work of C.S. has been supported by the Austrian Science Foundation (grants no. F65 and W1245), by the Fondation Sciences Mathématiques de Paris, and by Paris Science et Lettres. All authors participate in an Austrian-French PHC project Amadeus.