Raouf Boucekkine

Giorgio Fabbri

Salvatore Federico

Fausto Gozzi

MANAGING SPATIAL LINKAGES AND GEOGRAPHIC HETEROGENEITY IN DYNAMIC MODELS WITH TRANSBOUNDARY POLLUTION

Keywords: Transboundary pollution, spatiotemporal modeling, geographic heterogeneity, infinite dimensional optimal control, optimal spatiotemporal short-term dynamics JEL classification: Q53, R11, C61, R12, O41

We construct a spatiotemporal frame for the study of spatial economic and ecological patterns generated by transboundary pollution. Space is continuous and polluting emissions originate in the intensity of use of the production input. Pollution flows across locations following a diffusion process. The objective functional of the economy is to set the optimal production policy over time and space to maximize welfare from consumption, taking into account a negative local pollution externality and the diffusive nature of pollution. Our framework allows for space and time dependent preferences and productivity, and does not restrict diffusion speed to be space-independent. Accordingly, we develop a methodology to investigate the environmental and economic implications of spatiotemporal heterogeneity. We propose a method for an analytical characterization of the optimal paths. An application to technological spillovers is proposed for illustration. We focus on the determination of the optimal short-term spatiotemporal dynamics induced by the resulting non-autonomous problems.

Introduction

Transboundary pollution is readily recognized as a major topic in economics, notably in economic geography and environmental economics. A key issue posed by transboundary pollution is the treatment of the related externality problem: pollution generated in a certain place may flow integrally or partially into another distant place. Such a problem has been addressed by [START_REF] Candel-Sanchez | The externalities problem of transboundary and persistent pollution[END_REF] in a quite simple theoretical framework, the main objective being the identification of compensation mechanisms to the externality problem posed. Yet, just like in the latter paper, the spatial dimension has not been often accounted for in the related theories developed, and has mostly remained metaphorical, most of the time using the two-country model narrative (see [START_REF] Unteroberdoerster | Trade and transboundary pollution: Spatial separation reconsidered[END_REF]. Indeed, a large majority of papers in this topic is empirical (see for example, [START_REF] Gibson | The effects of road pricing on driver behavior and air pollution[END_REF]Carnovale, 2015, or Henderson, 1996). This said, an increasing number of theoretical works in the topic of transboundary pollution are introducing an explicit spatial dimension, that's they are based on a well defined spatial setting. An overwhelming part of it is static and it usually entails an ad hoc assumption on the level of pollution at given location. This is clear for example in [START_REF] Hutchinson | State enforcement of federal standards: Implications for interstate pollution[END_REF] and in Arnott et al. (2008). This is of course acceptable in certain contexts, and it is even more acceptable if the alternative (more) accurate specifications are intractable. This paper is a methodological contribution to this important area of economics at the intersection of geographic and environmental economics. Since spatial heterogeneity is admittedly key in shaping (local) pollution and the induced spatial distribution of pollution, we propose a methodology which closely incorporates a large set of spatial heterogeneities while the diffusion process of pollution over time and space is accurately modeled through a diffusion partial differential equation as in the pioneering works of Perez-Barahona and [START_REF] Camacho | Land use dynamics and the environment[END_REF] and de Frutos and Martin-Herran (2019a).

In short, we propose a new spatiotemporal framework allowing to incorporate geographic heterogeneity to a large extent while modelling pollution diffusion as a physical diffusion problem over time and space. Moreover, we are not only able to put together all these admittedly complex ingredients, we find a way for our setting to produce closed-form solutions for any of these spatial heterogeneity traits, and analytical characterizations of the induced long-term spatial distributions as well. This is done at the cost of a simplification in the shape of the production function (linearity assumption) but the economic insight from the analytical exploration of spatial heterogeneity impact on economic decision making and pollution outcomes well outweigh the latter cost in our view.

As one can infer from above, we are firstly interested in transboundary (diffusive) pollution like air or water pollution. The problem of pollution control is deeply intricate in this case. As our main objective is methodological in this paper, we shall not address the strategic aspects originating in the externality problem outlined in the beginning of the introduction. 1 We rather concentrate on a second set of issues, those inherent in the fact that the impact of air pollution is first of all local, and its magnitude and persistence depend pretty much on the local conditions. If a central planner at a country or international level has to set a pollution control policy for the benefit of all the individuals concerned, then she should take as much as possible into account the heterogeneity across locations, in addition to the fact that air pollution, being transboundary, requires the internalization of spatial externalities.

There are several spatial heterogeneity features to account for. Obvious ones are technological heterogeneity and heterogeneity in preferences (which covers cultural discrepancies with respect to the environment among others). But there are also more geographic and ecological differences, which matter a lot both in the diffusion of pollution across locations and in its local impact. The self-cleaning capacity of Nature may vary from a region to a close one, the local topography, land use and infrastructures may speed up pollution diffusion or slow it down... etc. These issues are quite known 1 Several papers have been written in the last decade on this topic from 2-country settings [START_REF] Boucekkine | Environmental quality versus economic performance: A dynamic game approach[END_REF] to continuous space modelling (see in particular, de Frutos andMartin-Herran, 2019b, de Frutos et al, 2021) through multi-country frameworks (for example, [START_REF] Dockner | International pollution control: cooperative versus noncooperative strategies[END_REF].

across disciplines (see Tiwari and Closs, 2010, for an excellent book on air pollution), and, as indicated in the beginning of this introduction, they are also very much recognized in the literature for decades. Several recent papers have been devoted to transboundary pollution. 2 None of these papers however poses the latter problem in a full-fledged analytical spatiotemporal frame incorporating the above mentioned technological, preference, geographic and ecological spatial discrepancies.

Clearly, taking up the challenge would involve plenty of technical problems, with a very likely lack of tractability and the forced use of numerical solutions. Contrary to other disciplines (like quantitative geography, climate science or ecology) in which the use of black-box disaggregated models is routine, the economists, being more interested in identifying mechanisms, are more keen to develop parsimonious models. In this paper, we build up a spatiotemporal optimal control framework allowing to encompass the heterogeneity traits outlined above in the presence of transboundary pollution, and still producing a comprehensive analytical characterization. In particular, with respect to the aforementioned recent literature on transboundary pollution, we considerably enrich the geography of the model. On the technological ground, we allow productivity both time and space-dependent: any pattern of technology diffusion across time and space can be accommodated. As for preferences, we make them time and space-dependent as well. Finally, to account for the local geographic and ecological conditions discrepancy, we allow for a space dependent self-cleaning capacity and pollution diffusion speed. This generality has a methodological implication, since the heterogeneities in space and time make very difficult to use the dynamic programming method (see e.g. Boucekkine et al., 2019a[START_REF] Boucekkine | Growth and agglomeration in the heterogeneous space: a generalized AK approach[END_REF] or the maximum 2 For example, [START_REF] Camacho | Land use dynamics and the environment[END_REF] and [START_REF] Camacho | Diffusion of soil pollution in an agricultural economy. The emergence of regions, frontiers and spatial patterns[END_REF] studied respectively the problem of atmospheric transboundary pollution in the context of an optimal land use problem and of soil pollution. Other authors have also provided valuable contributions to the analysis of the spatiotemporal deep nature of the transboundary pollution control problem; see for instance [START_REF] Augeraud-Véron | Optimal control for a groundwater pollution ruled by a convection-diffusion-reaction problem[END_REF], 2019a, 2019b, 2019c), Boucekkine et al. (2019a), La [START_REF] Torre | Transboundary pollution externalities: Think globally, act locally?[END_REF] or Grass and Ueckner (2017).

principle (e.g. [START_REF] Brito | Optimal control in space and time and the management of environmental resources[END_REF][START_REF] Camacho | On the dynamics of capital accumulation across space[END_REF]or Brock et al., 2014) in order to produce analytical solutions. Instead, we apply a functional transformation technique observing that the objective functional can be rewritten in a way that allows for a direct maximization method, ultimately finding the explicit form of the optimal control. 3 An extremely appealing feature of this method is that it builds on a pivotal spatial function (denoted α(•) in Section 3), which admits a neat economic interpretation: it corresponds at any location x to the discounted sum of future disutility stream of a unit of pollutant initially located at x. Indeed, in our spatiotemporal framework with transboundary pollution, a unit of pollutant has a different effect on social welfare depending on where it is initially located and how it is going to spread over space in the future. Our alternative method has the virtue to put forward this deep economic concept, which makes it definitely transparent and useful for economic analysis.

In a companion paper [START_REF] Boucekkine | From firm to globallevel pollution control: The case of transboundary pollution[END_REF], we apply this methodology to the problem of optimal pollution control in a particular spatiotemporal case, the main objective being the derivation and analysis of the longrun distributions of some relevant economic and environmental variables depending on technological, ecological and culture spatial heteorgeneities.

Here we develop our methodology in a much more general spatiotemporal framework, and provide with an additional application, enhancing the applicability of our method to more complex spatiotemporal contexts with a wider set of nontrivial heterogeneities and non-autonomous problems. In particular, this paper generalizes and complements the companion paper in two essential dimension. First, we allow for full spatial heterogeneity in structural preference, technological and physical parameters. This notably concerns the pollution diffusion parameter σ, the unitary emission of investment η, and the intertemporal elasticity of substitution is consumption, γ. In the companion paper, these parameters are constant, thus for example preventing to consider emissions depending on production and 3 A somehow related method is used in a different economic context by [START_REF] Barucci | Technology adoption and accumulation in a vintagecapital model[END_REF] but the problem there did not include a diffusive process.

time-increasing environmental efficiency of the technology in use, which admittedly limits the realism of the exercises. Second, instead of analyzing the implications for the long-run distributions depending on the particular spatial heterogeneity considered, as in the companion paper, we shall in a complementary way run our methodology on non-autonomous problems and study spatiotemporal transition dynamics. As a useful illustration, we tackle a problem of technological dynamics over space and time adapting an initial non-spatial frame due to [START_REF] Nocco | The rise and fall of regional inequalities with technological differences and knowledge spillovers[END_REF].

