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Mellin transform for fractional integrals with general analytic kernel

  • Received: 27 November 2021 Revised: 21 January 2022 Accepted: 10 February 2022 Published: 14 March 2022
  • MSC : Primary: 58F15, 58F17; Secondary: 53C35

  • Many different operators of fractional calculus have been proposed, which can be organized in some general classes of operators. According to this study, the class of fractional integrals and derivatives can be classified into two main categories, that is, with and without general analytical kernel (introduced in 2019). In this article, we define the Mellin transform for fractional differential operator with general analytic kernel in both Riemann-Liouville and Caputo derivatives of order ς0 and ϱ be a fixed parameter. We will also establish relation between Mellin transform with Laplace and Fourier transforms.

    Citation: Maliha Rashid, Amna Kalsoom, Maria Sager, Mustafa Inc, Dumitru Baleanu, Ali S. Alshomrani. Mellin transform for fractional integrals with general analytic kernel[J]. AIMS Mathematics, 2022, 7(5): 9443-9462. doi: 10.3934/math.2022524

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  • Many different operators of fractional calculus have been proposed, which can be organized in some general classes of operators. According to this study, the class of fractional integrals and derivatives can be classified into two main categories, that is, with and without general analytical kernel (introduced in 2019). In this article, we define the Mellin transform for fractional differential operator with general analytic kernel in both Riemann-Liouville and Caputo derivatives of order ς0 and ϱ be a fixed parameter. We will also establish relation between Mellin transform with Laplace and Fourier transforms.



    Recently, an important research activity on mean field games (MFGs for short) has been initiated since the pioneering works [28,29,30] of Lasry and Lions (related ideas have been developed independently in the engineering literature by Huang-Caines-Malhamé, see for example [23,24,25]): it aims at studying the asymptotic behavior of stochastic differential games (Nash equilibria) as the number N of agents tends to infinity. In these models, it is assumed that the agents are all identical and that an individual agent can hardly influence the outcome of the game. Moreover, each individual strategy is influenced by some averages of functions of the states of the other agents. In the limit when N+, a given agent feels the presence of the others through the statistical distribution of the states. Since perturbations of the strategy of a single agent do not influence the statistical states distribution, the latter acts as a parameter in the control problem to be solved by each agent. The delicate question of the passage to the limit is one of the main topics of the book of Carmona and Delarue, [10]. When the dynamics of the agents are independent stochastic processes, MFGs naturally lead to a coupled system of two partial differential equations (PDEs for short), a forward in time Kolmogorov or Fokker-Planck (FP) equation and a backward Hamilton-Jacobi-Bellman (HJB) equation. The unknown of this system is a pair of functions: the value function of the stochastic optimal control problem solved by a representative agent and the density of the distribution of states. In the infinite horizon limit, one obtains a system of two stationary PDEs.

    A very nice introduction to the theory of MFGs is supplied in the notes of Cardaliaguet [9]. Theoretical results on the existence of classical solutions to the previously mentioned system of PDEs can be found in [19,20,21,28,29,30]. Weak solutions have been studied in [5,30,33,34]. The numerical approximation of these systems of PDEs has been discussed in [1,3,5].

    A network (or a graph) is a set of items, referred to as vertices (or nodes or crosspoints), with connections between them referred to as edges. In the recent years, there has been an increasing interest in the investigation of dynamical systems and differential equations on networks, in particular in connection with problems of data transmission and traffic management (see for example [12,14,17]). The literature on optimal control in which the state variable takes its values on a network is recent: deterministic control problems and related Hamilton-Jacobi equations were studied in [2,4,26,27,31,32]. Stochastic processes on networks and related Kirchhoff conditions at the vertices were studied in [15,16].

    The present work is devoted to infinite horizon stochastic mean field games taking place on networks. The most important difficulty will be to deal with the transition conditions at the vertices. The latter are obtained from the theory of stochastic control in [15,16], see Section 1.3 below. In [7], the first article on MFGs on networks, Camilli and Marchi consider a particular type of Kirchhoff condition at the vertices for the value function: this condition comes from an assumption which can be informally stated as follows: consider a vertex ν of the network and assume that it is the intersection of p edges Γ1,,Γp, ; if, at time τ, the controlled stochastic process Xt associated to a given agent hits ν, then the probability that Xτ+ belongs to Γi is proportional to the diffusion coefficient in Γi. Under this assumption, it can be seen that the density of the distribution of states is continuous at the vertices of the network. In the present work, the above mentioned assumption is not made any longer. Therefore, it will be seen below that the value function satisfies more general Kirchhoff conditions, and accordingly, that the density of the distribution of states is no longer continuous at the vertices; the continuity condition is then replaced by suitable compatibility conditions on the jumps across the vertex. Moreover, as it will be explained in Remark 11 below, more general assumptions on the coupling costs will be made. Mean field games on networks with finite horizon will be considered in a forthcoming paper.

    After obtaining the transmission conditions at the vertices for both the value function and the density, we shall prove existence and uniqueness of weak solutions of the uncoupled HJB and FP equations (in suitable Sobolev spaces). We have chosen to work with weak solutions because it is a convenient way to deal with existence and uniqueness in the stationary regime, but also because it is difficult to avoid it in the nonstationary case, see the forthcoming work on finite horizon MFGs. Classical arguments will then lead to the regularity of the solutions. Next, we shall establish the existence result for the MFG system by a fixed point argument and a truncation technique. Uniqueness will also be proved under suitable assumptions.

    The present work is organized as follows: the remainder of Section 1 is devoted to setting the problem and obtaining the system of differential equations and the transmission conditions at the vertices. Section 2 contains useful results, first about some linear boundary value problems with elliptic equations, then on a pair of linear Kolmogorov and Fokker-Planck equations in duality. By and large, the existence of weak solutions is obtained by applying Banach-Necas-Babuška theorem to a special pair of Sobolev spaces referred to as V and W below and Fredholm's alternative, and uniqueness comes from a maximum principle. Section 3 is devoted to the HJB equation associated with an ergodic problem. Finally, the proofs of the main results of existence and uniqueness for the MFG system of differential equations are completed in Section 1.

    A bounded network Γ (or a bounded connected graph) is a connected subset of Rn made of a finite number of bounded non-intersecting straight segments, referred to as edges, which connect nodes referred to as vertices. The finite collection of vertices and the finite set of closed edges are respectively denoted by V:={νi,iI} and E:={Γα,αA}, where I and A are finite sets of indices contained in N. We assume that for α,βA, if αβ, then ΓαΓβ is either empty or made of a single vertex. The length of Γα is denoted by α. Given νiV, the set of indices of edges that are adjacent to the vertex νi is denoted by Ai={αA:νiΓα}. A vertex νi is named a boundary vertex if (Ai)=1, otherwise it is named a transition vertex. The set containing all the boundary vertices is named the boundary of the network and is denoted by Γ hereafter.

    The edges ΓαE are oriented in an arbitrary manner. In most of what follows, we shall make the following arbitrary choice that an edge ΓαE connecting two vertices νi and νj, with i<j is oriented from νi toward νj: this induces a natural parametrization πα:[0,α]Γα=[νi,νj]:

    πα(y)=(αy)νi+yνjfor y[0,α]. (1)

    For a function v:ΓR and αA, we define vα:(0,α)R by

    vα(y):=vπα(y), for all y(0,α).

    The function vα is a priori defined only in (0,α). When it is possible, we extend it by continuity at the boundary by setting

    vα(0):=limy0+vα(y) and vα(α):=limyαvα(y).

    In that latter case, we can define

    v|Γα(x)={vα(π1α(x)),if xΓαV,vα(0)=limy0+vα(y),if x=νi,vα(α)=limyαvα(y),if x=νj. (2)

    Notice that v|Γα does not coincide with the original function v at the vertices in general when v is not continuous.

    Remark 1. In what precedes, the edges have been arbitrarily oriented from the vertex with the smaller index toward the vertex with the larger one. Other choices are of course possible. In particular, by possibly dividing a single edge into two, adding thereby new artificial vertices, it is always possible to assume that for all vertices νiV,

    either πα(0)=νi, for all αAi or πα(α)=νi, for all αAi. (3)

    This idea was used by Von Below in [35]: some edges of Γ are cut into two by adding artificial vertices so that the new oriented network ¯Γ has the property (3), see Figure 1 for an example.

    Figure 1. 

    Left: the network Γ in which the edges are oriented toward the vertex with larger index (4 vertices and 4 edges). Right: a new network ˜Γ obtained by adding an artificial vertex (5 vertices and 5 edges): the oriented edges sharing a given vertex ν either have all their starting point equal ν, or have all their terminal point equal ν

    .

    In Sections 1.2 and 1.3 below, especially when dealing with stochastic calculus, it will be convenient to assume that property (3) holds. In the remaining part of the paper, it will be convenient to work with the original network, i.e., without the additional artificial vertices and with the orientation of the edges that has been chosen initially.

    The set of continuous functions on Γ is denoted by C(Γ) and we set

    PC(Γ)={v:ΓR: for all αA,|vαC(0,α)  vα can be extended by continuity to [0,α].}.

    By the definition of piecewise continuous functions vPC(Γ), for all αA, it is possible to define v|Γα by (2) and we have v|ΓαC(Γα), vαC([0,α]).

    For mN, the space of m-times continuously differentiable functions on Γ is defined by

    Cm(Γ):={vC(Γ):vαCm([0,α]) for all αA}.

