
Left: the network
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 ς≥0 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 ς≥0 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
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
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
A bounded network
The edges
πα(y)=(ℓα−y)νi+yνjfor y∈[0,ℓα]. | (1) |
For a function
vα(y):=v∘πα(y), for all y∈(0,ℓα). |
The function
vα(0):=limy→0+vα(y) and vα(ℓα):=limy→ℓ−αvα(y). |
In that latter case, we can define
v|Γα(x)={vα(π−1α(x)),if x∈Γα∖V,vα(0)=limy→0+vα(y),if x=νi,vα(ℓα)=limy→ℓ−αvα(y),if x=νj. | (2) |
Notice that
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
either πα(0)=νi, for all α∈Ai or πα(ℓα)=νi, for all α∈Ai. | (3) |
This idea was used by Von Below in [35]: some edges of
Left: the network
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
PC(Γ)={v:Γ→R: for all α∈A,|vα∈C(0,ℓα) vα can be extended by continuity to [0,ℓα].}. |
By the definition of piecewise continuous functions
For
Cm(Γ):={v∈C(Γ):vα∈Cm([0,ℓα]) for all α∈A}. |
Notice that
For a positive integer
∂kv(x)=∂kvα(π−1α(x)) if x∈Γα∖V. | (4) |
For a vertex
∂αv(ν):={limh→0+vα(0)−vα(h)h,if ν=πα(0),limh→0+vα(ℓα)−vα(ℓα−h)h,if ν=πα(ℓα). | (5) |
For all
niα={1if νi=πα(ℓα),−1if νi=πα(0), | (6) |
we have
∂αv(νi)=niα∂v|Γα(νi)=niα∂vα(π−1α(νi)). | (7) |
Remark 2. Changing the orientation of the edge does not change the value of
If for all
Lp(Γ)={v:v|Γα∈Lp(Γα) for all α∈A}, |
Definition 1.1. For any integer
After rescaling the edges, it may be assumed that
As in Remark 1, we make the assumption (3) by possibly adding artificial nodes: if
Let us consider the linear differential operator:
Lu(x)=Lαu(x):=μα∂2u(x)+a|Γα(x)∂u(x),if x∈Γα, | (8) |
with domain
D(L):={u∈C2(Γ):∑α∈Aipiα∂αu(νi)=0, for all i∈I}. | (9) |
Remark 3. Note that in the definition of
Freidlin and Sheu proved in [15] that
1. The operator
∑α∈Aipiα∂αu(νi)=0 | (10) |
define such a process in a unique way, see also [16,Theorem 3.1]. The process can be written
2.There exist
(a) a one dimensional Wiener process
(b) continuous non-decreasing processes
(c) continuous non-increasing processes
such that
dxt=√2μαtdWt+aαt(xt)dt+dℓi,t+dhi,t,ℓi,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
u(Xt)=u(X0)+∑α∈A∫t01{Xs∈Γα∖V}(μα∂2u(Xs)+a(Xs)∂u(Xs)ds+√2μα∂u(Xs)dWs)+∑i∈I∑α∈Aipiα∂αu(νi)(ℓi,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
Consider the invariant measure associated with the process
E[u(Xt)]:=∫Γu(x)m(x)dx, for all u∈PC(Γ). | (13) |
We focus on functions
∫Γ(a(x)∂u(x)+μ∂2u(x))m(x)dx=0. | (14) |
Since for
−μα∂2m+∂(ma)=0 | (15) |
in the sense of distributions in the edges
−μα∂m|Γα=−m|Γαa|Γα+cα. | (16) |
So
∑i∈I∑α∈Aiμαm|Γα(νi)∂αu(νi)+∑β∈A∫Γβ∂u|Γβ(x)(−μβ∂m|Γβ(x)+a|Γβ(x)m|Γβ(x))dx=0. |
This and (16) imply that
∑i∈I∑α∈Aiμαm|Γα(νi)∂αu(νi)+∑β∈Acβ∫Γβ∂u|Γβ(x)dx=0. | (17) |
For all
1.
2.
Using such a test-function in (17) implies that for all
0=∑β∈Acβ∫Γβ∂u|Γβ(x)dx=∑j∈I∑α∈Ajcαnjαu|Γα(νj)=∑α∈Ainiαcα, | (18) |
where
For all
1.
2.