More precisely, we use our methodology to study, following the terminology of [START_REF] Krugman | First nature, second nature, and the metropolitan location[END_REF], a first nature cause of spatial externalities, that is given geographic differences in terms of efficiency either at production or at pollution abatement. Also as written just above, we adapt a non-spatial technology evolution model specified in [START_REF] Nocco | The rise and fall of regional inequalities with technological differences and knowledge spillovers[END_REF]. A key feature of our adaptation is that technological spillovers tend to decrease with distance to a given technological center. This is totally consistent with empirical evidence. For instance, [START_REF] Deltas | Similarity of R&D activities, physical proximity and R&D spillovers[END_REF] found that in increase in distance between the center and the recipient by 500 km reduces spillovers by 55-70%. It is important to note here that our method can also in principle support the study of second nature causes. As a matter of fact, it can be readily shown that the inclusion of an additional policy control, like pollution abatement, does not at all break down the analytical solution scheme despite the nonlinearities potentially conveyed by this addition. 4That speaks a lot about the flexibility of our approach. In this paper, we prefer to emphasize more the ability of our method to address a large set of geographic heterogenities, in particular in structural parameters, and to deliver properly optimal transitional spatiotemporal dynamics.

The paper is organized as follows. Section 2 presents the basic economic problem and defends its economic relevance. Section 3 develops the method used to solve the generic spatiotemporal optimal growth model under transboundary pollution. In particular, closed-form solutions of the optimal strategies are identified. We show also further analytical results on transition dynamics and asymptotics (that's the computation of asymptotic spatial distributions). Section 4 gives an illustration of the method in the presence of technological discrepancy across space using the spatiotemporal spillover mechanism explained just above. Of course, our aim in this ultimate section is not to exploit entirely the richness of our setting but only to indicate how far the analytical study can go. Section 5 concludes.

Appendix A contains the formal proofs.

The basic economic problem

The basic problem incorporates some elements of the models by Boucekkine et al. (2011), Camacho and[START_REF] Camacho | Land use dynamics and the environment[END_REF], Boucekkine et al. (2019a), andde Frutos andMartin-Herran (2019a,b). The originality of our contribution is to provide a full-fledged analytical spatiotemporal setting yet incorporating, as described in detail below, space and time-dependence of both preferences and production technology, as well as space-dependence of the pollution diffusion coefficient and of the selfcleaning capacity.

Let us now describe the basic economic problem. Consider a continuum of locations, say along the circle in R 2 . The choice of the circle is made for simplicity.5 Call it S 1 :

(1)

S 1 := x ∈ R 2 : |x| R 2 = 1 .
Each location uses a linear (Leontief) production function: at any location and focus on other state variables (see again Barro and Sala-i-Martin, 2004, Chapter 6). In our case, both arguments hold: given the complexity of our problem (infinite dimensional optimization) and the economic target (economic performance in the heterogenous space with transboundary pollution), we find it convenient to shut down capital accumulation to be able to develop a comprehensive enough analytical spatiotemporal framework to approach the latter economic objectives.

x in time t ≥ 0, production is (2) y(t, x) = a(t, x) i(t,
At any location, output is produced, consumed and locally invested (no trade across locations), implying:

(3)

c(t, x) + i(t, x) = y(t, x),
where c(t, x) is consumption at location x at time t. The unique link among locations is transboundary pollution. We target air pollution and consider a broad specification incorporating ecological efficiency at any location x and time t. Precisely, the accumulation of pollution spatial profile is assumed to evolve according to the following parabolic partial differential equation (PDE):

(4)

       ∂p ∂t (t, x) = ∂ ∂x σ(x) ∂p ∂x (t, x) -δ(x)p(t, x) + η(t, x)i(t, x), (t, x) ∈ R + × S 1 , p(0, x) = p 0 (x), x ∈ S 1 ,
where p(t, x) is the pollution stock at location x and time t.

First, notice the general shape of the pollutants' emissions term η(t, x)i(t, x) in the pollution spatiotemporal dynamics depicted above. Indeed, the unusual function η(t, x) is meant to reflect that the pollution impact of emissions arising from the use of one unit of input may not be the same over time and space. It can be readily observed that: (i) taking η(t, x) = 1 brings to consider the case where polluting emissions at location x are exactly equal to input use intensity; (ii) if we specify

η(t, x) = a(t, x)φ(t, a), the term η(t, x)i(t, x) reads as a(t, x)φ(t, x)i(t, x) = φ(t, x)y(t, x
), and we are able to give a specification of the model where emissions depend on output rather than on input used; (iii) the temporal dependence allows us, in general, to incorporate exogenous technological progresses in ecological efficiency (such as those conveyed by abatement activities); (iv) finally, independent spatial heterogeneities can also be taken into account with this more general specification.

Second, we underline the space-dependent form of the transboundary pollution diffusion term, that is ∂ ∂x (σ(x) ∂p ∂x (t, x)), where σ(x) is the pollution diffusion speed at location x. Indeed, there is a large bunch of works documenting the role of local conditions in air pollution diffusion (see Tiwary and Colls, 2010, Chapter 1). Therefore, this generalization significantly increases the relevance of our analytical frame. Moreover, we notice that the pollution diffusion also depends on nature local regeneration capacity (the term δ(x)p(t, x)), and on current emissions (as captured by the term

η(t, x)i(t, x)).
Finally, since our setting is spatiotemporal, an initial spatial distribution of pollution is needed, and it is given by function p 0 (x) defined on S 1 .

The whole state variable dynamics then follows the PDE (4). The infinite dimensional nature of the involved optimization problem derives from the latter characteristic of the state dynamics.

On the side of preferences, we choose the instantaneous per location utility to be of CRRA type with respect to consumption and linear with respect to pollution:

U (c(t, x), p(t, x)) = c(t, x) 1-γ(t,x) 1 -γ(t, x) -w(x)p(t, x),
where γ(t, x) ∈ (0, 1) ∪ (1, ∞) measures the inverse of the elasticity of intertemporal substitution in consumption at location x and time t, and w(x)

measures, for instance, local environmental awareness at location x. Observe that our negative pollution externality is local. Our framework is not designed primarily to study global warming and therefore global pollution, but the diffusion of air pollutants with local health impact like particles for example (again a comprehensive account of air pollutants can be found in Tiwari and Colls, 2010, Chapter 1). The local impact of pollution is also captured via w(x), which can be indeed interpreted as local awareness and sensitivity to environmental problems. It can also reflect specific priorities of the planner to cope with particular local conditions. Notice also that our frame allows for time and space varying preferences through the parameter γ(t, x). We consider a planner problem who has to maximize the following social welfare under the above specified technological constraints of the economy and the transboundary pollution faced:

J p 0 ; i := ∞ 0 e -ρt S 1 c(t, x) 1-γ(t,x) 1 -γ(t, x) -w(x)p(t, x) dx dt,
where ρ is the parameter at which the planner discounts time. By using equations ( 1) and ( 2), we can rewrite the functional in terms of the control variable i(t, x):

J p 0 ; i = ∞ 0 e -ρt S 1 (a(t, x) -1)i(t, x) 1-γ(t,x) 1 -γ(t, x) -w(x)p(t, x) dx dt.
(5)

Notice we do not incorporate population density in our analysis, and notably in the functional, nor do we introduce mortality (possibly related to pollution). We fundamentally focus on handling spatial heterogeneity abstracting away from demography, which is an already daunting task. This does not mean population mobility is irrelevant in the topic of transboundary pollution. As cleverly pointed out by [START_REF] Haavio | Transboundary pollution and household mobility: Are they equivalent?[END_REF], mobility may encourage people to disregard the environment.6 . In our model, individuals are immobile and pay full attention to their environment. This is reflected by the term w(x)p(t, x) in our objectif functional. Further refinements may be possible by for example allowing the planner to move people and/or tax migrations while still keeping the close-form solutions. At the minute, one could simply interpret the social welfare function above as a Benthamite welfare function summing individual welfare over locations and time with one infinite-lived individual at each location.

Theoretical analysis

In this section we give a precise description of our results, specifying hypotheses and formalizing the statements. To increase the readability of the text we postpone the proof in Appendix A and we divide the section in four parts.

To apply the functional transformation technique that we exploit to solve the optimal control problem (Theorem 3.6), we start by rewriting the problem in a suitable Hilbert space formalism (Subsection 3.1), then we identify a spatial function α that will be essential in the transformed expression of the functional (Subsection 3.2), and finally we will characterize the optimal control and the corresponding social welfare (Subsection 3.3). Subsection 3.4 contains transitional and long-run analysis of the dynamics via series expansions.

3.1. Infinite dimensional formulation and preliminary results. On the space support S 1 introduced in (1) we consider the metrics induced by the Euclidean metrics of R 2 . In this way S 1 can be isometrically identified with 2πR/Z and the (class of) functions S 1 → R with 2π-periodic function R → R; differentiation of functions S 1 → R is defined according to this identification. Consequently, the initial pollution distribution and the space dependent parameters δ, σ and w are measurable functions

p 0 , δ, σ, w : S 1 → R + ;
similarly the time and space dependent parameters γ, a, η are measurable functions γ : R + ×S 1 → (0, 1)∪(1, +∞), a : R + ×S 1 → (1, +∞), η : R + ×S 1 → (0, +∞).

We proceed now to our infinite dimensional reformulation of the problem.

We will use the framework of Lebesgue and Sobolev spaces, for more details we refer to [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF]. The infinite dimensional space H, where we will reformulate our maximization, is the Lebesgue space L 2 (S 1 ; R), i.e.7

H := L 2 (S 1 ; R) := f : S 1 → R measurable :

S 1 |f (x)| 2 dx < ∞ ,
endowed with the usual inner product f, g = S 1 f (x)g(x)dx, which makes it a Hilbert space. We denote by • the associated norm, by H + the nonnegative cone of H, i.e.

H + := {f ∈ H : f ≥ 0},
and by 1 the constant function equal to 1 on S 1 . Moreover, we introduce the Sobolev space8

(6) W 2,2 (S 1 ; R) := f ∈ L 2 (S 1 ; R) : f is twice weakly differentiable and f , f ∈ L 2 (S 1 ; R) .
Some degree of regularity of the parameters will be necessary in the analysis. We will work with the following assumptions.

Assumption 3.1.