    Notice that vCm(Γ) is assumed to be continuous on Γ, and that its restriction v|Γα to each edge Γα belongs to Cm(Γα). The space Cm(Γ) is endowed with the norm vCm(Γ):=αAkmkvαL(0,α). For σ(0,1), the space Cm,σ(Γ), contains the functions vCm(Γ) such that mvαC0,σ([0,α]) for all αA; it is endowed with the norm vCm,σ(Γ):=vCm(Γ)+supαAsupyzy,z[0,α]|mvα(y)mvα(z)||yz|σ.

    For a positive integer m and a function vCm(Γ), we set for km,

    kv(x)=kvα(π1α(x)) if xΓαV. (4)

    For a vertex ν, we define αv(ν) as the outward directional derivative of v|Γα at ν as follows:

    αv(ν):={limh0+vα(0)vα(h)h,if ν=πα(0),limh0+vα(α)vα(αh)h,if ν=πα(α). (5)

    For all iI and αAi, setting

    niα={1if νi=πα(α),1if νi=πα(0), (6)

    we have

    αv(νi)=niαv|Γα(νi)=niαvα(π1α(νi)). (7)

    Remark 2. Changing the orientation of the edge does not change the value of αv(ν) in (5).

    If for all αA, vα is Lebesgue-integrable on (0,α), then the integral of v on Γ is defined by Γv(x)dx=αAα0vα(y)dy. The space

    Lp(Γ)={v:v|ΓαLp(Γα) for all αA},

    p[1,], is endowed with the norm vLp(Γ):=(αAvαpLp(0,α))1p if 1p<, and maxαAvαL(0,α) if p=+. We shall also need to deal with functions on Γ whose restrictions to the edges are weakly-differentiable: we shall use the same notations for the weak derivatives. Let us introduce Sobolev spaces on Γ:

    Definition 1.1. For any integer s1 and any real number p1, the Sobolev space Ws,p(Γ) is defined as follows: Ws,p(Γ):={vC(Γ):vαWs,p(0,α)αA}, and endowed with the norm vWs,p(Γ)=(sk=1αAkvαpLp(0,α)+vpLp(Γ))1p. We also set Hs(Γ)=Ws,2(Γ).

    After rescaling the edges, it may be assumed that α=1 for all αA. Let μα,αA and piα,iI,αAi be positive constants such that αAipiα=1. Consider also a real valued function aPC(Γ) such that for all αA, a|Γα is Lipschitz continuous.

    As in Remark 1, we make the assumption (3) by possibly adding artificial nodes: if νi is such an artificial node, then (Ai)=2, and we assume that piα=1/2 for αAi. The diffusion parameter μ has the same value on the two sides of an artificial vertex. Similarly, the function a does not have jumps across an artificial vertex.

    Let us consider the linear differential operator:

    Lu(x)=Lαu(x):=μα2u(x)+a|Γα(x)u(x),if xΓα, (8)

    with domain

    D(L):={uC2(Γ):αAipiααu(νi)=0, for all iI}. (9)

    Remark 3. Note that in the definition of D(L), the condition at boundary vertices boils down to a Neumann condition.

    Freidlin and Sheu proved in [15] that

    1. The operator L is the infinitesimal generator of a Feller-Markov process on Γ with continuous sample paths. The operators Lα and the transmission conditions at the vertices

    αAipiααu(νi)=0 (10)

    define such a process in a unique way, see also [16,Theorem 3.1]. The process can be written (Xt,αt) where XtΓαt. If Xt=νi, iI, αt is arbitrarily chosen as the smallest index in Ai. Setting xt=παt(Xt) defines the process xt with values in [0,1].

    2.There exist

    (a) a one dimensional Wiener process Wt,

    (b) continuous non-decreasing processes i,t, iI, which are measurable with respect to the σ-field generated by (Xt,αt),

    (c) continuous non-increasing processes hi,t, iI, which are measurable with respect to the σ-field generated by (Xt,αt),

    such that

    dxt=2μαtdWt+aαt(xt)dt+di,t+dhi,t,i,t increases only when Xt=νi and xt=0,hi,t decreases only when Xt=νi and xt=1. (11)

    3. The following Ito formula holds: for any real valued function uC2(Γ):

    u(Xt)=u(X0)+αAt01{XsΓαV}(μα2u(Xs)+a(Xs)u(Xs)ds+2μαu(Xs)dWs)+iIαAipiααu(νi)(i,t+hi,t). (12)

    Remark 4. The assumption that all the edges have unit length is not restrictive, because we can always rescale the constants μα and the piecewise continuous function a. The Ito formula in (12) holds when this assumption is not satisfied.

    Consider the invariant measure associated with the process Xt. We may assume that it is absolutely continuous with respect to the Lebesgue measure on Γ. Let m be its density:

    E[u(Xt)]:=Γu(x)m(x)dx, for all uPC(Γ). (13)

    We focus on functions uD(L). Taking the time-derivative of each member of (13), Ito's formula (12) and (10) lead to E[1{XtV}(au(Xt)+μ2u(Xt))]=0. This implies that

    Γ(a(x)u(x)+μ2u(x))m(x)dx=0. (14)

    Since for αA, any smooth function on Γ compactly supported in ΓαV clearly belongs to D(L), (14) implies that m satisfies

    μα2m+(ma)=0 (15)

    in the sense of distributions in the edges ΓαV, αA. This implies that there exists a real number cα such that

    μαm|Γα=m|Γαa|Γα+cα. (16)

    So m|Γα is C1 regular, and (16) is true pointwise. Using this information and recalling (14), we find that, for all uD(L),

    iIαAiμαm|Γα(νi)αu(νi)+βAΓβu|Γβ(x)(μβm|Γβ(x)+a|Γβ(x)m|Γβ(x))dx=0.

    This and (16) imply that

    iIαAiμαm|Γα(νi)αu(νi)+βAcβΓβu|Γβ(x)dx=0. (17)

    For all iI, it is possible to choose a function uD(L) such that

    1. u(νj)=δi,j for all jI;

    2. αu(νj)=0 for all jI and αAj.

    Using such a test-function in (17) implies that for all iI,

    0=βAcβΓβu|Γβ(x)dx=jIαAjcαnjαu|Γα(νj)=αAiniαcα, (18)

    where niα is defined in (6).

    For all iI and α,βAi, it is possible to choose a function uD(L) such that

    1. u takes the same value at each vertex of Γ, thus Γδu|Γδ(x)dx=0 for all δA;

    2. αu(νi)=1/piα, βu(νi)=1/piβ and all the other first order directional derivatives of u at the vertices are 0.

    Using such a test-function in (17) yields

    m|Γα(νi)γiα=m|Γβ(νi)γiβ,for all α,βAi,νiV,

    in which

    γiα=piαμα,for all iI,αAi. (19)

    Next, for iI, multiplying (16) at x=νi by niα for all αAi, then summing over all αAi, we get αAiμααm(νi)niα(m|Γα(νi)a|Γα(νi)cα)=0, and using (18), we obtain that

    αAiμααm(νi)niαa|Γα(νi)m|Γα(νi)=0,for all iI. (20)

    Summarizing, we get the following boundary value problem for m (recalling that the coefficients niα are defined in (6)):

    {μα2m+(ma)=0,x(ΓαV),αA,αAiμααm(νi)niαa|Γα(νi)m|Γα(νi)=0,νiV,m|Γα(νi)γiα=m|Γβ(νi)γiβ,α,βAi,νiV. (21)

    Consider a continuum of indistinguishable agents moving on the network Γ. The state of a representative agent at time t is a time-continuous controlled stochastic process Xt as defined in Section 1.2, where the control is the drift at, supposed to be of the form at=a(Xt). The function Xa(X) is the feedback. Let m(,t) be the probability measure on Γ that describes the distribution of states at time t.

    For a representative agent, the optimal control problem is of the form:

    ρ:=infaslim infT+1TEx[T0L(Xs,as)+V[m(,s)](Xs)ds], (22)

    where Ex stands for the expectation conditioned by the event X0=x. The running cost depends separately on the control and on the distribution of states.

    ● The contribution of the control involves the Lagrangian L, i.e., a real valued function defined on (αAΓαV)×R. If xΓαV and aR, L(x,a)=Lα(π1α(x),a), where Lα is a continuous real valued function defined on [0,α]×R. We assume that lim|a|infyΓαLα(y,a)/|a|=+.

    ● The contribution of the distribution of states involves the coupling cost operator, which can either be nonlocal, i.e., V:P(Γ)C2(Γ) (where P(Γ) is the set of Borel probability measures on Γ), or local, i.e., V[m](x)=F(m(x)) for a continuous function F:R+R, assuming that m is absolutely continuous with respect to the Lebesgue measure and identifying with its density.

    Further assumptions on L and V will be made below.

    Let us assume that there is an optimal feedback law, i.e. a function a defined on Γ which is sufficiently regular in the edges of the network, such that the optimal control at time t is given by at=a(Xt). Then, almost surely if XtΓαV, dπ1α(Xt)=aα(π1α(Xt))dt+2μαdWt. An informal way to describe the behavior of the process at the vertices is as follows: if Xt hits νiV, then it enters Γα, αAi with probability piα>0.

