We discuss local and global existence and uniqueness for the price formation
free boundary model with homogeneous Neumann boundary conditions introduced by
Lasry & Lions in 2007. The results are based on a transformation of the problem
to the heat equation with nonstandard boundary conditions. The free boundary
becomes the zero level set of the solution of the heat equation. The
transformation allows us to construct an explicit solution and discuss the
behavior of the free boundary. Global existence can be verified under certain
conditions on the free boundary and examples of non-existence are given

Caffarelli, Luis A.

Markowich, Peter A.

Wolfram, Marie-Therese

English

arXiv

Luis A. Caffarelli

Peter A. Markowich

Marie-Therese Wolfram

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 349 (2011) 841–844Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comMathematical Analysis/Partial Differential EquationsOn a price formation free boundary model by Lasry and Lions:The Neumann problemSur un modèle de formation des prix à frontière libre introduit par Lasry et Lions :Le problème de NeumannLuis A. Caffarelli a, Peter A. Markowich b,c, Marie-Therese Wolfram ca Department of Mathematics, Institute for Computational Engineering and Sciences, University of Texas at Austin, USAb DAMTP, University of Cambridge, Cambridge CB3 0WA, UKc Faculty of Mathematics, University of Vienna, 1090 Vienna, Austriaa r t i c l e i n f o a b s t r a c tArticle history:Received 6 June 2011Accepted after revision 6 July 2011Available online 27 July 2011Presented by Pierre-Louis LionsWe discuss local and global existence and uniqueness for the price formation free boundarymodel with homogeneous Neumann boundary conditions introduced by Lasry and Lionsin 2007. The results are based on a transformation of the problem to the heat equationwith nonstandard boundary conditions. The free boundary becomes the zero level set ofthe solution of the heat equation. The transformation allows us to construct an explicitsolution and discuss the behavior of the free boundary. Global existence can be verifiedunder certain conditions on the free boundary and examples of non-existence are given.© 2011 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éNous discutons l’existence locale et globale, ainsi que l’unicité des solutions pour lemodèle de formation des prix à frontière libre avec des conditions aux bords de Neumannhomogènes introduit par Lasry et Lions en 2007. Nos résultats sont basés sur unetransformation de ce problème en une équation de la chaleur avec des conditions aux bordsnon standard. La frontière libre devient la ligne de niveau zéro de la solution de l’équationde la chaleur. Cette transformation nous permet de construire une solution explicite et dediscuter le comportement de la frontière libre. L’existence globale peut être vérifiée souscertaines conditions sur la frontière libre, et nous donnons des exemples de non-existence.© 2011 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionIn this Note we discuss the mathematical analysis of a price formation mean field model, as stated in [6]. The equationmodels the trading of an economic good between a large group of buyers and a large group of vendors. It consists of anon-linear parabolic free boundary evolution equation which describes the evolution of the densities of buyers and vendorsand also determines the price of the good. Here we consider the problem on the real interval (−L, L), where L denotes themaximum and −L the minimum price. The model is given by the free boundary value problemE-mail addresses: caffarel@math.utexas.edu (L.A. Caffarelli), P.A.Markowich@damtp.cam.ac.uk (P.A. Markowich), marie-therese.wolfram@univie.ac.at(M.