(i) p 0 ∈ L 2 (S 1 ; R + ), δ ∈ C(S 1 ; R + ), σ ∈ C 1 (S 1 ; (0, +∞)), w ∈ C(S 1 ; (0, +∞));
(ii) the function γ : R + × S 1 → (0, 1) ∪ (1, +∞) is measurable and there exists κ ∈ (0, 1) such that, for every

(t, x) ∈ R + × S 1 , either (Case (A)) κ ≤ γ(t, x) ≤ 1 -κ or (Case (B)) 1 + κ ≤ γ(t, x) ≤ 1 κ .
(iii) There exist L > 0, and g ≥ 0 such that, for every

(t, x) ∈ R + × S 1 , a(t, x) -1 η(t, x) 1 γ(t,x) -1 ≤ Le gt ;
(iv) ρ > g.

Notice that the above Assumption 3.1 (iii) and (iv), which may seem quite technical, are indeed the translation, in this case, of the standard assumptions needed to guarantee "well posedness" (i.e. finiteness of the value function and existence/uniqueness of the optimal strategy) exactly as in the standard Ramsey-type AK model, see e.g. [START_REF] Freni | Existence of optimal strategies in linear multisector models[END_REF] on this.

Hereafter, our arguments will make use of the theory of unbounded linear operators and semigroups of linear operators, for which we refer to [START_REF] Engel | One-parameter Semigroups for Linear Evolution Equations[END_REF]. Denote by L(H) the space of bounded linear operators on H. We consider the differential operator L : D(L) ⊂ H → H, where

D(L) = W 2,2 (S 1 ; R); (Lϕ)(x) = (σϕ ) (x) -δ(x)ϕ(x), ϕ ∈ D(L).
Proposition 3.2. Let Assumption 3.1 hold. Then L generates a strongly continuous contraction semigroup (e tL ) t≥0 ⊂ L(H). Moreover, ρ belongs to the resolvent set of L, i.e. ρ -L :

D(L) → H is invertible with bounded inverse (ρ -L) -1 : H → D(L) and (7) (ρ -L) -1 h = ∞ 0 e -(ρ-L)t h dt ∀h ∈ H. Proof. See Appendix A Given i : R + × S 1 → R + , define I : R + → H + , I(t) := i(t, •).
Morever, define (2007), given I ∈ L 1 loc (R + ; H + ), we define the mild solution to (8) as ( 9)

Ψ : R + → H + , Ψ(t) := η(t, •).
P (t) = e tL p 0 + t 0 e (t-s)L Ψ(s)I(s)ds, t ≥ 0. Setting A(t) := a(t, •), Γ(t) := γ(t, •), and (A(t) -1)I(t) 1-Γ(t) 1 -Γ(t) (x) := (a(t, x) -1)i(t, x) 1-γ(t,x) 1 -γ(t, x) , x ∈ S 1 ,
the functional ( 5) is rewritten in this formalism as

(10) J(p 0 , I) = ∞ 0 e -ρt (A(t) -1)I(t) 1-Γ(t) 1 -Γ(t) , 1 -w, P (t) dt.
We introduce the following set of admissible controls

A := I ∈ L 1 loc (R + ; H + ) : ∞ 0 e -ρt Ψ(t)I(t) dt < ∞ .
The following result shows that the functional is well defined on A.

Proposition 3.3. Let Assumption 3.1 hold. The functional J(p 0 , I) is well defined for all p 0 ∈ H and I ∈ A.

Proof. See Appendix A.

Finally, we define the value function as the optimal value of J over A,

i.e.

v(p 0 ) := sup I∈A J(p 0 ; I).

Note that this function may possibly be infinite. The function

(11) α := (ρ -L) -1 w ∈ H.
will play a key role in the transformation of the functional J that we will perform: it represents the core of the solution. Moreover, as recalled in the introduction, it admits a clear economic interpretation: it corresponds at any location x to the discounted sum of future disutility stream of a unit of pollutant initially located at x. Hence in the next subsection we will investigate some properties of it and clarify its economic interpretation.

3.2. The function α and its properties.

3.2.1. The equation for α and its mathematical properties. By definition α

is the unique solution in W 2,2 (S 1 ; R) of the abstract ODE (12) (ρ -L) α = w.
More explicitly, α, as defined in (11), is the unique solution in W 2,2 (S 1 ; R)

to (13) ρα(x) - d dx σ(x) d dx α(x) + δ(x)α(x) = w(x), x ∈ S 1 ,
meaning that it verifies (13) pointwise almost everywhere in S 1 . The latter ODE can be viewed as on ODE on the interval (0, 2π) with zero-order and first-order periodic boundary conditions9 , that is

       ρα(x) - d dx σ(x) dα dx (x) + δ(x)α(x) = w(x), x ∈ (0, 2π), α(0) = α(2π), α (0) = α (2π),
where α (0) and α (2π) are, respectively, the right derivative at 0 and the left derivative at 2π of α.

As better argued below the value of α at a certain spatial point x has the meaning of the sum of all future (discounted) disutility of a unit of pollutant initially located at x.

By Sobolev embedding W 2,2 (S 1 ; R) ⊂ C 1 (S 1 ; R), so α ∈ C 1 (S 1 ; R).
With a little more subtle analysis something more can be said about the properties of α as shown in next proposition.

Proposition 3.4. Let Assumption 3.1 hold. Then α ∈ C 2 (S 1 ; R) and

0 < min S 1 w ρ + δ ≤ α(x) ≤ max S 1 w ρ + δ ∀x ∈ S 1 .
Proof. See Appendix A.

We have the following interesting result on the dependence of α on the diffusion coefficient σ when the latter is constant over space.

Proposition 3.5. Let Assumption 3.1 hold. Denote by α σ o the solution to (13) when σ(•) ≡ σ o > 0. We have

(14) lim σ o →0 + α σ o (x) = w(x) ρ + δ(x)
, lim

σ o →+∞ α σ o (x) = S 1 w(x)dx S 1 (ρ + δ(x))dx , ∀x ∈ S 1 . Proof. See Appendix A.
As will be clearer shortly (Subsection 3.3), the function α has a key role both in expressing the functional in its transformed form and in describing the optimal behavior of the planner. For this reason, understanding its behavior is interesting to describe the behavior of the model.

3.2.2.

The economic interpretation of α. We now exploit the equation for α and the Propositions 3.4-3.5 to describe the economic intuition about the function α. We said that α(x) has the meaning of the sum of all future (discounted) disutilities of a unit of pollutant initially located at x. First of all observe that, if we take the non-spatial version of the model or, equivalently, a specification of the model where all the parameters are constant in space, the value of such disutility is w ρ+δ . Indeed in this case the sum of all future (discounted) disutilities of a unit of pollutant would simply be (we take the integral since we are in the continuous time case) (15) +∞ 0 e -ρt e -δt w dt, where we recall that δ is the natural decay of the pollution. This result, not surprisingly, would be the same if we consider a specification of the model where w and δ are not anymore are constant in space, but no diffusion is present (i.e. σ ≡ 0). Indeed this corresponds to the case where the pollution does not move among the locations and accumulates in the production site. As recalled above, in the model the only link among the locations is the transboundary pollution. Letting the diffusivity to zero means to reset this channel of interdependence and therefore the model reduces to an independent (uni-dimensional) optimization problem to each point of the space. Since the pollutant which is produced at the location x remains there forever then, similarly to (15), the sum of all future (discounted) disutilities of a unit of pollutant would be

+∞ 0 e -ρt e -δ(x)t w(x) dt = w(x) ρ + δ(x)
The above, as expected, is also the pointwise expression appearing in Proposition 3.5, equation ( 14), when σ o → 0 + . When we add the space to the model a second order term, depending on σ, appears in the equation which defines α, see (13). This comes of course from the spatial diffusion process of the pollution. The solution of that equation (except when both w and δ do not depend on time) is given by a space-heterogeneous function α. This heterogeneity is due to the fact that, in the model, a unit of pollutant has different effect on the social utility depending on where it is located and how it is going to spread in the future. In terms of pollution-disutility, locations are indeed different for two reasons: the different decay δ(x) of pollutions and the different unitary instantaneous disutilities w(x). In the general spatial case, the function α at a point x is the (total/social) future discounted disutility of a unit of pollutant initially located at point x. To explain this fact we denote by ∆ {x} the Dirac delta in a certain spatial point x, and observe that (see equation (37) in the Appendix where we substitute p 0 with ∆ {x} ) we formally have:

α(x) = α, ∆ {x} = ∞ 0 e -ρt w, e Lt ∆ {x} dt i.e. (16) α(x) = ∞ 0 e -ρt S 1 w(ξ)ϕ(t, ξ; x) dξ dt
where ϕ(t, ξ; x) = e Lt ∆ {x} (ξ) is the solution of the parabolic equation

   ∂ϕ ∂t (t, ξ) = ∂ ∂ξ σ(ξ) ∂ϕ ∂ξ (t, ξ) -δ(ξ)ϕ(t, ξ), ϕ(0, ξ) = ∆ {x} (ξ) ,
i.e. the spatial density (with respect to the variable x) at time t of a pollutant initially concentrated at point x, once one takes into account the diffusion process and the natural decay. Thus, the term S 1 w(ξ)ϕ(t, ξ; x) dξ measures the instantaneous disutility all over the space and the whole expression in the right hand side of ( 16) is the total spatial (temporally discounted) future social disutility of a unit of pollutant initially concentrated at x.

Finally the second limit of Proposition 3.5, equation ( 14), correspond to the infinite diffusivity benchmark that is the case where, at each moment, the speed of the diffusion process is so fast that the pollution is instantaneously redistributed uniformly throughout the space. For this reason, whatever the specific value of δ or of w in the precise point of the emission is not relevant but only global averages of the parameters matters. Finally it's worth pointing out that all the results of the paper, including the possibility to extract explicit solutions, do depend on the geographic specification that we chose (the circle). Of course, they would be different under alternative specifications (for instance considering a segment with Dirichlet of Neumann boundary condition). This fact is once more clear if we look at the interpretation of the function α which is the solution of an hyperbolic ODE with periodic boundary condition.

Characterization of the optimal control.

3.3.1. Optimal spatiotemporal paths. We describe now how previous results can be used: first to rewrite the functional in a transformed form; second to explicitly find the optimal solution of the problem, the related optimal trajectory and the welfare function. The main results are described first (Theorem 3.6) in the Hilbert space formalism introduced above and then restated (Corollary 3.7) using a more readable PDE notation.

Theorem 3.6. Let Assumption 3.1 hold.