Using such a test-function in (17) yields
m|Γα(νi)γiα=m|Γβ(νi)γiβ,for all α,β∈Ai,νi∈V, |
in which
γiα=piαμα,for all i∈I,α∈Ai. | (19) |
Next, for
∑α∈Aiμα∂αm(νi)−niαa|Γα(νi)m|Γα(νi)=0,for all i∈I. | (20) |
Summarizing, we get the following boundary value problem for
{−μα∂2m+∂(ma)=0,x∈(Γα∖V),α∈A,∑α∈Aiμα∂αm(νi)−niαa|Γα(νi)m|Γα(νi)=0,νi∈V,m|Γα(νi)γiα=m|Γβ(νi)γiβ,α,β∈Ai,νi∈V. | (21) |
Consider a continuum of indistinguishable agents moving on the network
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
● The contribution of the control involves the Lagrangian
● The contribution of the distribution of states involves the coupling cost operator, which can either be nonlocal, i.e.,
Further assumptions on
Let us assume that there is an optimal feedback law, i.e. a function
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
{−μα∂2v+H(x,∂v)+ρ=V[m](x),x∈(Γα∖V),α∈A,∑α∈Aiγiαμα∂αv(νi)=0,νi∈V,v|Γα(νi)=v|Γβ(νi),α,β∈Ai,νi∈V,∫Γv(x)dx=0. | (23) |
We refer to [28,30] for the interpretation of the value function
Let us comment the different equations in (23):
1. The Hamiltonian
H(x,p)=supa{−ap−Lα(π−1α(x),a)}. |
The Hamiltonians
2. The second equation in (23) is a Kirchhoff transmission condition (or Neumann boundary condition if
3. The third condition means in particular that
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,
{−μα∂2v+H(x,∂v)+ρ=V([m]),x∈Γα∖V,α∈A,μα∂2m+∂(m∂pH(x,∂v))=0,x∈Γα∖V,α∈A,∑α∈Aiγiαμα∂α(νi)=0,νi∈V,∑α∈Ai[μα∂αm(νi)+niα∂pHα(νi,∂v|Γα(νi))m|Γα(νi)]=0,νi∈V,v|Γα(νi)=v|Γβ(νi),α,β∈Ai,νi∈V,m|Γα(νi)γiα=m|Γβ(νi)γiβ,α,β∈Ai,νi∈V,∫Γv(x)dx=0,∫Γm(x)dx=1,m≥0. | (24) |
At a vertex
Remark 5. In [7], the authors assume that
Let
Consider the Hamiltonian
Hα∈C1([0,ℓα]×R); | (25) |
Hα(x,⋅)is convex in p for each x∈[0,ℓα]; | (26) |
Hα(x,p)≥C0|p|q−C1 for (x,p)∈[0,ℓα]×R; | (27) |
|∂pHα(x,p)|≤C2(|p|q−1+1) for (x,p)∈[0,ℓα]×R. | (28) |
Remark 6. The Hamiltonian
Remark 7. From (28), there exists a positive constant
|Hα(x,p)|≤Cq(|p|q+1),for all (x,p)∈[0,ℓα]×R. | (29) |
Below, we shall focus on local coupling operators
V[˜m](x)=F(m(x)) with F∈C([0,+∞);R), | (30) |
for all
F(r)≥−M,for all r∈[0,+∞). | (31) |
Let us introduce two function spaces on
Definition 1.2. We define two Sobolev spaces,
W:={w:Γ→R:wα∈H1(0,ℓα) for all α∈A,w|Γα(νi)γiα=w|Γβ(νi)γiβ for all i∈I,α,β∈Ai} | (32) |
which is also a Hilbert space, endowed with the norm
Remark 8. We point out that, following Definition 1.1, functions in
Definition 1.3. Let the functions
{ψα is affine on (0,ℓα),ψ|Γα(νi)=γiα, if α∈Ai,ψ is constant on the edges Γα which touch the boundary of Γ. | (33) |
{ϕα is affine on (0,ℓα),ϕ|Γα(νi)=1γiα, if α∈Ai,ϕ is constant on the edges Γα which touch the boundary of Γ. | (34) |