    Under suitable assumptions, the Ito calculus recalled in Section 1.2 and the dynamic programming principle lead to the following ergodic Hamilton-Jacobi equation on Γ, more precisely the following boundary value problem:

    {μα2v+H(x,v)+ρ=V[m](x),x(ΓαV),αA,αAiγiαμααv(νi)=0,νiV,v|Γα(νi)=v|Γβ(νi),α,βAi,νiV,Γv(x)dx=0. (23)

    We refer to [28,30] for the interpretation of the value function v and the ergodic cost ρ.

    Let us comment the different equations in (23):

    1. The Hamiltonian H is a real valued function defined on (αAΓαV)×R. For xΓαV and pR,

    H(x,p)=supa{apLα(π1α(x),a)}.

    The Hamiltonians H|Γα×R are supposed to be C1 and coercive with respect to p uniformly in x (see Section 1.4.1).

    2. The second equation in (23) is a Kirchhoff transmission condition (or Neumann boundary condition if νiΓ); it is the consequence of the assumption on the behavior of Xs at vertices. It involves the positive constants γiα defined in (19).

    3. The third condition means in particular that v is continuous at the vertices.

    4. The fourth equation is a normalization condition.

    If (23) has a smooth solution, then it provides a feedback law for the optimal control problem, i.e.

    a(x)=pH(x,v(x)).

    At the MFG equilibrium, m is the density of the invariant measure associated with the optimal feedback law, so, according to Section 1.2, it satisfies (21), where a is replaced by a=pH(x,v(x)). We end up with the following system:

    {μα2v+H(x,v)+ρ=V([m]),xΓαV,αA,μα2m+(mpH(x,v))=0,xΓαV,αA,αAiγiαμαα(νi)=0,νiV,αAi[μααm(νi)+niαpHα(νi,v|Γα(νi))m|Γα(νi)]=0,νiV,v|Γα(νi)=v|Γβ(νi),α,βAi,νiV,m|Γα(νi)γiα=m|Γβ(νi)γiβ,α,βAi,νiV,Γv(x)dx=0,Γm(x)dx=1,m0. (24)

    At a vertex νi, iI, the transmission conditions for both v and m consist of dνi=(Ai) linear relations, which is the appropriate number of relations to have a well posed problem. If νiΓ, there is of course only one Neumann like condition for v and for m.

    Remark 5. In [7], the authors assume that γiα=γiβ for all iI, α,βAi. Therefore, the density m does not have jumps across the transition vertices.

    Let (μα)αA be a family of positive numbers, and for each iI let (γiα)αAi be a family of positive numbers such that αAiγiαμα=1.

    Consider the Hamiltonian H:Γ×RR. We assume that for all αA, we can define Hα:[0,α]×RR and H|Γα:Γα×RR by (2) and, for some positive constants C0,C1,C2 and q(1,2],

    HαC1([0,α]×R); (25)
    Hα(x,)is convex in p for each x[0,α]; (26)
    Hα(x,p)C0|p|qC1 for (x,p)[0,α]×R; (27)
    |pHα(x,p)|C2(|p|q1+1) for (x,p)[0,α]×R. (28)

    Remark 6. The Hamiltonian H is discontinuous at the vertices in general, although Hα is C1 up to the endpoints of [0,α].

    Remark 7. From (28), there exists a positive constant Cq such that

    |Hα(x,p)|Cq(|p|q+1),for all (x,p)[0,α]×R. (29)

    Below, we shall focus on local coupling operators V, namely

    V[˜m](x)=F(m(x)) with FC([0,+);R), (30)

    for all ˜m which are absolutely continuous with respect to the Lebesgue measure and such that d˜m(x)=m(x)dx. We shall also suppose that F is bounded from below, i.e., there exists a positive constant M such that

    F(r)M,for all r[0,+). (31)

    Let us introduce two function spaces on Γ, which will be the key ingredients in order to build weak solutions of (24).

    Definition 1.2. We define two Sobolev spaces, V:=H1(Γ), see Definition 1.1, and

    W:={w:ΓR:wαH1(0,α) for all αA,w|Γα(νi)γiα=w|Γβ(νi)γiβ for all iI,α,βAi} (32)

    which is also a Hilbert space, endowed with the norm wW=(αAwα2H1(0,α))12.

    Remark 8. We point out that, following Definition 1.1, functions in V are continuous on Γ. By contrast, functions in W are discontinuous in general.

    Definition 1.3. Let the functions ψW and ϕPC(Γ) be defined as follows:

    {ψα is affine on (0,α),ψ|Γα(νi)=γiα, if αAi,ψ is constant on the edges Γα which touch the boundary of Γ. (33)
    {ϕα is affine on (0,α),ϕ|Γα(νi)=1γiα, if αAi,ϕ is constant on the edges Γα which touch the boundary of Γ. (34)

    Note that both functions ψ,ϕ are positive and bounded. We set ¯ψ=maxΓψ, ψ_=minΓψ, ¯ϕ=maxΓϕ, ϕ_=minΓϕ.

    Remark 9. One can see that vVvψ is an isomorphism from V onto W and wWwϕ is the inverse isomorphism.

    Definition 1.4. Let the function space WW be defined as follows:

    W:={m:ΓR:mαC1([0,α]) for all αA,m|Γα(νi)γiα=m|Γβ(νi)γiβ for all iI,α,βAi}. (35)

    Remark 10. A function mW is in general discontinuous at the vertices of Γ, although for any αA, mα is C1 in [0,α].

    Definition 1.5. A solution of the Mean Field Games system (24) is a triple (v,ρ,m)C2(Γ)×R×W such that (v,ρ) is a classical solution of

    {μα2v+H(x,v)+ρ=F(m),in ΓαV,αA,αAiγiαμααv(νi)=0,if νiV, (36)

    (note that v is continuous at the vertices from the definition of C2(Γ)), and m satisfies

    αAΓα[μαmu+(mpH(x,v))u]dx=0,for all uV, (37)

    where V is given in Definition 1.2.

    We are ready to state the main result:

    Theorem 1.6. If assumptions (25)-(28) and (30)-(31) are satisfied, then there exists a solution (v,m,ρ)C2(Γ)×W×R of (24). If F is locally Lipschitz continuous, then vC2,1(Γ). Moreover if F is strictly increasing, then the solution is unique.

    Remark 11. The proof of the existence result in [7] is valid only in the case when the coupling cost F is bounded.

    Remark 12. The existence result in Theorem 1.6 holds if we assume that the coupling operator V is non local and regularizing, i.e., V is a continuous map from P to a bounded subset of F, with F:={f:ΓR:f|ΓαC0,σ(Γα)}. The proof, omitted in what follows, is similar to that of Lemma 4.1 below.

    This section contains elementary results on the solvability of some linear boundary value problems on Γ. To the best of our knowledge, these results are not available in the literature.

    We recall that the constants μα and γiα are defined in Section 1.2. Let λ be a positive number. We start with very simple linear boundary value problems, in which the only difficulty is the Kirchhoff condition:

    {μα2v+λv=f,in ΓαV,αA,v|Γα(νi)=v|Γβ(νi),α,βAi,iI,αAiγiαμααv(νi)=0,iI, (38)

    where fW, W is the topological dual of W.

    Remark 13. We have already noticed that, if νiΓ, the last condition in (38) boils down to a standard Neumann boundary condition αv(νi)=0, in which α is the unique element of Ai. Otherwise, if νiVΓ, the last condition in (38) is the Kirchhoff condition discussed above.

    Definition 2.1. A weak solution of (38) is a function vV such that

    Bλ(v,w)=f,wW,W,for all wW, (39)

    where Bλ:V×WR is the bilinear form defined as follows:

    Bλ(v,w)=αAΓα(μαvw+λvw)dx.

    Remark 14. Formally, (39) is obtained by testing the first line of (38) by wW, integrating by part the left hand side on each Γα and summing over αA. There is no contribution from the vertices, because of the Kirchhoff conditions on the one hand and on the other hand the jump conditions satisfied by the elements of W.

    Remark 15. By using the fact that Γα are line segments, i.e. one dimensional sets and solving the differential equations, we see that if v is a weak solution of (38) with fPC(Γ), then vC2(Γ).

    Let us first study the homogeneous case, i.e. f=0.

    Lemma 2.2. The function v=0 is the unique solution of the following boundary value problem

    {2v+λv=0,in ΓαV,αA,v|Γα(νi)=v|Γβ(νi),α,βAi,iI,αAiγiαμααv(νi)=0,iI, (40)

    Proof. Let Ii:={kI:ki;νkΓα for some αAi} be the set of indices of the vertices which are connected to νi. By Remark 1, it is not restrictive to assume (in the remainder of the proof) that for all kIi, Γα=Γαik=[νi,νk] is oriented from νi to νk.

    For kIi, Γα=[νi,νk], using the parametrization (1), the linear differential eqaution (40) in the edge Γα is

    vα(y)+λvα(y)=0,in (0,α),

    whose solution is

    vα(y)=ζαcosh(λy)+ξαsinh(λy), (41)

    with

    {ζα=vα(0)=v(νi),ζαcosh(λα)+ξαsinh(λα)=vα(α)=v(νk).

    It follows that αv(νi)=λξα=λsinh(λα)[v(νk)v(νi)cosh(λα)]. Hence, the transmission condition in (40) becomes: for all iI,

    0=αAiγiαμααv(νi)=kIiλγiαikμαikcosh(λαik)sinh(λαik)v(νi)kIiλγiαikμαiksinh(λαik)v(νk).