(i) The functional (10) can be rewritten as

(17) J p 0 ; I = -α, p 0 + ∞ 0 e -ρt (A(t) -1)I(t) 1-Γ(t) 1 -Γ(t) , 1 -α, Ψ(t)I(t) dt.
(ii) The control I * given by

I * (t)(x) := (η(t, x)α(x)) -1 γ(t,x) (a(t, x) -1) 1 γ(t,x) -1 (18)
belongs to A and is the unique optimal control of the problem.

(iii) The optimal state at time t ≥ 0, that is P * (t), is given by (19) P * (t) := e tL p 0 + t 0 e (t-s)L Ψ(s)I * (s)ds.

(iv) The value function is finite and affine in p 0 ; more precisely,

v(p 0 ) = J p 0 ; I * = -α, p 0 + q, where q := ∞ 0 e -ρt (A(t) -1)I * (t) 1-Γ(t) 1 -Γ(t) , 1 -α, Ψ(t)I * (t) dt.
Proof. See Appendix A.

In the following corollary we summarize the results we have obtained so far rephrasing them in the PDE setting, where we use the identification of S 1 with the real interval [0, 2π] with the identification of the extremes 0 and 2π.

Corollary 3.7. Let Assumption 3.1 hold.

(i) The optimal investment production input is given by

i * (t, x) = (η(t, x)α(x)) -1 γ(t,x) (a(t, x) -1) 1 γ(t,x) -1 . ( 20
)
where α is the unique solution to the following ODE

       ρα(x) - d dx σ(x) dα dx (x) + δ(x)α(x) = w(x), x ∈ (0, 2π), α(0) = α(2π), α (0) = α (2π).
(ii) The dynamics of the pollution profile p * along the optimal path is the unique solution to the following parabolic PDE

(21)                ∂p * ∂t (t, x) = ∂ ∂x σ(x) ∂p * ∂x (t, x) -δ(x)p * (t, x) + α(x) -1 γ(t,x) a(t, x) -1 η(t, x) 1 γ(t,x) -1 , p * (t, 0) = p * (t, 2π), ∂p * ∂x (t, 0) = ∂p * ∂x (t, 2π), t ≥ 0, p * (0, x) = p 0 (x), x ∈ [0, 2π].
(iii) The social welfare is

(22) v(p 0 ) = - 2π 0 α(x)p 0 (x)dx + ∞ 0 e -ρt 2π 0 γ(t, x) 1 -γ(t, x) a(t, x) -1 η(t, x)α(x) 1 γ(t,x) -1 dx dt,
In the previous statements we have seen the explicit solution of the optimal problem of the planner. A first, eminently technical, observation concerns the transformation of the functional. In fact, the first result of Theorem 3.6 is that the functional (10) can be rewritten in the form (17).

The new form is particularly useful as, in this expression, the state P no longer appears and it is only given in terms of the control I; this fact greatly simplifies the analysis. Thanks to this transformation, it is indeed much easier to find the expression of the optimal investment and the subsequent results.

3.3.2. Economic insight. Looking at the expressions that appear in Corollary 3.7, it is immediately evident that all the heterogeneities of the problem enter directly and in a non-trivial way in the solution. Partly, they appear explicitly in the expressions of the optimal spatial profile of the investment or of the parabolic equation describing the evolution of the spatial distribution of pollution and partly they contribute to these expressions through the expression of the function α. In this way the model delivers optimal outcomes that are sensitive to heterogeneities of different nature: environmental heterogeneities (the ability to regenerate of the ecological context measured in each place by δ), productive heterogeneities (the productivity which depends exogenously on both the location and the time) and preferences heterogeneities (both through the spatial heterogeneity of the disutility of pollution w and through the spatial heterogeneity of the elasticity of intertemporal substitution). This is clear for example in the optimal spatiotemporal path of investment given in the corollary above: it depends directly on technological efficiency both at production and at depollution (a(t, x) and η(t, x) respectively), and it depends also indirectly through the α function (and equation ( 13)) on other local conditions like self-cleaning capacity of Nature (δ(x)), proenvironmental awarness (w(x)) and pollution diffusion (σ(x)). Last but not least, as any optimal outcome of an intertemporal problem, investment also depends on the typical inherent parameters, the time discount parameter (ρ) and the elasticity of intertemporal substitution (through γ(t, x)), which happens to be time and space-dependent in our framework.

Another remarkable implication of our setting shows up in the optimal dynamics of pollution, equation ( 20). As one can see, the technological determinants of the latter entirely play through a single "sufficient" statistic, the ratio a(t,x)-1 η(t,x) . Of course, the larger productivity at production, the larger pollution accumulation, while depollution efficiency works in the opposite direction. In our model, the two technological channels interact directly in the expression of optimal pollution stock dynamics. A technological evolution in which the dynamics of productivity at production (here captured by the dynamics of a(t, x) -1) are one-to-one offset by those of depollution will leave the law of motion of pollution stock unaffected, and therefore entirely determined by non-technological factors, which are far from innocuous. One of the virtues of our framework is precisely to highlight the role of these non-technological factors. In particular, all the factors affecting pollution absorption capacity (that is, δ(x)) are increasingly important over time: they point at the crucial importance of land use policies over the world (not only at the Amazon).

As in study in sustainable development under pollution, we should not only look at short-term effect but also at the large. We show here below that our setting also allows for extracting some interesting asymptotic convergence results in the same analytical manner.

3.4. Transitional and long-run analysis of P * . In this section we analyze both the transitional dynamics of P * (t) and its limit behavior as t → ∞. Then the pollution profile along the optimal trajectory can then be expressed as a convergent series in H: Proof. See Appendix A.

(
The previous result allows to express the solution of the equation ( 21)

along the optimal path in terms of a series of space functions not dependent on time multiplied by time-dependent coefficients. This expression, which can be used in general to simulate the model, takes a particularly familiar form in some specific cases. For example, if the diffusivity σ and the natural decay of pollution δ are constant in the space variable, a standard Fourier series is obtained, and the functions e n are sins and cosines.

Observe also that ( 23) can also be used to express the total pollution S 1 p * (t, x)dx as a function of time, namely,

S 1 p * (t, x)dx = n∈N e n , 1 p * n (t), t ≥ 0.
where p * n (t) is given in (24). Indeed, again using this result, an even more precise description of the total pollution at time t can be given whenever the following when δ is constant in space as shown by the following proposition. Proposition 3.9. Let Assumption 3.1 hold and assume that the function δ is constant, i.e. δ(•) ≡ δ 0 ≥ 0. Then (26)

S 1 p * (t, x)dx = S 1 p 0 (x)dx e -δ 0 t + t 0 e -δ 0 (t-s) S 1 η(s, x)i * (s, x)dx ds.
Proof. See Appendix A.

3.4.2. Limit behaviour in the time-homogeneous case. We consider now the special case when the productivity coefficients and the ecological efficiency of the production process are time-independent: a(t, x) = a(x), η(t, x) = η(x); similarly for the inverse of the elasticity of intertemporal substitution, i.e. γ(t, x) = γ(x). In this case, the expressions of the optimal control are time independent

I * (t)(x) ≡ Ī * (x) = (η(x)α(x)) -1 γ(x) (a(x) -1) 1-γ(x) γ(x) , (27)
and we have a direct characterization of the long-run profile of the pollution stock along the optimal path as described in the following proposition.

Proposition 3.10. Let Assumption 3.1 hold. Assume that the coefficients a, γ, η are time-independent and that δ ≡ 0. Then we have

lim t→∞ P * (t) = p * ∞ in H,
where p * ∞ is the unique solution to the ODE (28)

       d dx σ(x) dp * ∞ dx (x) -δ(x)p * ∞ (x) + α(x) -1 γ(x) a(x) -1 η(x) 1 γ(x) -1 = 0, p * ∞ (0) = p * ∞ (2π), dp * ∞ dx (0) = dp * ∞ dx (2π).
Proof. See Appendix A.

In the next section, we explore the optimal spatiotemporal transitional dynamics in a case where the convergence theorem above does not apply.

We shall indeed, as advocated in the introduction, rather concentrate on the applicability of our methodology to non-atonomous problems, in particular those involving non-stationary technological environments, that is a(t, x)

effectively depending on the time index t. Our methodology also provides with closed-form solutions for the latter non-autonomous cases, see Section 3.4.1 above. An illustration is given below.10

4. An illustration: Technology evolution over space and time

We now propose an application of our method to technology diffusion across space and time. In our setting, two technological exogenous variables are considered: one represents the typical efficiency of the production process in the final good sector, that is a(t, x), and the second reflects essentially the advances in the depollution activity, as reflected by η(t, x).

We will concentrate on the former for simplicity, given the illustrative purpose of this section. We could have performed similar work on the latter.

More precisely, we look at the economic and environmental implications of spatiotemporal diffusion of technology using the North-South diffusion advocated by [START_REF] Nocco | The rise and fall of regional inequalities with technological differences and knowledge spillovers[END_REF].11

Consider an economy with a technological core (say the "North") at x = π, which is therefore the technological center. Concretely, we assume an initial distribution for productivity in the final sector, a(0, x), with a peak at x = π. 12 As argued above, we suppose that knowledge spillovers from the technological center are more difficult for peripheral locations. We capture this fact by defining a learning capabilities function ψ with the following linear form

(29) ψ(x) = a -b|x -π| if |x -π| < a/b 0 if |x -π| ≥ a/b
where a and b are two non-negative constants and |x -π| is the distance between x and π on S 1 .

We normalize, at all times, the level of the technology in the center π to a certain fixed level a π > 1, 13 and following [START_REF] Nocco | The rise and fall of regional inequalities with technological differences and knowledge spillovers[END_REF], we suppose that the dynamics a(t, x) of the technology at any point x has the form 14

(30) ∂a(t, x) ∂t = (a(t, x) -a π ) 3 + ψ(x)(a π -a(t, x))
with the initial value a(0, x) such that 1 < a(0, x) < a π for all x.

Remark that in the specification we use here the temporal evolution of a is determined separately for any spatial point.

This equation is made of two parts. The first part (a(t, x) -a π ) 3 describes the fact that, in absence of knowledge spillovers, the differences in technology between the center and the periphery tends to increase due to the accumulation process. The second part ψ(x)(a π -a(t, x)) represents knowledge spillovers which are higher when technological differences are larger and depends on the distance with the center through the learning capabilities function ψ introduced above.