Note that both functions
Remark 9. One can see that
Definition 1.4. Let the function space
W:={m:Γ→R:mα∈C1([0,ℓα]) for all α∈A,m|Γα(νi)γiα=m|Γβ(νi)γiβ for all i∈I,α,β∈Ai}. | (35) |
Remark 10. A function
Definition 1.5. A solution of the Mean Field Games system (24) is a triple
{−μα∂2v+H(x,∂v)+ρ=F(m),in Γα∖V,α∈A,∑α∈Aiγiαμα∂αv(νi)=0,if νi∈V, | (36) |
(note that
∑α∈A∫Γα[μα∂m∂u+∂(m∂pH(x,∂v))u]dx=0,for all u∈V, | (37) |
where
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
Remark 11. The proof of the existence result in [7] is valid only in the case when the coupling cost
Remark 12. The existence result in Theorem 1.6 holds if we assume that the coupling operator
This section contains elementary results on the solvability of some linear boundary value problems on
We recall that the constants
{−μα∂2v+λv=f,in Γα∖V,α∈A,v|Γα(νi)=v|Γβ(νi),α,β∈Ai,i∈I,∑α∈Aiγiαμα∂αv(νi)=0,i∈I, | (38) |
where
Remark 13. We have already noticed that, if
Definition 2.1. A weak solution of (38) is a function
Bλ(v,w)=⟨f,w⟩W′,W,for all w∈W, | (39) |
where
Bλ(v,w)=∑α∈A∫Γα(μα∂v∂w+λvw)dx. |
Remark 14. Formally, (39) is obtained by testing the first line of (38) by
Remark 15. By using the fact that
Let us first study the homogeneous case, i.e.
Lemma 2.2. The function
{−∂2v+λv=0,in Γα∖V,α∈A,v|Γα(νi)=v|Γβ(νi),α,β∈Ai,i∈I,∑α∈Aiγiαμα∂αv(νi)=0,i∈I, | (40) |
Proof. Let
For
−v″α(y)+λvα(y)=0,in (0,ℓα), |
whose solution is
vα(y)=ζαcosh(√λy)+ξαsinh(√λy), | (41) |
with
{ζα=vα(0)=v(νi),ζαcosh(√λℓα)+ξαsinh(√λℓα)=vα(ℓα)=v(νk). |
It follows that
0=∑α∈Aiγiαμα∂αv(νi)=∑k∈Ii√λγiαikμαikcosh(√λℓαik)sinh(√λℓαik)v(νi)−∑k∈Ii√λγiαikμαiksinh(√λℓαik)v(νk). |
Therefore, we obtain a system of linear equations of the form
{Mii=∑k∈Iiγiαikμαikcosh(√λℓαik)sinh(√λℓαik)>0,Mik=−γiαikμαiksinh(√λℓαik)≤0,k∈Ii,Mik=0,k∉Ii. |
For all
Let us now study the non-homogeneous problems (38).
Lemma 2.3. For any
Proof. First of all, we claim that for
Bλ(v,w)+λ0(v,w)=⟨f,w⟩W′,W | (42) |
has a unique solution
Bλ(v,ˆw)+λ0(v,ˆw)=∑α∈A∫Γα[μα|∂v|2ψ+μα(v∂v)∂ψ+(λ+λ0)v2ψ]dx≥∑α∈A∫Γα[μαψ_2|∂v|2+(λ0ψ_−μα¯∂ψ22ψ_)v2]dx. | (43) |
When
infv∈Vsupw∈WBλ(v,w)+λ0(v,w)‖v‖V‖w‖W≥μ_ψ_2Cψ. | (44) |
Using a similar argument for any
infw∈Wsupv∈VBλ(v,w)+λ0(v,w)‖w‖W‖v‖V≥μ_ϕ_2Cϕ. | (45) |
From (44) and (45), by the Banach-Necas-Babuška lemma (see [13]), for
Now, we fix
There exists ¯v∈W′∖{0} such that (Id−λ0(I∘¯Rλ0))¯v=0. | (46) |
For any g∈W′, there exists a unique ¯v∈W′ such that (Id−λ0(I∘¯Rλ0))¯v=g. | (47) |
We claim that (47) holds. Indeed, assume by contradiction that (46) holds. Then there exists
Then, (47) implies that there exists a positive constant
Consider