-T. Wolfram).1631-073X/$ – see front matter © 2011 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2011.07.006842 L.A. Caffarelli et al. / C. R. Acad. Sci. Paris, Ser. I 349 (2011) 841–844ft − fxx = λ(t)(δ(x − p(t) + a) − δ(x − p(t) − a)), x ∈ (−L, L), t > 0, (1a)λ(t) = − fx(p(t), t), f(p(t), t) = 0, (1b)f (x,0) = f I , p(0) = p0, for some p0 in (−L + a, L − a), (1c)fx(±L, t) = 0, t > 0, (1d)with 0 < a < L and compatibility conditions at time t = 0: f I (p0) = 0 and f I (x) > 0 for x < p0 and f I (x) < 0 for x > p0.Throughout this paper we assume that f I is in L2(−L, L) and denote the L2 norm on (−L, L) by ‖ · ‖. System (1) hasbeen studied in a number of papers, see [3,7,2]. Here we present global (non-)existence results of a solution of (1) on thebounded interval (−L, L).2. Analysis of the Neumann problem – transformation to the heat equationIn the sequel we denote the positive and negative part of a function f defined for x ∈ (−L, L) by f + := max( f ,0),f − := max(− f ,0). We extend f + , f − by the value 0 outside the interval (−L, L).Proposition 2.1. Let f be a solution of the modified price formation problem consisting of Eq. (1a) with conditions (1b), initial datum(1c) and modified boundary conditions (bc):fx(−L, t) ={0, p(t) > −L + a,fx(−L + a, t), p(t)  −L + a, fx(L, t) ={0, p(t) < L − a,fx(L − a, t), p(t)  L − a. (2)Then the functionF (x, t) ={∑∞n=0 f +(x + na, t), x < p(t),−∑∞n=0 f −(x − na, t), x > p(t), (3)is a solution of the following BVP for the heat equationFt = Fxx, for all x ∈ (−L, L), t > 0, (4a)Fx(±L, t) = Fx(±L ∓ a, t), t > 0, (4b)F (x, t = 0) = F I (x), for all x ∈ (−L, L), (4c)where F I is constructed from f I according to (3). Conversely, if F is a solution of (4) then f (x, t) = F (x, t)− F +(x+a, t)+ F −(x−a, t)satisfies (1a), (1b), (1c), (2).The construction of the function F is motivated by the fact that f + ( f −) has jump-discontinuities at x = p(t) − a(x = p(t) + a) and x = p(t) of equal magnitude but opposite signs. The summation procedure then moves the jumps in thederivatives of f outward. Note that the sum in (3) actually consists only of finitely many terms and that the free boundaryp = p(t) of the price formation problem becomes the zero level set of the heat solution and vice versa. In principle the freeboundary can leave and reenter the interval (−L +a, L −a) without impeding the existence of a global solution of (1a), (1b),(1c), (2).We shall now construct an ‘explicit’ solution of (4) by separation of variables. We set F (x, t) = ϕ(x)ψ(t) and find ϕ′′/ϕ =ψ̇/ψ = −z2. The boundary conditions give the equation G(z) := cos(zL) − cos(z(L − a)) = 0 with the eigenfunctions ϕ(x) =A sin(zx) and H(z) := sin(zL) − sin(z(L − a)) = 0 with eigenfunctions ϕ = B cos(zx). We easily compute G(z) = 0 iff z =(2π l)/a, (2π l)/(2L − a) and H(z) = 0 iff z = (2π l)/a, (π(2l − 1))/(2L − a) for l ∈ Z. Next we conclude that H = H(z) isa sine-type function of type L with simple and separated zeros. Therefore {exp(i 2π la x),exp(i π(2l−1)2L−a x)} is a Riesz basis inL2(−L, L), see [1]. Also G = G(z) is a sine-type function of type L, its zero are separated except z = 0, which is a zero oforder 2. Hence {exp(i 2π la x),exp(i 2π l2L−a x), x}l∈Z is a Riesz basis in L2(−L, L), see Theorem D in [5]. By separately consideringeven and odd parts we conclude that {cos( 2π la x), sin( 2π la x), sin( 2π l2L−a x), cos( π(2l−1)2L−a x),1, x}, l = 1,2, . . . is a Riesz basis ofeigenfunctions of the heat equation (4). The solution of (4) isF (x, t) =∞∑l=1[(Al sin(ω1,lx) + Bl cos(ω1,lx))e−ω21,lt + Cl sin(ω2,lx)e−ω22,lt + Dl cos(ω3,lx)e−ω23,lt] + A0x + B0, (5)with ω1,l = (2π l)/a, ω2,l = (2π l)/(2L − a), ω3,l = ((2l − 1)π)/(2L − a). Note that Al , Bl , Cl , Dl , A0, B0 can be determineduniquely for every initial datum F I ∈ L2(−L, L).Theorem 2.1 (Global existence). The BVP (1a), (1b), (1c), (2) has a unique global solution f = f (x, t) for t > 0. Furthermore the freeboundary p = p(t) is a smooth graph p(t) ∈ (−L, L) for all t > 0.L.A. Caffarelli et al. / C. R. Acad. Sci. Paris, Ser. I 349 (2011) 841–844 843The proof follows from the construction described above. The min–max principle implies that p = p(t) is a graph fort > 0 and the Hopf principle implies that p is smooth. Finite-time oscillations of p(t) are excluded by the x-analyticity ofsolutions of the heat equations (see [2]). Since 0 <∫ −L+a−L F (x, t)dx and 0 >∫ LL−a F (x, t)dx are conserved in time (by theequation and the boundary conditions) we conclude that p(t) ∈ (−L, L) for all t > 0.Clearly, a solution of (1a), (1b), (1c), (2) is a solution of (1) on a time interval [0, T ] iff p(t) ∈ (−L +a, L −a) for t ∈ [0, T ].Then the total mass of buyers MB =∫ p(t)−∞ f (x, t)dx and vendors MV = −∫ ∞p(t) f (x, t)dx are time conserved quantities.Theorem 2.2. The BVP (1) has a global solution conserving the total mass of buyers and vendors iff the zero level set p of the solutionof (4) satisfies p(t) ∈ (−L + a, L − a) for all t > 0. Then the free boundary p(t) converges to p∞ ∈ (−L + a, L − a).The proof of Theorem 2.2 is based on the following lemmas:Lemma 2.3. The solution F converges exponentially fast to F∞ = A∞x + B∞ in L2(−L, L).Proof. From the Riesz base property we deduce that there exist c1, c2 ∈ R+ such that c1‖F I‖2  ∑∞l=1(A2l + B2l + C2l + D2l )+A20 + B20  c2‖F I‖2. For F̃ = F − (A0x + B0) we deducec1‖ F̃‖2 ∞∑l=1[(A2l + B2l + C2l + D2l)]e−2γlt  e−2γ1t∞∑l=1(A2l + B2l + C2l + D2l) c2e−2γ1t‖F I‖2,with the sequence γl = min((4π2l2)/a2, (4π2l2)/(2L − a)2, ((2l − 1)2π2)/(2L − a)2), l = 1,2, . . . . Lemma 2.4. The solution (5) satisfies |Fx(x, t)|  supx∈(−L,L) |(F I )x| and |F (x, t)|  c, ∀x ∈ (−L, L), t > 0.Proof. The function V = Fx satisfies the heat equation with V (−L, t) = V (−L + a, t), V (L, t) = V (L − a, t) and initial con-dition V (x, t = 0) = (F I )x(x). If V assumes its maximum on the cylinder [−L, L] × [0, T ) at either boundary x = ±L, thenit must also assume a maximum (with the same value) in the interior (due to bc on V ). This contradicts the maximumprinciple, thus V must assume its maximum at t = 0. The same arguments hold for the minimum.Since F (x, t) = F (−L, t) + ∫ x−L Fx(y, t)dy we deduce aF (−L, t) = − ∫ −L+a−L F (x, t)dx + ∫ −L+a−L ∫ x−L Fx(y, t)dy dx = I1 + I2.We know that Fx is bounded, therefore |I2|  K . In addition ddt∫ −L+a−L F (x, t)dx = 0 and therefore∫ −L+a−L F (x, t)dx =∫ −L+a−L F I (x)dx. Thus F (−L, t) as well as F = F (x, t) are bounded uniformly on (−L, L) × R+ . Proof of Theorem 2.2. We know that F converges exponentially fast to F∞ = A∞x + B∞ , and that there exists a smoothgraph p = p(t) such that F (p(t), t) = 0. Now we assume that p(t) ∈ (−L + a, L − a) for t > 0, choose any sequence tn → ∞and conclude that there is a subsequence tnk such that p(tnk ) → p∞ ∈ [−L +a, L −a]. Let ϕ be a test function in D(p∞, L). Ifk is sufficiently large we conclude f (x, tnk ) < 0 for x ∈ suppϕ . Therefore∫ L−L f (x, tnk )ϕ(x)dx < 0. Since f (·, t) → f∞ there isa