    Therefore, we obtain a system of linear equations of the form MU=0 with M=(Mij)1i,jN, N=(I), and U=(v(ν1),,v(νN))T, where M is defined by

    {Mii=kIiγiαikμαikcosh(λαik)sinh(λαik)>0,Mik=γiαikμαiksinh(λαik)0,kIi,Mik=0,kIi.

    For all iI, since cosh(λαik)>1 for all kIi, the sum of the entries on each row is positive and M is diagonal dominant. Thus, M is invertible and U=0 is the unique solution of the system. Finally, by solving the ODE in each edge Γβ with vβ(0)=vβ(β)=0, we get that v=0 on Γ.

    Let us now study the non-homogeneous problems (38).

    Lemma 2.3. For any f in W, (38) has a unique weak solution v in V, see Definition 1.2. Moreover, there exists a constant C such that vVCfW.

    Proof. First of all, we claim that for λ0>0 large enough and any fW, the problem

    Bλ(v,w)+λ0(v,w)=f,wW,W (42)

    has a unique solution vV. Let us prove the claim. Let vV, then ˆw:=vψ belongs to W, where ψ is given by Definition 1.3. Let us set ¯ψ:=maxΓ|ψ| and ψ_:=minΓψ>0, (ψ is bounded, see Definition 1.3); we get

    Bλ(v,ˆw)+λ0(v,ˆw)=αAΓα[μα|v|2ψ+μα(vv)ψ+(λ+λ0)v2ψ]dxαAΓα[μαψ_2|v|2+(λ0ψ_μα¯ψ22ψ_)v2]dx. (43)

    When λ0μα2+μα¯ψ22ψ_2 for all αA, we obtain that Bλ(v,ˆw)+λ0(v,ˆw)μ_ψ_2v2Vμ_ψ_2CψvVˆwW, using the fact that, from Remark 9, there exists a positive constant Cψ such that vψWCψvV for all vV. This yields

    infvVsupwWBλ(v,w)+λ0(v,w)vVwWμ_ψ_2Cψ. (44)

    Using a similar argument for any wW and ˆv=wϕ, where ϕ is given in Definition 1.3, we obtain that for λ0 large enough, there exist a positive constant Cϕ such that

    infwWsupvVBλ(v,w)+λ0(v,w)wWvVμ_ϕ_2Cϕ. (45)

    From (44) and (45), by the Banach-Necas-Babuška lemma (see [13]), for λ0 large enough, for any fW, there exists a unique solution vV of (42) and vVCfW for a positive constant C. Hence, our claim is proved.

    Now, we fix λ0 large enough and we define the continuous linear operator ¯Rλ0:WV where ¯Rλ0(f)=v is the unique solution of (42). Since the injection I from V to W is compact, then I¯Rλ0 is a compact operator from W into W. By the Fredholm alternative (see [18]), one of the following assertions holds:

    There exists ¯vW{0} such that (Idλ0(I¯Rλ0))¯v=0. (46)
    For any gW, there exists a unique ¯vW such that (Idλ0(I¯Rλ0))¯v=g. (47)

    We claim that (47) holds. Indeed, assume by contradiction that (46) holds. Then there exists ¯v0 such that ¯vV and I¯Rλ0¯v=¯vλ0. Therefore, ¯vV, and Bλ(¯vλ0,w)+λ0(¯vλ0,w)=(¯v,w), for all wW. This yields that Bλ(¯v,w)=0 for all wW and by Lemma 2.2, we get that ¯v=0, which leads us to a contradiction. Hence, our claim is proved.

    Then, (47) implies that there exists a positive constant C such that for all fW, (38) has a unique weak solution v and that vVCfW, see [11] for the details.

    Consider bPC(Γ). This paragraph is devoted to the following boundary value problem including a Kolmogorov equation

    {μα2v+bv=0,in ΓαV,αA,v|Γα(νi)=v|Γβ(νi),α,βAi,iI,αAiγiαμααv(νi)=0,iI. (48)

    Definition 2.4. A weak solution of (48) is a function vV such that

    A(v,w)=0,for all wW,

    where A:V×WR is the bilinear form defined by

    A(v,w):=αAΓα(μαvw+bvw)dx.

    As in Remark 15, if v is a weak solution of (48), then vC2(Γ).

    The uniqueness of solutions of (48) up to the addition of constants is obtained by using a maximum principle:

    Lemma 2.5. For bPC(Γ), the solutions of (48) are the constant functions on Γ.

    Proof of Lemma 2.5. First of all, any constant function on Γ is a solution of (48). Now let v be a solution of (48) then vC2(Γ). Assume that the maximum of v over Γ is achieved in Γα; by the maximum principle, it is achieved at some endpoint νi of Γα. Without loss of generality, using Remark 1, we can assume that πβ(νi)=0 for all βAi. We have βv(νi)0 for all βAi because νi is the maximum point of v. Since all the coefficients γiβ,μβ are positive, by the Kirchhoff condition if νi is a transition vertex, or by the Neumann boundary condition if νi is a boundary vertex, we infer that βv(νi)=0 for all βAi. This implies that vβ is a solution of the first order linear homogeneous differential equation u+bβu=0, on [0,β], with u(0)=0. Therefore, vβ0 and v is constant on Γβ for all βAi. We can propagate this argument, starting from the vertices connected to νi. Since the network Γ is connected and v is continuous, we obtain that v is constant on Γ.

    This paragraph is devoted to the dual boundary value problem of (48); it involves a Fokker-Planck equation:

    {μα2m(bm)=0,in ΓαV,αA,m|Γα(νi)γiα=m|Γβ(νi)γiβ,α,βAi,iI,αAi[niαb|Γα(νi)m|Γα(νi)+μααm(νi)]=0,iI, (49)

    where bPC(Γ), with

    m0,Γmdx=1. (50)

    First of all, let λ0 be a nonnegative constant; for all hV, we introduce the modified boundary value problem

    {λ0mμα2m(bm)=h,in ΓαV,αA,m|Γα(νi)γiα=m|Γβ(νi)γiβ,α,βAi,iI,αAi[niαb|Γα(νi)m|Γα(νi)+μααm(νi)]=0,iI. (51)

    Definition 2.6. For λR, consider the bilinear form Aλ:W×VR defined by

    Aλ(m,v)=αAΓα[λmv+(μαm+bm)v]dx.

    A weak solution of (51) is a function mW such that

    Aλ0(m,v)=h,vV,V,for all vV.

    A weak solution of (49) is a function mW such that

    A0(m,v):=αAΓα(μαm+bm)vdx=0,for all vV. (52)

    Remark 16. Formally, to get (52), we multiply the first line of (49) by vV, integrate by part, sum over αA and use the third line of (49) to see that there is no contribution from the vertices.

    Theorem 2.7. For any bPC(Γ),

    ● (Existence) There exists a solution ˆmW of (49)-(50) satisfying

    ˆmWC,0ˆmC, (53)

    where the constant C depends only on b and {μα}αA. Moreover, ˆmαC1(0,α) for all αA. Hence, ˆmW.

    ● (Uniqueness) ˆm is the unique solution of (49)-(50).

    ● (Strictly positive solution) ˆm is strictly positive.

    Proof of existence in Theorem 2.7. We divide the proof of existence into three steps:

    Step 1. Let λ0 be a large positive constant that will be chosen later. We claim that for ¯mL2(Γ) and h:=λ0¯mL2(Γ)V, (51) has a unique solution mW. This allows us to define a linear operator as follows:

    T:L2(Γ)W,T(¯m)=m,

    where m is the solution of (51) with h=λ0¯m. We are going to prove that T is well-defined and continuous, i.e, for all ¯mL2(Γ), (51) has a unique solution that depends continuously on ¯m. For wW, set ˆv:=wϕV where ϕ is given by Definition 1.3. We have

    Aλ0(w,ˆv)=αAΓα[λ0ϕw2+(μαw+bw)(wϕ)]dx=αAΓα[(λ0ϕ+bϕ)w2+(μαϕ+bϕ)ww+μαϕ(w)2]dx.

    It follows that when λ0 is large enough (larger than a constant that only depends on b,ϕ and μα), Aλ0(w,ˆv)ˆCλ0w2W for some positive constant ˆCλ0. Moreover, by Remark 9, there exists a positive constant ˆCϕ such that for all wW, we have wϕVCϕwW. This yields

    infwWsupvVAλ0(w,v)vVwWˆCλ0Cϕ.

    Using similar arguments, for λ0 large enough, there exist two positive constants Cλ0 and Cψ such that

    infvVsupwWAλ0(w,v)wWvVCλ0Cψ.

    From Banach-Necas-Babuška lemma (see [13]), there exists a constant ¯C such that for all ¯mL2(Γ), there exists a unique solution m of (51) with h=λ0¯m and mW¯C¯mL2(Γ). Hence, the map T is well-defined and continuous from L2(Γ) to W.

    Step 2. Let K be the set defined by

    K:={mL2(Γ):m0 and Γmdx=1}.

    We claim that T(K)K which means Γm=1 and m0. Indeed, using v=1 as a test-function in (51), we have Γmdx=Γ¯mdx=1. Next, consider the negative part m of m defined by m(x)=1{m(x)<0}m(x). Notice that mW and mϕV, where ϕ is given by Definition 1.3. Using mϕ as a test-function in (51) yields

    αAΓα[(λ0ϕ+bϕ)(m)2+μα(m)2ϕ+(μαϕ+bϕ)mm]dx=Γλ0¯mmϕdx.