{−μα∂2v+b∂v=0,in Γα∖V,α∈A,v|Γα(νi)=v|Γβ(νi),α,β∈Ai,i∈I,∑α∈Aiγiαμα∂αv(νi)=0,i∈I. | (48) |
Definition 2.4. A weak solution of (48) is a function
A⋆(v,w)=0,for all w∈W, |
where
A⋆(v,w):=∑α∈A∫Γα(μα∂v∂w+b∂vw)dx. |
As in Remark 15, if
The uniqueness of solutions of (48) up to the addition of constants is obtained by using a maximum principle:
Lemma 2.5. For
Proof of Lemma 2.5. First of all, any constant function on
This paragraph is devoted to the dual boundary value problem of (48); it involves a Fokker-Planck equation:
{−μα∂2m−∂(bm)=0,in Γα∖V,α∈A,m|Γα(νi)γiα=m|Γβ(νi)γiβ,α,β∈Ai,i∈I,∑α∈Ai[niαb|Γα(νi)m|Γα(νi)+μα∂αm(νi)]=0,i∈I, | (49) |
where
m≥0,∫Γmdx=1. | (50) |
First of all, let
{λ0m−μα∂2m−∂(bm)=h,in Γα∖V,α∈A,m|Γα(νi)γiα=m|Γβ(νi)γiβ,α,β∈Ai,i∈I,∑α∈Ai[niαb|Γα(νi)m|Γα(νi)+μα∂αm(νi)]=0,i∈I. | (51) |
Definition 2.6. For
Aλ(m,v)=∑α∈A∫Γα[λmv+(μα∂m+bm)∂v]dx. |
A weak solution of (51) is a function
Aλ0(m,v)=⟨h,v⟩V′,V,for all v∈V. |
A weak solution of (49) is a function
A0(m,v):=∑α∈A∫Γα(μα∂m+bm)∂vdx=0,for all v∈V. | (52) |
Remark 16. Formally, to get (52), we multiply the first line of (49) by
Theorem 2.7. For any
● (Existence) There exists a solution
‖ˆm‖W≤C,0≤ˆm≤C, | (53) |
where the constant
● (Uniqueness)
● (Strictly positive solution)
Proof of existence in Theorem 2.7. We divide the proof of existence into three steps:
Step 1. Let
T:L2(Γ)⟶W,T(¯m)=m, |
where
Aλ0(w,ˆv)=∑α∈A∫Γα[λ0ϕw2+(μα∂w+bw)∂(wϕ)]dx=∑α∈A∫Γα[(λ0ϕ+b∂ϕ)w2+(μα∂ϕ+bϕ)w∂w+μαϕ(∂w)2]dx. |
It follows that when
infw∈Wsupv∈VAλ0(w,v)‖v‖V‖w‖W≥ˆCλ0Cϕ. |
Using similar arguments, for
infv∈Vsupw∈WAλ0(w,v)‖w‖W‖v‖V≥Cλ0Cψ. |
From Banach-Necas-Babuška lemma (see [13]), there exists a constant
Step 2. Let
K:={m∈L2(Γ):m≥0 and ∫Γmdx=1}. |
We claim that
∑α∈A−∫Γα[(λ0ϕ+b∂ϕ)(m−)2+μα(∂m−)2ϕ+(μα∂ϕ+bϕ)m−∂m−]dx=∫Γλ0¯mm−ϕdx. |
We can see that the right hand side is non-negative. Moreover, for
Step 3. We claim that
∑α∈A∫Γα[(λ0ϕ+b∂ϕ)m2+μα(∂m)2ϕ+(μα∂ϕ+bϕ)m(∂m)]dx=∫Γλ0¯mmϕdx. | (54) |
Since
∫Γ¯mmϕdx≤∫Γ¯mdx‖m‖L∞(Γ)¯ϕ=‖m‖L∞(Γ)¯ϕ≤C‖m‖W. |
Hence, from (54), for
‖m‖W≤λ0CC1. | (55) |
Therefore,
Finally, from the differential equation in (49), for all
ˆm′α+bαˆmα=Cα,for all x∈(0,ℓα). | (56) |
It follows that
Remark 17. Let
Proof of the positivity in Theorem 2.7. From (50),
ˆm′α+bαˆmα=0,on [0,ℓα], |
with
Therefore, it is enough to consider
Now, from Remark 1, we may assume without loss of generality that
Case 1. if
ˆm|Γβ(νi)=γiβγiαˆm|Γα(νi)=0,for all β∈Ai. | (57) |
This yields that
Cβ=ˆm′β(0)+bβ(0)ˆmβ(0)=0. |
Therefore,
Case 2. if
Finally, we have found that
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
Aλ0(w,v)=⟨f,v⟩V′,V for all v∈V. | (58) |
and
Sλ0:L2(Γ)⟶W,f⟼w, |
where
In order to characterize