subsequence tnkl such that f (x, tnkl ) converges to f∞(x) pointwise a.e. in (−L, L). The function | f (x, tnk )|  K on [−L, L]for all k. Then we deduce from Lebesgue’s dominated convergence theorem that∫ L−L f (x, tnk )ϕ(x)dx →∫ L−L f∞ϕ(x)  0.Since f∞ = A∞x + B∞ + (A∞(x − a) + B∞)− − (A∞(x + a) + B∞)+ we conclude f∞  0 for x > p∞ and f∞  0for x < p∞ . From Lebesgue’s dominated convergence theorem, we conclude −∫ Lp(tnkl)f (x, t)dx → − ∫ Lp∞ f∞(x)dx  0.Analogously∫ p∞−L f∞ dx  0.Next we show that F∞ has a unique zero in (−L, L). If F∞ = A∞x + B∞  0 on (−L, L) then F −∞ = 0, F +∞ = F∞and therefore f∞ is constant in (−L, L), which is a contradiction. The same argument holds for F∞  0. Therefore thefunction f∞ is given by f∞(x) = ±α for x ∈ (−L, p∞ − a) and x ∈ (p∞ + a, L) respectively and f∞ = −α/a(x − p∞) forx ∈ [p∞ − a, p∞ + a], with α ∈ R+ . We conclude that p∞ is unique and that p(t) → p∞ as t → ∞, since every sequencehas a subsequence which converges to the same limit. Theorem 2.5. Let f I be such that MB/MV /∈ [a/(4L − 3a), (4L − 3a)/a], where MB , MV denotes the initial mass of buyers andvendors. Then (1) does not have a global-in-time solution, which conserves both buyers and vendors masses.Proof. The result follows since α, p∞ cannot be adjusted so that p∞ ∈ [−L + a, L − a], where MB =∫ p∞−L f∞ dx, MV =− ∫ Lp∞ f∞ dx. Note that the choice of L, which corresponds to the maximally attainable price, is more or less arbitrary but a bad (toosmall) choice of L might impede global existence. In this case the model clearly looses its ‘practical’ significance. We remarkthat global existence results for (1) (with a free boundary which remains in (−L + a, L − a)) for initial data which are smallperturbations of stationary solutions are straightforward, without using the analytical machinery of [4].844 L.A. Caffarelli et al. / C. R. Acad. Sci. Paris, Ser. I 349 (2011) 841–844AcknowledgementsP.M. acknowledges support by the King Abdullah University of Science and Technology, the Leverhulme Trust and theRoyal Society. L.C. acknowledges support from the Division of Mathematical Sciences of the NSF, M.T.W. from the AustrianScience Foundation FWF.References[1] S.A. Avdonin, I. Joó, Riesz basis of exponentials and sine-type functions, Acta Math. Hungar. 51 (1) (1988) 3–14.[2] L.A. Caffarelli, P.A. Markowich, J.-F. Pietschmann, On a price formation free boundary problem by Lasry & Lions, C. R. Acad. Sci. Paris, Ser. I 349 (11–12)(2011) 621–624.[3] L. Chayes, M. González, M.P. Gualdani, I. Kim, Global existence and uniqueness of solutions to a model of price formation, SIAM J. Math. Anal. 41 (5)(2009) 2107–2135.[4] M. González, M.P. Gualdani, Asymptotics for a free-boundary problem in price formation, Nonlinear Anal. 74 (2011) 3269–3294.[5] M. Horváth, On the Muckenhoupt condition, Period. Math. Hungar. 18 (1) (1987) 53–58.[6] J.-M. Lasry, P.-L. Lions, Mean field games, Jpn. J. Math. 2 (1) (2007) 229–260.[7] P.A. Markowich, N. Matevosyan, J.-F. Pietschmann, M.-T. Wolfram, On a parabolic free boundary equation modeling price formation, Math. ModelsMethods Appl. Sci. 11 (19) (2009) 1929–1957.

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On a price formation free boundary model by Lasry and Lions: The Neumann problem

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On a price formation free boundary model by Lasry & Lions: The Neumann
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On a price formation free boundary model by Lasry & Lions: The Neumann problem

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