4.1. Spatiotemporal technological catching-up dynamics with spatially homogenous diffusion. We represent three possible dynamics of the system depending on the strength and on the form of the learning capability function and then on the knowledge spillovers intensity. The choice of other parameters is the same in all the figures. We keep the value of various parameters constant constant in space to emphasize the (possible)

13 It is a π = 1.2 in the numerical simulations.

14 The solution to (30) might become smaller than 1 depending on the value of ψ(x). This does not happens for the parameters and the time-horizon we use in our figures but, to be consistent with our theoretical results, one can bound the Nocco's construction in order to remain in the range allowed by our hypotheses for all times. Since the optimal values of the input at any time t found in Theorem 3.6 only depends on the value of a at the same time t this manipulation is irrelevant in the illustrations which follow.

endogenous divergence dynamics driven by technological accumulation process. Their values are the following: the time discount ρ is 0.03, the diffusion coefficient σ is 0.5, the nature local regeneration capacity δ is 0.2, the pollution impact of emissions η is constant in time (and space) and equal to 1 as the local environmental awareness w. Finally the inverse of the elasticity of intertemporal substitution in consumption γ is time and space independent and equal to 0.5. The values of these parameters do of course matter in the shape of the optimal spatiotemporal dynamics calculated, but our here aim is to compare the outcomes of three different learning capabilities schemes, and this comparison is unaffected by the latter parameters' choices. In Figure 1 we study the no-learning benchmark. In this case we take a (and b) equal to 0 in (29) so that no knowledge spillovers is at work. This specific form of ψ is represented in the first image of Figure 1. As already said, the second graphic represents the initial spatial distribution of the technology. The third graphic displays the evolution of the technological level with respect to the center's, that is for any location x, the ratio a(t, x)/a π where a(t, x) follows the dynamics described by ( 30). In the absence of spillover effects a strong divergence dynamics arises and the relative distance among technologies in the core and in the periphery increases over time. This kind of behavior is reproduced in the investment and output dynamics (fourth and fifth graphics of Figure 1). The sixth graphic of Figure 1 gives the dynamics of the stock of pollution 15 . Not surprisingly the regions where input and then emissions are higher are the more polluted and differences accentuate over time due to the divergence process in input (and then emission) among locations over time. Observe that since pollution is transboundary, the relative difference between core and periphery in terms of pollution stock is milder than the relative difference of emissions.

In Figure 2 we have a low-spillover scenario, a situation where the learning/spillover dynamics is active but it is not very strong. In particular, for regions which are too far from the core the learning function ψ turns to be zero. More precisely in the definition of ψ given in (29) we fix now 15 The initial datum for the pollution stock profile is found by considering an history of past emissions always equal to the initial emission profile appearing in the fifth graphic. In Figure 3 we consider the high-spillover scenario: knowledge spillover are strong in all locations. 16 Now the knowledge spillovers are strong enough to offset the divergent forces at work in all locations. We see this fact in the spatial dynamics of all the variables: relative level of technology, input and production and also in the dynamics of the spatial profile of the pollution stock which, in the long run converges toward the same value in all the regions. The same set-up is replicated in Figure 4 where we look at the dynamics for a longer time interval. In other words, pollution diffuses less quickly, say, on the right halfcircle. To make things simple, and in particular to save space, we just give the main implications of this new specification for the optimal short-term spatiotemporal dynamics in the no-learning case (corresponding to Figure 1 above). As we will see in a minute, this is enough to illustrate how effective our methodlogy works in the much less commonly analyzed (but more realistic) cases. Figure 6 delivers the counterpart outcomes to Figure 1 with homogenous diffusion speed. It's easy to see that similar core-periphery features, already commented above, still occur. This is hardly surprising. To get a much better sense of the differences brought by heterogenous diffusion speed, one has to resort to finer outcomes. Figure 7 compares the surface sections at t = 15 of optimal pollution in the heterogenous vs homogenous diffusion cases. That is it gives the spatial distributions of pollution in the two cases at t = 15 with the dashed line representing the homogenous diffusion case. One can observe that spatial heterogeneity brings a clear additional asymmetry to the pollution shape over space though the average diffusion speed is the same in the homogenous and heterogenous diffusion cases. Figure 7 shows the polution distribution to be more skewed to the right in the heterogenous case. This is indeed reasonable as we are assuming that the diffusion speed is on average lower in the right half-circle, eventually leading to less pollution mobility in the latter region. This is the kind of sensible results which can be fully internalized in our framework in the determination of optimal depollution policies for example. This is only an illustration of the kind of outputs our methodology can generate as we get exploiting step by step the very rich set of spatial and temporal heterogeneities allowed.

Conclusion

In this paper, we have provided with a generic spatiotemporal non-linearquadratic framework for transboundary pollution control. The objective functional to be maximized is a Benthamite social welfare function depending on the intertemporal stream of consumption at any location, and internalizing the spatial externalities resulting from pollution diffusion. The essential contribution of this work is to identify optimal pollution control policies with a very large account of geographic heterogeneity: (i) heterogeneity in productivity and in ecological efficiency of the production process, which also includes the broad spatio-temporal characteristics of the exogenous technological process; (ii) heterogeneity in preferences, notably in the intertemporal elasticity of substitution and in the disutility from the pollution, and finally: (iii) the heterogeneity in the environmental/ecological context, in particular in terms of speed of diffusion of pollutants and local regeneration capacity.

Despite the complexity of the problem, we have been able to produce a solution method which has two unexpected virtues (given the complexity of the task). First, it allows for closed-form solutions, and second, the solutions produced are based on a neatly singled out spatial function with a clear economic interpretation. In this paper, we have specially focused on one of the most complicated aspects in the study of this class of models:

the analysis of the inherent non-autonomous problems and the determination of the optimal spatiotemporal transition dynamics using our analytical results. We show how our methodology can be effective in this respect with an application to a problem with technological spatiotemporal dynamics, generalizing and complementing previous companion work in [START_REF] Boucekkine | From firm to globallevel pollution control: The case of transboundary pollution[END_REF].

by parts yields

Lϕ, ϕ = S 1 (Lϕ)(x)ϕ(x)dx = - S 1 σ(x)|ϕ (x)| 2 dx - S 1 δ(x))|ϕ(x)| 2 dx ≤ 0 (31) Since C ∞ (S 1 ; R) is a core for L, (32) extends to all functions ϕ ∈ D(L),
showing that the operator L is dissipative. Similarly, a double integration by parts shows that Using again the fact that e (t-s)L is a contraction and Assumption 3.1, we have, for each s ≥ 0, T ≥ 0

(32) Lϕ, η = ϕ, Lη , ∀ϕ, η ∈ C ∞ (S 1 ; R). Again, since C ∞ (S 1 ; R) is a core for L, (
T s e -(ρ-L)(t-s) Ψ(s)I(s)dt ≤ ∞ s e -ρ(t-s) Ψ(s)I(s) dt ≤ 1 ρ Ψ(s)I(s) .
Hence, by definition of A, the claim follows sending T to +∞.

Proof of Proposition 3.4. The fact that α solves ( 13) and the fact that, by Assumption 3.1, we have σ(•) > 0 yield

α (x) = 1 σ(x)
[(ρ + δ(x))α(x) -σ (x)α (x) -w(x)] , for a.e. x ∈ S 1 .

Since α ∈ C 1 (S 1 ; R), it follows, by Assumption 3.1, that α ∈ C 2 (S 1 ; R). Now, let x * ∈ S 1 be a minimum point of α over S 1 . Then α (x * ) ≥ 0.

Plugging this into (13) we get

(ρ + δ(x * ))α(x * ) = σ(x * )α (x * ) + w(x * ) ≥ w(x * ),
and the estimate from below follows. The estimate from above can be obtaind symmetrically.

Proof of Proposition 3.5. Case σ → 0 + . First, notice that under the above assumptions (13) reads as

(34) ρα σ o (x) -σ o α σ o (x) + δ(x)α σ o (x) = w(x), x ∈ S 1 , By Proposition 3.4 we have α * (x) := lim inf σ o →0 + α σ o (ζ) : σ o ≤ σ o , ζ ∈ S 1 , |ζ -x| ≤ 1/σ o ≥ min S 1 w ρ + δ , α * (x) := lim sup σ o →0 + α σ o (ζ) : σ o ≤ σ o , ζ ∈ S 1 , |ζ -x| ≤ 1/σ o ≤ max S 1 w ρ + δ .
Clearly α * ≥ α * . By stability of viscosity solutions (see e.g. [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF], the latter functions are, respectively, (viscosity) super-and sub-solution to the limit equation ρα 0 (x) + δ(x)α 0 (x) = w(x), whose unique solution is

α 0 (x) = w(x) ρ + δ(x)
. for some α ∞ ∈ C(S 1 ; R). Again by stability viscosity solutions, α ∞ must solve, in the viscosity sense, the limit equation

α ∞ (x) = 0, x ∈ S 1 .
Hence, it must be α ∞ ≡ c 0 for some c 0 ≥ 0. To find the value of c 0 we may integrate (34) over S 1 getting S 1

(ρ + δ(x))α σ o (x)dx = S 1 w(x)dx.

Letting σ o → +∞ above, we get c 0 = S 1 w(x)dx S 1 (ρ + δ(x))dx .

As this value does not depend on the sequence σ n chosen, the claim follows.

Proof of Theorem 3.6. (i) Using ( 9) it is possible to rewrite the second part of (10). We first set e -(ρ-L)t := e -ρt e tL , t ≥ 0, and we write We look now at the last term of the last line of (36). It can be rewritten by exchanging the integrals as follows: (ii) After writing explicitely the inner products in (10), the integral can be optimized pointwisely. We end up, for (t, x) ∈ R + × S 1 fixed, with the optimization sup ι≥0 (a(t, x) -1)ι 1-γ(t,x) 1 -γ(t, x)

-α(x)η(t, x)ι ; so, we easily get the claimed expression (18) of the candidate unique optimal control I * . On the other hand, we need to verify that I * ∈ A. Indeed, by Assumption 3.1, we have Ψ(t)(x)I * (t)(x) = α(x) -1 γ(t,x) η(t, x) 1-1 γ(t,x) (a(t, x) -1)

1 γ(t,x) -1 = α(x) -1 γ(t,x)
a(t, x) -1 η(t, x)

1 γ(t,x) -1
.