    We can see that the right hand side is non-negative. Moreover, for λ0 large enough (larger than the same constant as above, which only depends on b,ϕ and μα), the left hand side is non-positive. This implies that m=0, and hence m0. Therefore, the claim is proved.

    Step 3. We claim that T has a fixed point. Let us now focus on the case when ¯mK. Using mϕ as a test function in (51) yields

    αAΓα[(λ0ϕ+bϕ)m2+μα(m)2ϕ+(μαϕ+bϕ)m(m)]dx=Γλ0¯mmϕdx. (54)

    Since H1(0,α) is continuously embedded in L(0,α), there exists a positive constant C (independent of ¯mK) such that

    Γ¯mmϕdxΓ¯mdxmL(Γ)¯ϕ=mL(Γ)¯ϕCmW.

    Hence, from (54), for λ0 large enough, there exists a positive constant C1 such that C1m2Wλ0CmW. Thus

    mWλ0CC1. (55)

    Therefore, T(K) is bounded in W. Since the bounded subsets of W are relatively compact in L2(Γ), ¯T(K) is compact in L2(Γ). Moreover, we can see that K is closed and convex in L2(Γ). By Schauder fixed point theorem, see [18,Corollary 11.2], T has a fixed point ˆmK which is also a solution of (49) and ˆmWλ0C/C1.

    Finally, from the differential equation in (49), for all αA, (ˆmα+bαˆmα)=0 on (0,α). Hence, there exists a constant Cα such that

    ˆmα+bαˆmα=Cα,for all x(0,α). (56)

    It follows that ˆmαC([0,α]), for all αA. Hence ˆmαC1([0,α]) for all αA. Thus, ˆmW.

    Remark 17. Let mW be a solution of (49). If b,bPC(Γ), standard arguments yield that mαC2([0,α]) for all αA. Moreover, by Theorem 2.7, there exists a constant C which depends only on b,{bα}αA and μα such that mαC2(0,α)C for all αA.

    Proof of the positivity in Theorem 2.7. From (50), ˆm is non-negative on Γ. Assume by contradiction that there exists x0Γα for some αA such that ˆm|Γα(x0)=0. Therefore, the minimum of ˆm over Γ is achieved at x0Γα. If x0ΓαV, then ˆm(x0)=0. In (56), we thus have Cα=0, and hence ˆmα satisfies

    ˆmα+bαˆmα=0,on [0,α],

    with ˆmα(π1α(x0))=0. It follows that ˆmα0 and ˆm|Γα(νi)=ˆm|Γα(νj)=0 if Γα=[νi,νj].

    Therefore, it is enough to consider x0V.

    Now, from Remark 1, we may assume without loss of generality that x0=νi and πβ(νi)=0 for all βAi. We have the following two cases.

    Case 1. if x0=νi is a transition vertex, then, since ˆm belongs to W, we get

    ˆm|Γβ(νi)=γiβγiαˆm|Γα(νi)=0,for all βAi. (57)

    This yields that νi is also a minimum point of ˆm|Γβ for all βAi. Thus βˆm(νi)0 for all βAi. From the transmission condition in (49) which has a classical meaning from the regularity of ˆm, βˆm(νi)=0, since all the coefficients μβ are positive. From (56), for all βAi, we have

    Cβ=ˆmβ(0)+bβ(0)ˆmβ(0)=0.

    Therefore, ˆmβ(y)+bβ(y)ˆmβ(y)=0, for all y[0,β] with ˆmβ(0)=0. This implies that ˆmβ0 for all βAi. We can propagate the arguments from the vertices connected to νi. Since Γ is connected, we obtain that ˆm0 on Γ.

    Case 2. if x0=νi is a boundary vertex, then the Robin condition in (49) implies that αˆm(νi)=0 since μα is positive. From (56), we have Cα=0. Therefore, ˆmα(y)+bα(y)ˆmα(y)=0, for all y[0,α] with ˆmα(0)=0. This implies that ˆm(νj)=0 where νj is the other endpoint of Γα. We are back to Case 1, so ˆm0 on Γ.

    Finally, we have found that ˆm0 on Γ, in contradiction with Γˆmdx=1.

    Now we prove uniqueness for (49)-(50).

    Proof of uniqueness in Theorem 2.7. The proof of uniqueness is similar to the argument in [7,Proposition 13]. As in the proof of Lemma 2.3, we can prove that for λ0 large enough, there exists a constant C such that for any fV, there exists a unique wW which satisfies

    Aλ0(w,v)=f,vV,V for all vV. (58)

    and wWCfV. This allows us to define the continuous linear operator

    Sλ0:L2(Γ)W,fw,

    where w is a solution of (58). Then we define Rλ0=JSλ0 where J is the injection from W in L2(Γ), which is compact. Obviously, Rλ0 is a compact operator from L2(Γ) into L2(Γ). Moreover, mW is a solution of (49) if and only if mker(Idλ0Rλ0). By Fredholm alternative, see [18], dimker(Idλ0Rλ0)=dimker(Idλ0Rλ0).

    In order to characterize Rλ0, we now consider the following boundary value problem for gL2(Γ)W:

    {λ0vμα2v+bv=g,in ΓαV,αA,v|Γα(νi)=v|Γβ(νi)α,βAi,iI,αAiγiαμααv(νi)=0,iI. (59)

    A weak solution of (59) is a function vV such that

    Tλ0(v,w):=αAΓα(λ0vw+μαvw+bwv)dx=Γgwdx,for all wW.

    Using similar arguments as in the proof of existence in Theorem 2.7, we see that for λ0 large enough and all gL2(Γ), there exists a unique solution vV of (59). Moreover, there exists a constant C such that vVCgL2(Γ) for all gL2(Γ). This allows us to define a continuous operator

    Tλ0:L2(Γ)V,gv.

    Then we define ˜Rλ0=ITλ0 where I is the injection from V in L2(Γ). Since I compact, ˜Rλ0 is a compact operator from L2(Γ) into L2(Γ). For any gL2(Γ), set v=Tλ0g. Noticing that Tλ0(v,w)=Aλ0(w,v) for all vV,wW, we obtain that

    (g,Rλ0f)L2(Γ)=Tλ0(v,Sλ0f)=Aλ0(Sλ0f,v)=(f,v)L2(Γ)=(f,˜Rλ0g)L2(Γ).

    Thus Rλ0=˜Rλ0. But ker(Idλ0˜Rλ0) is the set of solutions of (48), which, from Lemma 2.5, consists of constant functions on Γ.

    This implies that dimker(Idλ0Rλ0)=1 and then that dimker(Idλ0Rλ0)=dimker(Idλ0Rλ0)=1.

    Finally, since the solutions m of (49) are in ker(Idλ0Rλ0) and satisfy the normalization condition Γmdx=1, we obtain the desired uniqueness property in Theorem 2.7.

    This section is devoted to the following boundary value problem including a Hamilton-Jacobi equation:

    {μα2v+H(x,v)+λv=0,in ΓαV,αA,v|Γα(νi)=v|Γβ(νi),α,βAi,iI,αAiγiαμααv(νi)=0,iI, (60)

    where λ is a positive constant and the Hamiltonian H:Γ×RR is defined in Section 1, except that, in (60) and the whole Section 3.1 below, the Hamiltonian contains the coupling term, i.e, H(x,v) in (60) plays the role of H(x,v)F(m(x)) in (24).

    Definition 3.1  ● A classical solution of (60) is a function vC2(Γ) which satisfies (60) pointwise.

    ● A weak solution of (60) is a function vV such that

    αAΓα(μαvw+H(x,v)w+λvw)dx=0for all wW.

    Proposition 1. Assume that

    HαC([0,α]×R), (61)
    |H(x,p)|C2(1+|p|2)for all xΓ,pR, (62)

    where C2 is a positive constant. There exists a classical solution v of (60). Moreover, if Hα is locally Lipschitz with respect to both variables for all αA, then the solution v belongs to C2,1(Γ).

    Remark 18. Assume (61) and that vH2(Γ)V is a weak solution of (60). From the compact embedding of H2(0,α) into C1,σ([0,α]) for all σ(0,1/2), we get vC1,σ(Γ). Therefore, from the differential equation in (60) μα2vα()=Hα(,vα())+λvα()C([0,α]). It follows that v is a classical solution of (60).

    Remark 19. Assume now that H is locally Lipschitz continuous and that vH2(Γ)V is a weak solution of (60). From Remark 18, vC1,σ(Γ) for σ(0,1/2) and the function λvαHα(,vα) belongs to C0,σ([0,α]). Then, from the first line of (60), vC2,σ(Γ). This implies that vαLip[0,α] and using the differential equation again, we see that vC2,1(Γ).

    Let us start with the case when H is a bounded Hamiltonian.

    Lemma 3.2. Assume (61) and for some CH>0,

    |H(x,p)|CH,for all(x,p)Γ×R. (63)

    There exists a classical solution v of (60). Moreover, if Hα is locally Lipschitz in [0,α]×R for all αA then the solution v belongs to C2,1(Γ).