{λ0v−μα∂2v+b∂v=g,in Γα∖V,α∈A,v|Γα(νi)=v|Γβ(νi)α,β∈Ai,i∈I,∑α∈Aiγiαμα∂αv(νi)=0,i∈I. | (59) |
A weak solution of (59) is a function
Tλ0(v,w):=∑α∈A∫Γα(λ0vw+μα∂v∂w+bw∂v)dx=∫Γgwdx,for all w∈W. |
Using similar arguments as in the proof of existence in Theorem 2.7, we see that for
Tλ0:L2(Γ)⟶V,g⟼v. |
Then we define
(g,Rλ0f)L2(Γ)=Tλ0(v,Sλ0f)=Aλ0(Sλ0f,v)=(f,v)L2(Γ)=(f,˜Rλ0g)L2(Γ). |
Thus
This implies that
Finally, since the solutions
This section is devoted to the following boundary value problem including a Hamilton-Jacobi equation:
{−μα∂2v+H(x,∂v)+λv=0,in Γα∖V,α∈A,v|Γα(νi)=v|Γβ(νi),α,β∈Ai,i∈I,∑α∈Aiγiαμα∂αv(νi)=0,i∈I, | (60) |
where
Definition 3.1 ● A classical solution of (60) is a function
● A weak solution of (60) is a function
∑α∈A∫Γα(μα∂v∂w+H(x,∂v)w+λvw)dx=0for all w∈W. |
Proposition 1. Assume that
Hα∈C([0,ℓα]×R), | (61) |
|H(x,p)|≤C2(1+|p|2)for all x∈Γ,p∈R, | (62) |
where
Remark 18. Assume (61) and that
Remark 19. Assume now that
Let us start with the case when
Lemma 3.2. Assume (61) and for some
|H(x,p)|≤CH,for all(x,p)∈Γ×R. | (63) |
There exists a classical solution
Proof of Lemma 3.2. For any
{−μα∂2v+λv=−H(x,∂u),if x∈Γα∖V,α∈A,v|Γα(νi)=v|Γβ(νi),α,β∈Ai,i∈I,∑α∈Aiγiαμα∂αv(νi)=0,i∈I, | (64) |
has a unique weak solution
‖v‖V≤C‖H(x,∂u)‖L2(Γ)≤CCH|Γ|1/2, | (65) |
where
μ_‖∂2v‖L2(Γ)≤λ‖v‖L2(Γ)+‖H(x,∂u)‖L2(Γ)≤λ‖v‖V+CH|Γ|1/2≤(λC+1)CH|Γ|1/2, | (66) |
where
Next, we claim that
{un→u,in {V},vn=T(un),for all {n},v=T(u), | (67) |
we need to prove that
From Schauder fixed point theorem, see [18,Corollary 11.2],
Assume now that
Lemma 3.3. Assume (61). If
{−μα∂2v+H(x,∂v)+λv≥−μα∂2u+H(x,∂u)+λu,ifx∈Γα∖V,α∈A,∑α∈Aiγiαμα∂αv(νi)≥∑α∈Aiγiαμα∂αu(νi),ifνi∈V, | (68) |
then
Proof of Lemma 3.3. The proof is reminiscent of an argument in [8]. Suppose by contradiction that
Now consider the case when
−μα[∂2vα(0)−∂2uα(0)]≥Hα(0,∂uα(0))−Hα(0,∂vα(0))⏟=0+λ(uα(0)−vα(0))>0. |
Therefore,
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
{−μα∂2v+Hn(x,∂v)+λv=0,x∈Γα∖V,α∈A,v|Γα(νi)=v|Γβ(νi), for all α,β∈Ai,i∈I,∑α∈Aiγiαμα∂αv(νi)=0,i∈I. | (69) |
We wish to pass to the limit as
Estimate in
Estimate in
∑α∈A∫Γα(μα∂vn∂wn+λvnwn)dx=−∫ΓHn(x,∂vn)wndx. |
Since
∑α∈A∫ΓαeKv2n[(μαψ)(∂vn)2+(μα2Kψ)v2n(∂vn)2+(μα∂ψ)vn∂vn+λψv2n]dx≤∫ΓeKv2n|Hn(x,∂vn)||vnψ|dx≤∫ΓceKv2nψ|vn|dx+∫ΓcψeKv2n|vn|ψ(∂vn)2dx≤∫ΓeKv2n(λψv2n+ψc24λ)dx+∑α∈A∫ΓαeKv2n[μα2ψ(∂vn)2+c22μαψ(∂vn)2v2n]dx, |
where we have used Young inequalities. Since
∑α∈A∫ΓαeKv2n[(μα2ψ)(∂vn)2+2ψ(μαK−c24μα)v2n(∂vn)2+(μα∂ψ)vn∂vn]dx≤c24λ∫ΓeKv2nψdx. | (70) |
Next, choosing
∑α∈A∫ΓαeKv2n[μα2ψ(∂vn)2+2ψv2n(∂vn)2+(μα∂ψ)vn∂vn]dx≤C |
for a positive constant
Estimate in
μ_|∂2vn|≤c+c|∂vn|2+λ|vn|,for all α∈A. |
Thus
Hence, for any
Assume now that
The proof is complete.