Since α is bounded from above and from below by positive constants, then so is α(x) -1 γ(t,x) by Assumption 3.1(ii). Consequently, by Assumption 3.1(iii), we get, for some C 0 > 0 0 ≤ Ψ(t)(x)I * (t)(x) ≤ C 0 e gt , ∀x ∈ S 1 .

Since ρ > g by Assumption 3.1(iv), we get I * ∈ A. We can write explicitly the Fourier coefficients p * n (t) by the following argument. By Proposition 3.2, Chapter 1, Part II of Bensoussan et al. (2007), the function P * defined in ( 19) is also a weak solution to (8) with I = I * , i.e., taking into account that L is self-adjoint, i.e. L = L , it holds P * (t), ϕ = p 0 , ϕ + Proof of Proposition 3.9. First we observe that, in this case, e 0 (•) ≡ 1 √ 2π , λ 0 = -δ 0 . Hence From the mild form of P given in (9) we now get, for t ≥ 0, P * (t), e 0 = e tL p 0 , e 0 + t 0 e (t-s)L Ψ(s)I * (s), e 0 ds = p 0 , e tL e 0 + t 0 Ψ(s)I * (s), e (t-s)L e 0 ds = p 0 , e -δ 0 t e 0 + t 0 Ψ(s)I * (s), e -δ 0 (t-s) e 0 ds, where we used that e -δ 0 t is the eigenvalue of e tL associated to e 0 . The claim immediately follows.

FinallyP

  , given h, k ∈ H, define (hk)(x) := h(x)k(x). Then, with the identification P (t) = p(t, •), we reformulate (4) in H as (t) = LP (t) + Ψ(t)I(t), t ≥ 0,P (0) = p 0 ∈ H,According to Definition 3.1(v), Chapter 1, Part II, ofBensoussan et al.

  3.4.1. Transitional dynamics through series expansion. Recall that a non identically zero function φ ∈ D(L) is called eigenfunction of L if there exists a real number (called eigenvalue) λ such that Lφ = λφ. Proposition 3.8. Let Assumption 3.1 hold. There exists a decreasing sequence {λ n } n∈N ⊂ (-∞, 0] such that λ n → -∞ and an orthonormal basis {e n } n∈N ⊂ H such that e n ∈ D(L) and Le n = λ n e n ∀n ∈ N.

Figure 1 .

 1 Figure 1. The no knowledge spillover case. Evolution of technological level, investment and output (with respect to the technological center) and pollution stock in time and space.

Figure 2 .

 2 Figure 2. The low knowledge spillover case. Evolution of technological level, investment and output (with respect to the technological center) and pollution stock in time and space.

a

  = 0.371 and b equal to 0.15. The corresponding learning function is represented in the first graphic of Figure2. We observe that the dynamics is now qualitatively different compared to the benchmark. In some locations, the magnitude of learning capabilities, ψ, and the initial technological level are high enough to give rise to a "catch up" so that in the long-run the values of the variables of interest (production, investment and pollution stock) are the same as in the core. Other regions, in particular the more peripheral ones with zero learning capabilities, remain far from the frontier, ultimately experiencing levels of investment and production lower and lower with respect to the core. So differently from the benchmark, we have a more articulated picture: some of the non-core locations reach in the long run the same level of economic activity as the core, others no.

Figure 3 .

 3 Figure 3. The high knowledge spillover case. Evolution of technological level, investment and output (with respect to the technological center) and pollution stock in time and space.

Figure 4 .

 4 Figure 4. The high knowledge spillover case with a longer time evolution.

Figure 5 .Figure 6 .

 56 Figure 5. The shape of the diffusion speed

Figure 7 .

 7 Figure 7. Comparison of the pollution spatial distributions, sections at t=15, the dashed curve for the benchmark case with constant diffusion speed.

  32) extends to all couples of functions in D(L), showing that L is self-adjoint, i.e. L = L , where L denotes the adjoint of L. Therefore, by[START_REF] Engel | One-parameter Semigroups for Linear Evolution Equations[END_REF] (in particular, Chapter II), L generates a strongly continuous contraction semigroup (e tL ) t≥0 ⊂ L(H); in particular, since ρ > 0, by standard theory of strongly continuous semigroup in Banach spaces (see, e.g. pages 82-83, Chapter II and Theorem 1.10, Chapter II of[START_REF] Engel | One-parameter Semigroups for Linear Evolution Equations[END_REF], it follows that ρ belongs to the resolvent set of L and that (7) holds.Proof of Proposition 3.3. First of all we observe that, by Assumption 3.1(ii), the first term in the functional is always positive (Case (A)) or always negative (Case (B)), possibly infinite. Hence to prove the claim it is enough to show that, given any p 0 ∈ H, the term ∞ 0 e -ρt w, P (t) dt is well defined and finite for every I ∈ A. We have ∞ 0 e -ρt w, P (t) dt = ∞ 0 e -ρt w, e tL p 0 + t 0 e (t-s)L Ψ(s)I(s)ds dt (33) Now, since w is bounded and e tL is a contraction, the integral ∞ 0 e -ρt w, e tL p 0 dt is finite. Moreover, for all T > 0 we get, by Fubini-Tonelliw, e -(ρ-L)(t-s) Ψ(s)I(s) ds dt = ρ-L)(t-s) Ψ(s)I(s)dt ds

  By standard comparison of viscosity solutions one has α* ≥ α 0 ≥ α * . It follows that ∃ lim σ o →0 + α σ o (x) = α * (x) = α * (x) = α 0 (x) ∀x ∈ S 1 . Case σ → +∞. First, we rewrite (34) as (35) α σ o (x) = 1 σ o [ρα σ o (x) + δ(x)α σ o (x) -w(x)] , x ∈ S 1 ,From this and from Proposition 3.4, we see that α σ o is equi-bounded and equi-uniformly continuous with respect to σ o ≥ 1. Hence, by Ascoli-Arzelà Theorem we have that, from each sequence σ n → +∞, we can extract a subsequence σ n k such that lim k→+∞ α σn k = α ∞ uniformly on x ∈ S 1 ,

∞ 0 e

 0 -ρt w, P (t) dt = ∞ 0 e -ρt w, e tL p 0 + t 0 e (t-s)L Ψ(s)I(s)ds dt = w, first term of the right hand side is the only one which depends on the initial datum and, by (7), it can be rewritten as ρ-L)t p 0 dt = w, (ρ -L) -1 p 0 = (ρ -L) -1 w, p 0 = α, p 0 , where α is defined in (11).

  w, e (t-s)L Ψ(s)I(s) ds dt = (ρ -L) -1 w, Ψ(s)I(s) ds Hence, we can finally rewrite (10) as (17).

(

  iii)-(iv) These claims immediately follow by straightforward computations. Proof of Proposition 3.8. Consider the operator (1 -L) -1 ∈ L(H). The range of this operator is the space D(L) = W 2,2 (S 1 ; R), which by Kondrachov's Theorem is embedded in H = L 2 (S 1 ; R) with compact embedding.It follows that (1 -L) -1 is compact; being also self-adjoint, by standard spectral theory in Hilbert spaces, there exists an orthonormal basis of eigenvectors for it, hence also for L. Hence, considering also that L is dissipative and unbounded, there exists a decreasing sequence{λ n } n∈N ⊂ (-∞, 0] such that λ n → -∞and an orthonormal basis {e n } n∈N ⊂ H such that (38) e n ∈ D(L) and Le n = λ n e n ∀n ∈ N. Consider the Fourier series expansion P * (t) = n∈N p * n (t)e n , where p * n (t) := P * (t), e n .

t 0 P

 0 * (s), Lϕ + Ψ(s)I * (s), ϕ ds ∀t ≥ 0, ∀ϕ ∈ D(L).In particular, taking into account (38), we have p * n (t) = P * (t), e n = p 0 , e n + t 0 λ n P * (s), e n + Ψ(s)I * (s), e n ds = p 0 , e n + t 0 λ n p * n (s) + ξ * n (s) ds ∀t ≥ 0, ∀n ∈ N.

Then

S 1 p

 1 * (t, x)dx = P * (t), 1 = n∈N p * n (t)e n , 1 = n∈N e n , 1 p * n (t), t ≥ 0,as claimed.

S 1 p

 1 * (t, x)dx = P * (t), 1 = √ 2π P * (t), e 0 .

The method works even with a decreasing returns abatement technology. This fairly straightforward extension is available upon request.

Our approach allows generalizations to compact finite dimensional manifolds without boundary (see, e.g.,[START_REF] Fabbri | Geographical structure and convergence: A note on geometry in spatial growth models[END_REF].

A good illustration of this is the impact of nomadism on environmental quality

Actually, rather than a space of functions, L 2 (S 1 ; R) is a space of equivalence classes of functions, with the equivalence relation identifying functions which are equal almost everywhere, i.e. out of a null Lebesgue measure set. For details we refer again to[START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF].

We refer to[START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF] for the notion of weak differentiability.

Falling into the Sturm-Liouville theory with periodic boundary conditions (see[START_REF] Coddington | Theory of ordinary differential equations[END_REF].

As explained in the introduction, the companion paper[START_REF] Boucekkine | From firm to globallevel pollution control: The case of transboundary pollution[END_REF] explores the counterpart long-run distributions for the different types of spatial heterogeneities using the counterpart of Proposition 3.10

above.11 Again, we could have chosen alternative specifications for technology diffusion. See a short survey in the 2005 Nocco's paper. The important common point to these specifications is that distance to the technological center matters in the strength of spillovers, which is an empirically established feature, as outlined in the

introduction. 12 These initial distributions are represented in the second graphics of Figures1, 2, 3. To make the point we keep the same initial distribution in all our simulations.

We set a = 0.571 and b equal to 0.15, the corresponding learning function is represented in the first graphic of Figure3) as before.