    Proof of Lemma 3.2. For any uV, from Lemma 2.3, the following boundary value problem:

    {μα2v+λv=H(x,u),if xΓαV,αA,v|Γα(νi)=v|Γβ(νi),α,βAi,iI,αAiγiαμααv(νi)=0,iI, (64)

    has a unique weak solution vV. This allows us to define the map T:VV by T(u):=v. Moreover, from Lemma 2.3, there exists a constant C such that

    vVCH(x,u)L2(Γ)CCH|Γ|1/2, (65)

    where |Γ|=ΣαAα. Therefore, from the differential equation in (64),

    μ_2vL2(Γ)λvL2(Γ)+H(x,u)L2(Γ)λvV+CH|Γ|1/2(λC+1)CH|Γ|1/2, (66)

    where μ_:=minαAμα. From (65) and (66), T(V) is a bounded subset of H2(Γ), see Definition 1.1. From the compact embedding of H2(Γ) into V, we deduce that ¯T(V) is a compact subset of V.

    Next, we claim that T is continuous from V to V. Assuming that

    {unu,in {V},vn=T(un),for all {n},v=T(u), (67)

    we need to prove that vnv in V. Since {vn} is uniformly bounded in H2(Γ), then, up to the extraction of a subsequence, vnˆv in C1,σ(Γ) for some σ(0,1/2). From (67), we have that unu in L2(Γα) for all αA. This yields that, up to another extraction of a subsequence, unu almost everywhere in Γα. Thus H(x,un)H(x,u) in L2(Γα) by Lebesgue dominated convergence theorem. Hence, ˆv is a weak solution of (64). Since the latter is unique, ˆv=v and we can conclude that the whole sequence vn converges to v. The claim is proved.

    From Schauder fixed point theorem, see [18,Corollary 11.2], T admits a fixed point which is a weak solution of (60). Moreover, recalling that vH2(Γ), we obtain that v is a classical solution of (60) from Remark 18.

    Assume now that H is locally Lipschitz. Since vαH2(0,α) for all αA, we may use Remark 19 and obtain that vC2,1(Γ).

    Lemma 3.3. Assume (61). If v,uC2(Γ) satisfy

    {μα2v+H(x,v)+λvμα2u+H(x,u)+λu,ifxΓαV,αA,αAiγiαμααv(νi)αAiγiαμααu(νi),ifνiV, (68)

    then vu.

    Proof of Lemma 3.3. The proof is reminiscent of an argument in [8]. Suppose by contradiction that δ:=maxΓ{uv}>0. Let x0Γα be a maximum point of uv. It suffices to consider the case when x0V, since if x0ΓV, then u(x0)>v(x0), u(x0)=v(x0), 2u(x0)2v(x0), and we obtain a contradiction with the first line of (68).

    Now consider the case when x0=νiV; from Remark 1, we can assume without restriction that πα(0)=νi. Since uv achieves its maximum over Γ at νi, we obtain that βu(νi)βv(νi), for all βAi. From Kirchhoff conditions in (68), this implies that βu(νi)=βv(νi), for all βAi. It follows that vα(0)=uα(0). Using the first line of (68), we get that

    μα[2vα(0)2uα(0)]Hα(0,uα(0))Hα(0,vα(0))=0+λ(uα(0)vα(0))>0.

    Therefore, uαvα is locally strictly convex in [0,α] near 0 and its first order derivative vanishes at 0. This contradicts the fact that νi is the maximum point of uv.

    We now turn to Proposition 1.

    Proof of Proposition 1. We adapt the classical proof of Boccardo, Murat and Puel in [6]. First of all, we truncate the Hamiltonian as follows:

    Hn(x,p)={H(x,p),if |p|n,H(x,p|p|n),if |p|>n.

    By Lemma 3.2, for all nN, since Hn(x,p) is continuous and bounded by C2(1+n2), there exists a classical solution vnC2(Γ) for the following boundary value problem

    {μα2v+Hn(x,v)+λv=0,xΓαV,αA,v|Γα(νi)=v|Γβ(νi), for all α,βAi,iI,αAiγiαμααv(νi)=0,iI. (69)

    We wish to pass to the limit as n tend to +; we first need to estimate vn uniformly in n, successively in L(Γ), H1(Γ) and H2(Γ).

    Estimate in L(Γ). Since |Hn(x,p)|c(1+|p|2) for all x,p, then φ=c/λ and ¯φ=c/λ are respectively a sub- and super-solution of (69). Therefore, from Lemma 3.3, we obtain |λvn|c.

    Estimate in V. For a positive constant K to be chosen later, we introduce wn:=eKv2nvnψW, where ψ is given in Definition 1.3. Using wn as a test function in (69) leads to

    αAΓα(μαvnwn+λvnwn)dx=ΓHn(x,vn)wndx.

    Since |Hn(x,p)|c(1+p2), we have

    αAΓαeKv2n[(μαψ)(vn)2+(μα2Kψ)v2n(vn)2+(μαψ)vnvn+λψv2n]dxΓeKv2n|Hn(x,vn)||vnψ|dxΓceKv2nψ|vn|dx+ΓcψeKv2n|vn|ψ(vn)2dxΓeKv2n(λψv2n+ψc24λ)dx+αAΓαeKv2n[μα2ψ(vn)2+c22μαψ(vn)2v2n]dx,

    where we have used Young inequalities. Since λ>0 and ψ>0, we deduce that

    αAΓαeKv2n[(μα2ψ)(vn)2+2ψ(μαKc24μα)v2n(vn)2+(μαψ)vnvn]dxc24λΓeKv2nψdx. (70)

    Next, choosing K>(1+c2/4μ_)/μ_ yields that

    αAΓαeKv2n[μα2ψ(vn)2+2ψv2n(vn)2+(μαψ)vnvn]dxC

    for a positive constant C independent of n, because vn is bounded by c/λ. Since ψ is bounded from below by a positive number and ψ is piecewise constant on Γ, we infer that αAΓαeKv2nv2n(vn)2˜C, where ˜C is a positive constant independent on n. Using this information and (70) again, we obtain that Γ(vn)2 is bounded uniformly in n. There exists a constant ¯C such that vnV¯C for all n.

    Estimate in H2(Γ). From the differential equation in (69) and (62), we have

    μ_|2vn|c+c|vn|2+λ|vn|,for all αA.

    Thus 2vn is uniformly bounded in L1(Γ). This and the previous estimate on vnL2(Γ) yield that vn is uniformly bounded in L(Γ), from the continuous embedding of W1,1(0,α) into C([0,α]). Therefore, from (69), we get that 2vn is uniformly bounded in L(Γ). This implies in particular that vn is uniformly bounded in W2,(Γ).

    Hence, for any σ(0,1), up to the extraction of a subsequence, there exists vV such that vnv in C1,σ(Γ). This yields that Hn(x,vn)H(x,v) for all xΓ. By Lebesgue's Dominated Convergence Theorem, we obtain that v is a weak solution of (60), and since vC1,σ(Γ), by Remark 18, v is a classical solution of (60).

    Assume now that H is locally Lipschitz. We may use Remark 19 and obtain that vC2,1(Γ).

    The proof is complete.

    For fPC(Γ), we wish to prove the existence of (v,ρ)C2(Γ)×R such that

    {μα2v+H(x,v)+ρ=f(x),in ΓαV,αA,v|Γα(νi)=v|Γβ(νi),α,βAi,iI,αAiγiαμααv(νi)=0,iI, (71)

    with the normalization condition

    Γvdx=0. (72)

    Theorem 3.4. Assume (25)-(27). There exists a unique couple (v,ρ)C2(Γ)×R satisfying (71)-(72), with |ρ|maxxΓ|H(x,0)f(x)|. There exists a constant ¯C which only depends upon fL(Γ),μα and the constants in (27) such that

    vC2(Γ)¯C. (73)

    Moreover, for some σ(0,1), if fαC0,σ([0,α]) for all αA, then (v,ρ)C2,σ(Γ)×R; there exists a constant ¯C which only depends upon fαC0,σ([0,α]),μα and the constants in (27) such that

    vC2.σ(Γ)¯C. (74)

    Proof of existence in Theorem 3.4. By Proposition 1, for any λ>0, the following boundary value problem

    {μα2v+H(x,v)+λv=f,in ΓαV,αA,v|Γα(νi)=v|Γβ(νi),α,βAi,iI,αAiγiαμααv(νi)=0,iI, (75)

    has a unique solution vλC2(Γ). Set C:=maxΓ|f()H(,0)|. The constant functions φ:=C/λ and ¯φ=C/λ are respectively sub- and super-solution of (75). By Lemma 3.3,

    Cλvλ(x)C,for all xΓ. (76)

    Next, set uλ:=vλminΓvλ. We see that uλ is the unique classical solution of

    {μα2uλ+H(x,uλ)+λuλ+λminΓvλ=f,in ΓαV,αA,u|Γα(νi)=u|Γβ(νi),α,βAi,iI,αAiγiαμααuλ(νi)=0,iI. (77)

    Before passing to the limit as λ tends 0, we need to estimate uλ in C2(Γ) uniformly with respect to λ. We do this in two steps:

    Step 1. Estimate of uλLq(Γ). Using ψ as a test-function in (77), see Definition 1.3, and recalling that λuλ+λminΓvλ=λvλ, we see that

    αAΓαμαuλψdx+Γ(H(x,uλ)+λvλ)ψdx=Γfψdx.