For
{−μα∂2v+H(x,∂v)+ρ=f(x),in Γα∖V,α∈A,v|Γα(νi)=v|Γβ(νi),α,β∈Ai,i∈I,∑α∈Aiγiαμα∂αv(νi)=0,i∈I, | (71) |
with the normalization condition
∫Γvdx=0. | (72) |
Theorem 3.4. Assume (25)-(27). There exists a unique couple
‖v‖C2(Γ)≤¯C. | (73) |
Moreover, for some
‖v‖C2.σ(Γ)≤¯C. | (74) |
Proof of existence in Theorem 3.4. By Proposition 1, for any
{−μα∂2v+H(x,∂v)+λv=f,in Γα∖V,α∈A,v|Γα(νi)=v|Γβ(νi),α,β∈Ai,i∈I,∑α∈Aiγiαμα∂αv(νi)=0,i∈I, | (75) |
has a unique solution
−C≤λvλ(x)≤C,for all x∈Γ. | (76) |
Next, set
{−μα∂2uλ+H(x,∂uλ)+λuλ+λminΓvλ=f,in Γα∖V,α∈A,u|Γα(νi)=u|Γβ(νi),α,β∈Ai,i∈I,∑α∈Aiγiαμα∂αuλ(νi)=0,i∈I. | (77) |
Before passing to the limit as
Step 1. Estimate of
∑α∈A∫Γαμα∂uλ∂ψdx+∫Γ(H(x,∂uλ)+λvλ)ψdx=∫Γfψdx. |
From (27) and (76),
∑α∈A∫Γαμα∂uλ∂ψdx+∑α∈A∫ΓαC0|∂uλ|qψdx≤∫Γ(f+C+C1)ψdx. |
On the other hand, since
∑α∈A∫Γαμα∂uλ∂ψdx+12∑α∈A∫ΓαC0|∂uλ|qψdx+C′>0,for all λ>0. |
Subtracting, we get
‖∂uλ‖Lq(Γ)≤˜C, | (78) |
where
Step 2. Estimate of
|uλ(x)|=|uλ(x)−uλ(xλ)|≤∫Γ|∂uλ|dx≤‖∂uλ‖Lq(Γ)|Γ|q/(q−1). |
From (78) and the latter inequality, we deduce that
|uλ(y)|≤|uλ(y)−uλ(νi)|+|uλ(νi)|≤2˜C|Γ|q/(q−1). |
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
‖uλ‖L∞(Γ)≤M˜C|Γ|q/(q−1). |
This bound is uniform with respect to
μ_|∂2uλ|≤|H(x,∂uλ)|+|λvλ|+|f|≤Cq(1+|∂uλ|q)+C+‖f‖L∞(Γ). |
Hence, from (78),
After the extraction of a subsequence, we may assume that when
Furthermore, if for some
Proof of uniqueness in Theorem 3.4. Assume that there exist two solutions
We now prove the uniqueness of
Remark 20. Since there exists a unique solution of (71)-(72), we conclude that the whole sequence
We first prove Theorem 1.6 when
Theorem 4.1. Assume (25)-(28), (30) and that
Proof of existence in Theorem 4.1. We adapt the proof of Camilli and Marchi in [7,Theorem 1]. For
Mσ={m:mα∈C0,σ([0,ℓα]) and m|Γα(νi)γiα=m|Γβ(νi)γiβ| for all i∈I and α,β∈Ai} |
which, endowed with the norm
‖m‖Mσ=‖m‖L∞(Γ)+maxα∈Asupy,z∈[0,ℓα],y≠z|mα(y)−mα(z)||y−z|σ, |
is a Banach space. Now consider the set
K={m∈Mσ:m≥0 and ∫Γmdx=1} |
and observe that
Next,
The image of