Acknowledgements

The work of Giorgio Fabbri is partially supported by the French National Research Agency in the framework of the "Investissements d'avenir" program (ANR-15-IDEX-02) and of the center of excellence LABEX MME-DII

Appendix A. Proofs Proof of Proposition 3.2. Due to Assumption 3.1, L is a closed, densely defined, unbounded linear operator on the space H (see, e.g. Lunardi, 1995, p. 71-75, Sections 3.1 and 3.1.1). A core for it is the space C ∞ (S 1 ; R) (see, e.g., Engel and Nagel, 1995, pages 69-70). Let ϕ ∈ C ∞ (S 1 ; R). Integration Proof of Proposition 3.10. In this case I * (•) ≡ Ī * ∈ H is time independent too. Since δ ≡ 0, we have λ 0 < 0. Let us write L = L 0 -λ 0 , where L 0 := L + λ 0 , and note that L 0 is dissipative by definition, hence e sL 0 is a contraction.

Then, setting Ψ := η(•) ∈ H, we can rewrite P * (t) = e λ 0 t e tL 0 p 0 + t 0 e λ 0 (t-s) e (t-s)L 0 Ψ Ī * ds = e λ 0 t e tL 0 p 0 + t 0 e λ 0 t e tL 0 Ψ Ī * ds, and take the limit above when t → ∞. Since e sL 0 is a contraction, the first term of the right hand side converges to 0, whereas the second one converges to

Then, the limit state P * ∞ ∈ H can be expressed using again Proposition 3.14, page 82 and Theorem 1.10, Chapter II of [START_REF] Engel | One-parameter Semigroups for Linear Evolution Equations[END_REF] as


V Sarou-Kanian

J.-C Rifflet

F Millot

UV -visible pyrometry of refractory oxides at high temperature

Keywords: optical pyrometry, high temperature, melting point, refractory oxide, UV

We report new experimental methods to determine temperatures of liquid and solid oxides in the range T=2300-3500 K. The development of a UV pyrometer working at λ=224 or 313 nm allows a comparison with more classical visible pyrometers. The principle consists of calibrating first a monochromatic UV or visible pyrometer aiming at a self-contained liquid pool of Y 2 O 3 or Sc 2 O 3 during its solidification respectively considered at T=2712 K and T=2740 K. This method has then been applied to the analysis of the free cooling curves of self contained liquid pools of 12 refractory simple di-and sesquioxides that have relatively wellknown properties. Particular attention has been paid to the comparison between recommended and measured solidification apparent temperatures at 5 different wavelengths between λ=224 and 800 nm. Comparable values are systematically obtained for apparent temperatures in the range [300-550 nm]. These values are in good agreement with recommended values for rare earth oxides.

INTRODUCTION

Optical pyrometry is a practical technique for measuring very high temperatures. However it also requires the knowledge of the emissivity of materials to get a relation between emitted light and temperature. Classes of materials are being studied for sharing specific optical properties and also because they can be heated in specific ways. Refractory oxides belong to such a class of materials and have been very much studied between 1930 and 1980. Sophisticated black body cavities heated in solar or more classical furnaces have been developed and have led to the recommendation or proposition of characteristic temperatures such as solidification of some oxides [1][2][3].

In the present work the advantages and drawbacks of using monochromatic pyrometry at different wavelengths in the range 0.2-0.8 µm are examined to evaluate the temperature of the surface or near surface of a refractory oxide material. Measurements are finally performed on well-known oxides that are compared with literature data. Conclusions that are specific to oxides can then be drawn from these studies.

THEORETICAL CONSIDERATIONS

There are few basic equations that have to be considered to relate the light emitted by a material and its temperature. The normal spectral radiance L λ (T) of a material at the temperature T and the wavelength λ is such as:

( ) ( ) ( ) ( ) ( ) 1 T C exp 1 C T T L T T L 2 5 1 - λ λ ⋅ ε = ⋅ ε = λ λ λ λ o (Eq. 1)
with ε λ (T) the normal spectral emissivity at T, ( )

T L o λ
the spectral radiance of the black body, and C 1 (=1.191 10 -16 W.m -2 .sr -1 ) and C 2 (=0.014388 m.K) respectively the first and second radiation constant.

The normal spectral emissivity is of great importance as it represents the optical properties of the material, and it has to be considered very carefully. From the first and second Kirchoff's laws, the normal spectral emissivity is expressed as:

( ) ( ) ( ) T T 1 T λ λ λ τ - ρ - = ε (Eq. 2)
with ρ λ (T) and τ λ (T) respectively the normal-hemispherical spectral reflectance and the normal-hemisperical spectral transmittance. In the case of a homogeneous and non-diffusive material, both properties are also related to the spectral reflectivity R λ (T) and the spectral transmissivity Tr λ (T) in the expressions [START_REF] Siegel | Thermal Radiation and Heat Transfer[END_REF]:

( ) ( ) ( ) ( ) [ ] T Tr T 1 T R T λ λ λ λ ⋅ τ + ⋅ = ρ (Eq. 3) ( ) ( ) [ ] ( ) ( ) ( ) T Tr T Tr T R 1 T R 1 T 2 2 2 λ λ λ λ λ ⋅ ⋅ - - = τ (Eq. 4)
R λ (T) depends on the refractive index n λ (Τ) and the extinction coefficient k λ (Τ). For normal incidence:

( ) ( ) ( ) ( ) ( ) ( ) ( ) T k 1 T n T k 1 T n T R 2 λ 2 λ 2 λ 2 λ λ + + + - = (Eq. 5)
while Tr λ (T) depends on the extinction coefficient and on the thickness d:

( ) ( ) [ ] ( ) [ ] d T exp / d T k 4 exp T Tr ⋅ α - = λ ⋅ ⋅ π - = λ λ λ (Eq. 6)
with α λ (T) the absorption coefficient (in m -1 ).

By combining the equations 2-6, the normal spectral emissivity can be expressed as:

( ) ( ) [ ] ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) λ ⋅ ⋅ π - ⋅ + + + - - λ ⋅ ⋅ π - - ⋅       + + + - - = ⋅ - - ⋅ - = ε λ λ λ λ λ λ λ / d T k 4 exp T k 1 T n T k 1 T n 1 / d T k 4 exp 1 T k 1 T n T k 1 T n 1 T Tr T R 1 T Tr 1 T R 1 T 2 λ 2 λ 2 λ 2 λ 2 λ 2 λ 2 λ 2 λ (Eq. 7)
All these equations indicate that the spectral emissivity of a material may depend on various parameters, and that it may depart from unity either if it is transparent, semi transparent or if it reflects the light. On the one hand the transparency may be due to a low absorption coefficient or/and a small thickness of the emitting material. On the other hand the reflexion process can originate from a high refractive index value or from a strong extinction coefficient as well.

The examination of the dielectric function of refractory materials (oxides, nitrides, carbides) at room temperature shows a strong electronic absorption in the UV range and a strong lattice vibrational absorption in the mid-IR. In this latter spectral range is located the Christiansen wavelength which corresponds to a refraction index n λ equal to 1 and an extinction coefficient k λ <<1. Therefore the reflectivity tends to zero, and if the material is sufficiently thick to be considered as opaque (Tr λ =0), the material behaves as a blackbody (ε λ =1) [START_REF] Rousseau | [END_REF]. The development of optical pyrometry at the Christiansen wavelength could be promising, however it strongly depends on the materials (λ Christiansen =6→50 µm).

Away from the Christiansen wavelength, the accuracy of the normal spectral emissivity has to be carefully examined. The relative error on ε λ (T) can be derived from Equation 1 as:

( ) ( ) ( ) T C exp 1 1 T T C T T 2 2 2 λ - - ⋅ ∆ λ = ε ε ∆ λ λ (Eq. 8)
In the UV-visible range, Wien's approximation (λT<<C 2 ) can be applied and the exponential term is negligible. Equation 8shows that the temperature error is proportional to the temperature squared, the wavelength and the relative error on spectral emissivity. As there is generally a poor knowledge of the effects of temperature and wavelength on the spectral emissivity of materials, the application of pyrometry at short wavelength seems to be advantageous. Figure 1 clearly demonstrates that the temperature error would not exceed tens of degrees in the UV range while it would rapidly reach one hundred degrees in the IR range.

Nevertheless, the smaller dynamic range of UV pyrometry allows exploring only a restricted region of temperature as compared to IR pyrometry 1 . The UV-visible range also represents advantages for pyrometric measurements as one can presume the existence of a spectral region allowing to have a sufficient opacity to make τ λ (T)→0, and k λ (T)<<1. Indeed by considering an absorption coefficient of 1000 cm -1 , that is k λ <10 -2 for λ<1 µm, 99% of the light is stopped after a thickness of 50 µm. In that case, the normal spectral emissivity only depends on the refractive index according to the following expression:

( ) ( ) ( ) ( ) ( ) 2 λ 1 T n T n 4 T R 1 T ε + ⋅ ≈ - = λ λ λ
(Eq. 9)

1 L°0 .3µm (3000K) / L°0 .3µm (2300K)>100 and L°5 µm (3000K) / L°5 µm (2300K)≈1.5.

These arguments made us choose the near UV pyrometry as more promising for a general approach of refractory oxides.

EXPERIMENTAL SET UP

The experimental set up is shown on Figure 2. A 800 Watt Coherent CO 2 laser operating at λ=10.6 µm heats the surface of the sample. Its beam diameter is 13 mm and the power of the laser is controlled in order to create a 5-10 mm liquid pool in the middle of the sample.

Pyrometers consist of a solar blind UV Hamamatsu photomultiplier tube and Radiotechnique photomultiplier tubes for the visible and near IR. Optical filters with a spectral width of 1-10% are used to select the wavelength domains. The effective wavelength of every pyrometer was calculated by convoluting the spectral response of the filter with that of the photomultiplier and the black body function at 2500 and 3000K. The signal of the visible and near IR pyrometers is chopped and amplified with a lock in amplifier whereas that of the UV pyrometer is directly amplified. The wavelength of the UV pyrometer is λ=224 nm for some experiments and it is changed to λ=313 nm for some others by changing the optical filter. The optical paths of the three visible pyrometers (λ=400-800 nm) are identical and they are also isochronous. After the switch off of the laser power, the cooling of the sample is recorded at 5000Hz with the four pyrometers during few seconds. For some cases, a high speed CCD camera (Kodak EktaPro 1000HRC, 1000 frames per second during 5 seconds) was used in order to observe the cooling events. Samples are obtained from 4N or 5N oxide powders. They are pressed uniaxially in the shape of a cylinder (diameter=13mm, height=5mm). They are finally sintered under air at 1500°C during 15 hours. As some samples are destroyed slowly under ambient air (for instance La 2 O 3 ), they are immediately examined after their preparation.