    From (27) and (76),

    αAΓαμαuλψdx+αAΓαC0|uλ|qψdxΓ(f+C+C1)ψdx.

    On the other hand, since q>1, ψψ_>0 and ψ is bounded, there exists a large enough positive constant C such that

    αAΓαμαuλψdx+12αAΓαC0|uλ|qψdx+C>0,for all λ>0.

    Subtracting, we get C02ψ_Γ|uλ|qdxΓ(f+C+C1)ψdx+C. Hence, for all λ>0,

    uλLq(Γ)˜C, (78)

    where ˜C:=[(2Γ(|f|+C+C1)ψdx+2C)/(C0ψ_)]1/q.

    Step 2. Estimate of uλC2(Γ). Since uλ=vλminΓvλ, there exists αA and xλΓα such that uλ(xλ)=0. For all λ>0 and xΓα, we have

    |uλ(x)|=|uλ(x)uλ(xλ)|Γ|uλ|dxuλLq(Γ)|Γ|q/(q1).

    From (78) and the latter inequality, we deduce that uλ|ΓαL(Γα)˜C|Γ|q/(q1). Let νi be a transition vertex which belongs to Γα. For all βAi, yΓβ,

    |uλ(y)||uλ(y)uλ(νi)|+|uλ(νi)|2˜C|Γ|q/(q1).

    Since the network is connected and the number of edges is finite, repeating the argument as many times as necessary, we obtain that there exists MN such that

    uλL(Γ)M˜C|Γ|q/(q1).

    This bound is uniform with respect to λ(0,1]. Next, from (77) and (29), we get

    μ_|2uλ||H(x,uλ)|+|λvλ|+|f|Cq(1+|uλ|q)+C+fL(Γ).

    Hence, from (78), 2uλ is bounded in L1(Γ) uniformly with respect to λ(0,1]. From the continuous embedding of W1,1(0,α) in C([0,α]), we infer that uλ|Γα is bounded in C(Γα) uniformly with respect to λ(0,1]. From the equation (77) and (76), this implies that uλ is bounded in C2(Γ) uniformly with respect to λ(0,1].

    After the extraction of a subsequence, we may assume that when λ0+, the sequence uλ converges to some function vC1,1(Γ) and that λminvλ converges to some constant ρ. Notice that v still satisfies the Kirchhoff conditions since uλ|Γα(νi)v|Γα(νi) as λ0+. Passing to the limit in (77), we get that the couple (v,ρ) satisfies (71) in the weak sense, then in the classical sense by using an argument similar to Remark 18. Adding a constant to v, we also get (72).

    Furthermore, if for some σ(0,1), f|ΓαC0,σ(Γα) for all αA, a bootstrap argument using the Lipschitz continuity of H on the bounded subsets of Γ×R shows that uλ is bounded in C2,σ(Γ) uniformly with respect to λ(0,1]. After a further extraction of a subsequence if necessary, we obtain (74).

    Proof of uniqueness in Theorem 3.4. Assume that there exist two solutions (v,ρ) and (˜v,˜ρ) of (71)-(72). First of all, we claim that ρ=˜ρ. By symmetry, it suffices to prove that ρ˜ρ. Let x0 be a maximum point of e:=˜vv. Using similar arguments as in the proof of Lemma 3.3, with λv and λu respectively replaced by ρ and ˜ρ, we get ρ˜ρ and the claim is proved.

    We now prove the uniqueness of v. Since Hα belongs to C1(Γα×R) for all αA, then e is a solution of μα2eα[10pHα(y,θvα+(1θ)˜vα)dθ]eα=0,in (0,α), with the same transmission and boundary condition as in (71). By Lemma 2.5, e is a constant function on Γ. Moreover, from (72), we know that Γedx=0. This yields that e=0 on Γ. Hence, (71)-(72) has a unique solution.

    Remark 20. Since there exists a unique solution of (71)-(72), we conclude that the whole sequence (uλ,λvλ) in the proof of Theorem 3.4 converges to (v,ρ) as λ0.

    We first prove Theorem 1.6 when F is bounded.

    Theorem 4.1. Assume (25)-(28), (30) and that F is bounded. There exists a solution (v,m,ρ)C2(Γ)×W×R to the mean field games system (24). If F is locally Lipschitz continuous, then vC2,1(Γ). If furthermore F is strictly increasing, then the solution is unique.

    Proof of existence in Theorem 4.1. We adapt the proof of Camilli and Marchi in [7,Theorem 1]. For σ(0,1/2) let us introduce the space

    Mσ={m:mαC0,σ([0,α]) and m|Γα(νi)γiα=m|Γβ(νi)γiβ| for all iI and α,βAi}

    which, endowed with the norm

    mMσ=mL(Γ)+maxαAsupy,z[0,α],yz|mα(y)mα(z)||yz|σ,

    is a Banach space. Now consider the set

    K={mMσ:m0 and Γmdx=1}

    and observe that K is a closed and convex subset of Mσ. We define a map T:KK as follows: given mK, set f=F(m). By Theorem 3.4, (71)-(72) has a unique solution (v,ρ)C2(Γ)×R. Next, for v given, we solve (49)-(50) with b()=pH(,v())PC(Γ). By Theorem 2.7, there exists a unique solution ¯mKW of (49)-(50). We set T(m)=¯m; we claim that T is continuous and has a precompact image. We proceed in several steps:

    T is continuous. Let mn,mK be such that mnmMσ0 as n+; set ¯mn=T(mn),¯m=T(m). We need to prove that ¯mn¯m in Mσ. Let (vn,ρn),(v,ρ) be the solutions of (71)-(72) corresponding respectively to f=F(mn) and f=F(m). Using estimate (73), we see that up to the extraction of a subsequence, we may assume that (vn,ρn)(¯v,¯ρ) in C1(Γ)×R. Since F(mn)|ΓαF(m)|Γα in C(Γα), Hα(y,(vn)α)Hα(y,¯vα) in C([0,α]), and since it is possible to pass to the limit in the transmission and boundary conditions thanks to the C1-convergence, we obtain that (¯v,¯ρ) is a weak (and strong by Remark 18) solution of (71)-(72). By uniqueness, (¯v,¯ρ)=(v,ρ) and the whole sequence (vn,ρn) converges.

    Next, ¯mn=T(mn),¯m=T(m) are respectively the solutions of (49)-(50) corresponding to b=pH(x,vn) and b=pH(x,v). From the estimate (53), since pH(x,vn) is uniformly bounded in L(Γ), we see that ¯mn is uniformly bounded in W. Therefore, up to the extraction of subsequence, ¯mnˆm in W and ¯mnˆm in Mσ, because W is compactly embedded in Mσ for σ(0,1/2). It is easy to pass to the limit and find that ˆm is a solution of (49)-(50) with b=pH(x,v). From Theorem 2.7, we obtain that ¯m=ˆm, and hence the whole sequence ¯mn converges to ¯m.

    The image of T is precompact. Since FC0(R+;R) is a uniformly bounded function, we see that F(m) is bounded in L(Γ) uniformly with respect to mK. From Theorem 3.4, there exists a constant ¯C such that for all mK, the unique solution v of (71)-(72) with f=F(m) satisfies vC2(Γ)¯C. From Theorem 2.7, we obtain that ¯m=T(m) is bounded in W by a constant independent of m. Since W is compactly embedded in Mσ, for σ(0,1/2) we deduce that T has a precompact image.

    End of the proof. We can apply Schauder fixed point theorem (see [18,Corollary 11.2]) to conclude that the map T admits a fixed point m. By Theorem 2.7, we get mW. Hence, there exists a solution (v,m,ρ)C2(Γ)×W×R to the mean field games system (24). If F is locally Lipschitz continuous, then vC2,1(Γ) from the final part of Theorem 3.4.

    Proof of uniqueness in Theorem 4.1. We assume that F is strictly increasing and that there exist two solutions (v1,m1,ρ1) and (v2,m2,ρ2) of (24). We set ¯v=v1v2,¯m=m1m2 and ¯ρ=ρ1ρ2 and write the equations for ¯v,¯m and ¯ρ

    {μα2¯v+H(x,v1)H(x,v2)+¯ρ(F(m1)F(m2))=0,in ΓαV,μα2¯m(m1pH(x,v1))+(m2pH(x,v2))=0,in ΓαV,¯v|Γα(νi)=¯v|Γβ(νi),¯m|Γα(νi)γiα=¯m|Γβ(νi)γiβ,α,βAi,iI,αAiγiαμαα¯v(νi)=0,iI,αAiniα[m1|Γα(νi)pH(νi,v1|Γα(νi))m2|Γα(νi)pH(νi,v2|Γα(νi))]+αAiμαα¯m(νi)=0,iI,Γ¯vdx=0,Γ¯mdx=0. (79)

    Multiplying the equation for ¯v by ¯m and integrating over Γα, we get

    Γαμα¯v¯m+[H(x,v1)H(x,v2)+¯ρ(F(m1)F(m2))]¯mdx[μα¯mα¯vα]α0=0. (80)

    Multiplying the equation for ¯m by ¯v and integrating over Γα, we get

    Γαμα¯v¯m+[m1pH(x,v1)m2pH(x,v2)]¯vdx[¯v|Γα(μα¯m|Γα+m1|ΓαpH(x,v1|Γα)m2|ΓαpH(x,v2|Γα))]α0=0. (81)

    Subtracting (80) to (81), summing over αA, assembling the terms corresponding to a same vertex νi and taking into account the transmission and the normalization condition for ¯v and ¯m, we obtain

    0=αAΓα(m1m2)[F(m1)F(m2)]dx+αAΓαm1[H(x,v2)H(x,v1)+pH(x,v1)¯v]dx+αAΓαm2[H(x,v1)H(x,v2)pH(x,v2)¯v]dx.