End of the proof. We can apply Schauder fixed point theorem (see [18,Corollary 11.2]) to conclude that the map
Proof of uniqueness in Theorem 4.1. We assume that
{−μα∂2¯v+H(x,∂v1)−H(x,∂v2)+¯ρ−(F(m1)−F(m2))=0,in Γα∖V,−μα∂2¯m−∂(m1∂pH(x,∂v1))+∂(m2∂pH(x,∂v2))=0,in Γα∖V,¯v|Γα(νi)=¯v|Γβ(νi),¯m|Γα(νi)γiα=¯m|Γβ(νi)γiβ,α,β∈Ai,i∈I,∑α∈Aiγiαμα∂α¯v(νi)=0,i∈I,∑α∈Ainiα[m1|Γα(νi)∂pH(νi,∂v1|Γα(νi))−m2|Γα(νi)∂pH(νi,∂v2|Γα(νi))]+∑α∈Aiμα∂α¯m(νi)=0,i∈I,∫Γ¯vdx=0,∫Γ¯mdx=0. | (79) |
Multiplying the equation for
∫Γαμα∂¯v∂¯m+[H(x,∂v1)−H(x,∂v2)+¯ρ−(F(m1)−F(m2))]¯mdx−[μα¯mα∂¯vα]ℓα0=0. | (80) |
Multiplying the equation for
∫Γαμα∂¯v∂¯m+[m1∂pH(x,∂v1)−m2∂pH(x,∂v2)]∂¯vdx−[¯v|Γα(μα∂¯m|Γα+m1|Γα∂pH(x,∂v1|Γα)−m2|Γα∂pH(x,∂v2|Γα))]ℓα0=0. | (81) |
Subtracting (80) to (81), summing over
0=∑α∈A∫Γα(m1−m2)[F(m1)−F(m2)]dx+∑α∈A∫Γαm1[H(x,∂v2)−H(x,∂v1)+∂pH(x,∂v1)∂¯v]dx+∑α∈A∫Γαm2[H(x,∂v1)−H(x,∂v2)−∂pH(x,∂v2)∂¯v]dx. |
Since
Proof of Theorem 1.6 for a general coupling
We now truncate the coupling function as follows:
Fn(r)={F(r),if |r|≤n,F(r|r|n),if |r|≥n. |
Then
Step 1.
∑α∈A∫Γαμα∂vn∂mndx+∫ΓH(x,∂vn)mndx+ρn=∫ΓFn(mn)mndx, | (82) |
and
∑α∈A∫Γαμα∂vn∂mndx+∫Γ∂pH(x,∂vn)mn∂vndx=0. | (83) |
Subtracting the two equations, we obtain
ρn=∫ΓFn(mn)mndx+∫Γ[∂pH(x,∂vn)∂vn−H(x,∂vn)]mndx. | (84) |
In what follows, the constant
ρn≥∫ΓFn(mn)mndx−C∫Γmndx=∫ΓFn(mn)mndx−C. | (85) |
Hence, since
Step 2.
∑α∈A∫Γαμα∂w∂udx=0,for all u∈V,and∫Γwdx=1. |
Multiplying the HJB equation of (24) by
∑α∈A∫Γαμα∂vn∂wdx⏟=0+∫ΓH(x,∂vn)wdx+ρn∫Γwdx⏟=1=∫ΓFn(mn)wdx. |
This implies, using (27), (53) and
ρn=∫ΓFn(mn)wdx−∫ΓH(x,∂vn)wdx≤‖w‖L∞(Γ)∫Γ(Fn(mn)+M)dx−M−∫Γ(C0|∂vn|q−C1)wdx≤C∫ΓFn(mn)dx+C−∫ΓC0|∂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
\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
Step 3. Prove that
\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}\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
Next, since
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
Step 4. Passage to the limit
From Step 1 and 2, since
\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
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