RESULTS AND DISCUSSION

Calibration of pyrometers

The photomultipliers used for the monochromatic pyrometer convert the luminance emitted by the material into a voltage U. This output voltage is related to Equation 1 as:

( ) ( ) 1 T C exp 1 T K U 2 - λ ⋅ ε ⋅ = λ λ (Eq. 10)
K λ represents the apparatus function taking into account the photoelectric conversion of the detector and the optical path.

The calibration of the pyrometers consists in determining K λ or the product K λ •ε λ (T) by using the known melting point of a high temperature material. Alumina (Al 2 O 3 ) is the only high temperature oxide which is an auxiliary reference in the international scale of temperatures with a congruent melting point at T=(2327 ± 6) K [1,6]. Figure 3 illustrates the cooling curves of alumina for four wavelengths (λ=313, 400, 550, 800 nm). as the laser radiation is very close to the Christiansen's wavelength, the energy is completely absorbed by alumina in few hundreds of microns (k 10.6 µm ≈ 10 -2 , [7]). This results in generating a strong temperature gradient between the extreme surface and the interior of the liquid pool. Furthermore it can be assumed that the absorption coefficient varies with the wavelength in such a way that the opacity increases as the wavelength decreases [START_REF] Anfimov | AIAA 28 th Thermophysics Conference[END_REF].

Therefore the UV pyrometer looks at the surface of an opaque liquid whereas the three visible pyrometers integrate the light coming from some semitransparent layer near the surface that present strong temperature gradients. Once the laser is switched off (transition 1/2), the surface becomes the coldest point of the emissive liquid which is strongly convective.

Therefore internal convection tends to reduce the temperature gradient near the surface and the four pyrometers indicate the same temperature during the cooling in the liquid state (region 2). Then the liquid-solid transition occurs with the formation of a solid layer (transition 2/3), and a plateau of several hundreds of milliseconds is observed which is used for calibration (region 3). The duration of the plateau seems to depend on the pyrometer wavelength. This behaviour indicates that the solid which is formed on top of the liquid is almost transparent, particularly in the UV range, and it allows measuring the solidification at the solid/liquid interface. The solidification of the sample stops the internal convection and new temperature gradients form. In that case the UV pyrometer detects for longer time the solidification plateau at 2327 K temperature of the solid/liquid interface. Finally a decrease of the temperature is observed after a plateau of 1 s corresponding to the end of the solidification process (region 4).

Although the liquid-solid transition Al 2 O 3 seemed practical to calibrate the pyrometers, it could not be used because of its relatively low temperature (T=2327 K). Indeed it did not allow to measure temperatures as high as 3500 K particularly in the UV range without changing the gain values of the amplifiers (→ the main drawback of the UV pyrometry, see note 1, p. 2). Therefore it has been decided to examine a new reference suitable for higher temperatures. Scandia (Sc 2 O 3 ) and yttria (Y 2 O 3 ) were good candidates for distinct reasons.

The melting point of Y 2 O 3 has been carefully studied within a task force on secondary standards for temperature measurement of the IUPAC commission [START_REF] Foëx | High Temp[END_REF]. The averaged value obtained form the ten laboratories is (2712 ± 12) K. On the contrary, the results on the melting point of Sc 2 O 3 are rather scattered between 2680 K and 2762 K [3]. An interesting value of T=2753 K is the one reported by Coutures et al. [START_REF] Coutures | [END_REF] which is based on a pyrometer calibration using the solidification of Y 2 O 3 in almost blackbody conditions. With the same approach, the melting point of Sc 2 O 3 was determined by calibrating the pyrometers on the melting point of Y 2 O 3 (T=2712 K). Good agreement was observed between the different pyrometers: 2740 ± 5 K (Figure 4) and with the value reported by Coutures et al. [START_REF] Coutures | [END_REF]. The first point is that the phase transitions (liquid/solid and solid/solid) are no more a nice plateau as it is for alumina but instead an abrupt change in the shape of the curve that some authors have qualified by the term "pyrometric accident". Its origin is not to be found in thermodynamic differences with alumina since the heat of solidification is almost the same for Al 2 O 3 , Y 2 O 3 and Sc 2 O 3 [11]. It may be suggested that the opacity could explain these differences. If the solid is no more transparent (as it is for alumina), the pyrometer only detects light emitted from the sample surface. The observation with the high speed camera showed that the solidification is very short time when the solid starts to form on the surface.

Therefore the pyrometer detects the cooling of the upper part of the solid surface (cold point) while liquid is still solidifying beneath this top solid layer (hot point). In fact the scattering of the values obtained for the melting point of Y 2 O 3 (Figure 4) may result from the relative uncertainty of the solidification event observed by the pyrometer. The consistency between two pyrometers during a particular experiment is much better than the consistency between two experiments even performed in the same conditions.

Temperature measurements on other oxides

Several simple oxides have quite high solidification temperatures. We have discarded from the list these that could be difficult to handle like BeO and those from the uranyl group. [12]. The last accident is certainly the one reported as the solidification temperature in literature although the equilibrium one is somewhere between the first and the second pyrometric accidents. The value reported here corresponds to the second accident in order to compare with literature. The 800 nm pyrometer (a very popular pyrometer) is departing from recommended values showing clearly the increasing influence of the semi-transparency on the apparent temperature values in the near and mid IR. In the medium range (λ=300-550 nm), the good agreement is obtained because the opacity is sufficient to exclude the artefacts of the semi transparency, and at the same time the refraction index is only slightly varying [14]. The errors on the melting temperature that it is expected for cubic hafnia (n=2.15) using scandia (n=2) [15] to calibrate the pyrometers vary from 7 K at λ=400 nm to 13 K at λ=600 nm using Equation 9and values of n λ in the literature. This error is in any case much smaller than the uncertainty of 35K reported in the literature [2]. Finally the temperature measured at λ=224 nm indicates a slight decrease of the apparent melting temperature of ZrO 2 and HfO 2 . This point could be explained by the difference between the change of the refraction index of these oxides and that of Y 2 O 3 and Sc 2 O 3 in this region where n λ varying more strongly.

The reproducibility of the measurements of the melting temperature shows (Figure 9) that the values can vary within 50 K. This uncertainty may be found in the non reproducibility of the experiment itself instead of that of the pyrometers which indicate much more coherent values between pyrometers during a single shot. It should then be proposed that the term of "pyrometric accident" is a better qualification than the more academic term "solidification plateau" for these oxides.

CONCLUSIONS

The pyrometry of refractory oxides is examined on a dozen of well-known compounds. It is observed the existence of a wavelength range from 300nm to 550nm that is particularly useful to get correct temperature measurements. It first consists in performing a calibration of the pyrometer with the melting points of Sc 2 O 3 (2740 ± 5 K) or Y 2 O 3 (2712 ± 12 K), and by considering that the spectral emissivity of any material is the same as the calibration one.

Although this latter assumption is generally not verified, this procedure does not introduce important errors because the corrections (10 K) that are expected are in the range of the precision of the method (10 K) as long as the surface of the emitting solid is sufficiently opaque at the considered wavelength. It is also observed that the main problem of the pyrometric measurement on refractory oxides is the non reproducibility of the solidification event (50K). Finally, in the case of silicates and aluminates, it is recommended to shift the wavelength range [300-550 nm] down to λ=224 nm since they are more transparent in the near UV than the sesquioxides as well as zirconia and hafnia.

Figure 1 :

 1 Figure 1 : Error on the temperature measurement as a function of the wavelength considering different values of relative error of the spectral emissivity for T=2500 K.

Figure 2 :

 2 Figure 2 : Experimental set-up.

Figure 3 :

 3 Figure 3 : Cooling curves of alumina for four different wavelengths. Regions -1: laser on; 1/2: laser cut-off; 2: liquid cooling; 2/3: liquid-solid transition; 3-4: solidification.

Figure 4 :

 4 Figure 4 : Melting point of Sc 2 O 3 using pyrometers at different wavelengths and calibrated with the melting point of Y 2 O 3 at T=2712 K.

Figure 5 :

 5 Figure 5 : Cooling curves of Y 2 O 3 and Sc 2 O 3 for the UV wavelengths. L=liquid; solid phases: H=hexagonal; C=cubic.

Figure 6 :

 6 Figure 6 : Cooling curves obtained with the UV pyrometer for sesquioxides from La 2 O 3 to Dy 2 O 3 .L=liquid; solid phases: X=cubic high temperature; H=hexagonal; T=tetragonal; B=monoclinic; C=cubic. *=laser cut-off; ∞=saturated signal.

Figure 7 :

 7 Figure 7 : Cooling curves obtained with the UV pyrometer for sesquioxides from Ho 2 O 3 to Lu 2 O 3 and for ZrO 2 and HfO 2 . L=liquid; solid phases: X=cubic high temperature; H=hexagonal; C=cubic.*=laser cut-off; ∞=saturated signal.

Figure 8 :

 8 Figure 8 : Averaged apparent temperatures of melting of few oxides compared with recommended values (lines).

Figure 9 :

 9 Figure 9: Melting temperature of rare-earth sesquioxides. •: This work measured at λ=224 nm and λ=313nm. : Literature [3].

  Others like CaO and MgO which evaporate at high temperature are also left aside. Rare earth oxides as well as HfO 2 and ZrO 2 have been considered. Examples of cooling curves obtained with the UV pyrometer (λ=224 nm or 313 nm) are presented on Figure6and Figure 7. The cooling curves have very similar shapes to that of Y 2 O 3 and Sc 2 O 3 , presenting "pyrometric accidents" instead of plateaus as it is observed in the case of Al 2 O 3 . The solidification of the liquid pool corresponds to the first accident on cooling and it is generally quite clearly observed. Then the following accidents are related to the solid-solid phase transitions. It can be noticed that the UV pyrometer detects the very small variation of temperature due to the phase transition as for Gd 2 O 3 and Dy 2 O 3 . For Yb 2 O 3 and Lu 2 O 3 , the solidification of a metastable solid is just later abruptly changed to the stable phase. This phenomenon has already been reported for Yb 2 O 3 by Misuno et al