    Since F is strictly monotone then the first sum is non-negative. Moreover, by the convexity of H and the positivity of m1,m2, the last two sums are non-negative. Therefore, we have that m1=m2. From Theorem 3.4, we finally obtain v1=v2 and ρ1=ρ2.

    Proof of Theorem 1.6 for a general coupling F. We only need to modify the proof of existence.

    We now truncate the coupling function as follows:

    Fn(r)={F(r),if |r|n,F(r|r|n),if |r|n.

    Then Fn is continuous, bounded below by M as in (31) and bounded above by some constant Cn. By Theorem 4.1, for all nN, there exists a unique solution (vn,mn,ρn)C2(Γ)×W×R of the mean field game system (24) where F is replaced by Fn. We wish to pass to the limit as n+. We proceed in several steps:

    Step 1. ρn is bounded from below. Multiplying the HJB equation in (24) by mn and the Fokker-Planck equation in (24) by vn, using integration by parts and the transmission conditions, we obtain that

    αAΓαμαvnmndx+ΓH(x,vn)mndx+ρn=ΓFn(mn)mndx, (82)

    and

    αAΓαμαvnmndx+ΓpH(x,vn)mnvndx=0. (83)

    Subtracting the two equations, we obtain

    ρn=ΓFn(mn)mndx+Γ[pH(x,vn)vnH(x,vn)]mndx. (84)

    In what follows, the constant C may vary from line to line but remains independent of n. From (26), we see that pH(x,vn)vnH(x,vn)H(x,0)C. Therefore

    ρnΓFn(mn)mndxCΓmndx=ΓFn(mn)mndxC. (85)

    Hence, since Fn+M0 and Γmndx=1, we get that ρn is bounded from below by MC independently of n.

    Step 2. ρn and ΓFn(mn)dx are uniformly bounded. By Theorem 2.7, there exists a positive solution wW of (49)-(50) with b=0. It yields

    αAΓαμαwudx=0,for all uV,andΓwdx=1.

    Multiplying the HJB equation of (24) by w, using integration by parts and the Kirchhoff condition, we get

    αAΓαμαvnwdx=0+ΓH(x,vn)wdx+ρnΓwdx=1=ΓFn(mn)wdx.

    This implies, using (27), (53) and Fn+M0,

    ρn=ΓFn(mn)wdxΓH(x,vn)wdxwL(Γ)Γ(Fn(mn)+M)dxMΓ(C0|vn|qC1)wdxCΓFn(mn)dx+CΓC0|vn|qwdx. (86)

    Thus, by (85), we have

    \begin{equation} -M-C\le\int_{\Gamma}F_{n}\left(m_{n}\right)m_{n}dx-C\le\rho_{n}\le C\int_{\Gamma}F_{n}\left(m_{n}\right)dx+C. \end{equation} (87)

    Let K>0 be a constant to be chosen later. We have

    \begin{align} \int_{\Gamma}F_{n}\left(m_{n}\right)dx & \le\dfrac{1}{K}\int_{m_{n}\ge K}\left[F_{n}\left(m_{n}\right)+M\right]m_{n}dx+\sup\limits_{0\le r\le K}F (r)\int_{m_{n}\le K}dx \\ & \le\dfrac{1}{K}\int_{\Gamma}F_{n}\left(m_{n}\right)m_{n}dx+\dfrac{M}{K}+C_K, \end{align} (88)

    where C_K is independent of n . Choosing K = 2C where C is the constant in (87), we get by combining (88) with (87) that \int_{\Gamma}F_n(m_n)m_n\le C . Using (88) again, we obtain \int_{\Gamma}F_n(m_n)dx\le C . Hence, from (87), we conclude that |\rho_n|+ |\int_{\Gamma}F_n(m_n)dx|\le C .

    Step 3. Prove that F_{n}\left(m_{n}\right) is uniformly integrable and v_{n} and m_{n} are uniformly bounded respectively in C^{1}\left(\Gamma\right) and W . Let E be measurable with |E| = \eta . By (88) with \Gamma is replaced by E , we have

    \begin{align*} \int_{E}F_n (m_n) m_n dx & \le\dfrac{1}{K}\int_{E\cap \left\{m_n\ge K \right\}}F_{n}\left(m_{n}\right)m_{n}dx+\dfrac{M}{K}+\sup\limits_{0\le r\le K}F(r)\int_{E\cap \left\{m_n\le K \right\}}\!\!\!\!\!\!dx\\ & \le \dfrac{C+M}{K}+C_{K}\eta, \end{align*}

    since \int_{E} F_n(m_n) m_n dx\le C and \sup_{0\le r \le K}F_{n}(r)\le \sup_{0\le r \le K}F(r): = C_K . Therefore, for all \varepsilon>0 , we may choose K such that (C+M)/K\le\varepsilon/2 and then \eta such that C_K \eta\le\varepsilon/2 and get

    \int_{E}F_{n}\left(m_{n}\right)dx\le \varepsilon,\quad\text{for all }{E}\text{ which satisfies }{\left|E\right|\le\eta},

    which proves the uniform integrability of \left\{F_n (m_n) \right\}_n .

    Next, since \rho_{n} and \int_{\Gamma}F_{n}\left(m_{n}\right)dx are uniformly bounded, we infer from (86) that \partial v_{n} is uniformly bounded in L^{q}\left(\Gamma\right) . Since by the condition \int_{\Gamma}v_ndx = 0 , there exists \overline{x}_n such that v_n(\overline{x}_n) = 0 , we infer from the latter bound that v_n is uniformly bounded in L^{\infty}(\Gamma) .

    Using the HJB equation in (24) and Remark 7, we get

    \mu_{\alpha}|\partial^2 v_n|\le |H(x,\partial v_n)| +|F_n(m_n)|+|\rho_n| \le C_q (|\partial v_n|^q+1)+|F_n(m_n)|+|\rho_n|.

    We obtain that \partial^{2}v_{n} is uniformly bounded in L^{1}\left(\Gamma\right) , which implies that v_n is uniformly bounded in C^1(\Gamma) . Therefore the sequence of functions C_q (|\partial v_n|^q+1)+|F_n(m_n)|+|\rho_n| is uniformly integrable, and so is \partial^{2}v_{n} . This implies that \partial v_{n} is equicontinuous. Hence, \left\{ v_{n}\right\} is relatively compact in C^{1}\left(\Gamma\right) by Arzelà-Ascoli's theorem. Finally, from the Fokker-Planck equation and Theorem 2.7, since \partial_{p}H\left(x,\partial v_{n}\right) is uniformly bounded in L^{\infty}\left(\Gamma\right) , we obtain that m_{n} is uniformly bounded in W .

    Step 4. Passage to the limit

    From Step 1 and 2, since \left\{ \rho_{n}\right\} is uniformly bounded, there exists \rho\in\mathbb{R} such that \rho_{n}\rightarrow\rho up to the extraction of subsequence. From Step 3, there exists m\in W such that m_{n}\rightharpoonup m in W and m_{n}\to m almost everywhere, up to the extraction of subsequence. Also from Step 3, since F_{n}\left(m_{n}\right) is uniformly integrable, from Vitali theorem, \lim_{n\rightarrow\infty}\int_{\Gamma}F_{n}\left(m_{n}\right)\tilde{w}dx = \int_{\Gamma}F\left(m\right)\tilde{w}dx,\quad\text{for all }\tilde{w}\in W . From Step 3, up to the extraction of subsequence, there exists v\in C^{1}\left(\Gamma\right) such that v_{n}\rightarrow v in C^{1}\left(\Gamma\right) . Hence, \left(v,\rho,m\right) satisfies the weak form of the MFG system:

    \sum\limits_{\alpha\in\mathcal{A}}\int_{\Gamma_{\alpha}}\mu_{\alpha}\partial v\partial\tilde{w}dx+\int_{\Gamma} \left(H\left(x,\partial v\right)+\rho\right)\tilde{w}dx = \int_{\Gamma}F\left(m\right)\tilde{w}dx,\quad\text{for all }\tilde{w}\in W,

    and

    \sum\limits_{\alpha\in\mathcal{A}}\int_{\Gamma_{\alpha}}\mu_{\alpha}\partial m\partial\tilde{v}dx+\int_{\Gamma}\partial_{p}H\left(x,\partial v\right)m\partial\tilde{v}dx = 0,\quad\text{for all }\tilde{v}\in V.

    Finally, we prove the regularity for the solution of (24). Since m\in W , m|_{\Gamma_\alpha}\in C^{0,\sigma} for some constant \sigma\in (0,1/2) and all \alpha\in \mathcal A . By Theorem 3.4, v\in C^2(\Gamma) ( v\in C^{2,\sigma}(\Gamma) if F is locally Lipschitz continuous). Then, by Theorem 2.7, we see that m\in \mathcal{W} . If F is locally Lipschitz continuous, this implies that v\in C^{2,1}(\Gamma) . We also obtain that v and m satisfy the Kirchhoff and transmission conditions in (24). The proof is done.



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