一类新型平均场偏微分方程的Sobolev解的概率解释
发布时间：2018-08-08 12:31
【摘要】：自2009年Buckdahn,Djehiche,Li和Peng[1]率先引入平均场倒向随机微分方程(简记为,MFBSDEs),这类方程就倍受关注。他们研究了 MFBSDEs和相应偏微分方程(简记为,PDEs)粘性解的关系。本文主要研究的是一类新型的平均场PDEs的弱解一 Sobolev解。与粘性解不同的是Sobolev解的存在唯一性不需要依赖于比较定理的结果,故方程的系数可以依赖于(?)。本文主要研究的方程形式如下:平均场SDE:平均场BSDE:以及新型平均场PDE:第一部分:主要的假设条件有:假设3.1:(A1)(i)函数b和σ关于(?),x满足Lipschitz条件。(ii)b(.,0,0)和σ(.,0,0)是F-循序可测连续函数且存在常数l0,使得对任意的0≤t≤T,(?),x(?)R~d(A2)(i)Φ是F(?)B(R)-可测随机变量,f(·,(?),x,(?),y,(?),z)是F-适应的可测过程,对任意的((?),x,(?),y,(?),z)∈ R~d × R~d × R~n × R~n × R~n×d × R~n×d 成立。且 f(t,(?),x,0,0,0,0)∈H_F~2(0,T;R~n)。(ii)f 关于 (?),x,(?),y,(?),z 满足 Lipschitz 条件。(iii)f和Φ满足线性增长条件,也就是说,存在c0,使得a.s.对任意的(?),x∈R~d，|f(t,(?),x,0,0,0,0)| + |Φ((?),x)| ≤c(1+|(?)|+|x|).(iv)G,θ,Ψ,κ:R~d→R~d,∧:R~d → R~d,Γ:R~n×d→R~n×d 的 Lipschitz 连续函数。(A3)给定((?))∈Rd × Rn × R×d,气对任意的s ∈[0,T],(x,y,z)→f(s,(?),x,(?),y,(?),z)∈Cb3,3,3(Rd × Rn × Rn×d,Rn).(A4)b ∈C61,3,3([0,T]× Rd × Rd,Rd)且 σ ∈Cb1,3,3([[0,T]× Rd ×RRd,Rd×d)。同时,我们给出值函数的定义为u(t,x)=Ytt,x。那么,在假设3.1下,平均场PDE(3)存在唯一解,且满足以下关系式Yst,x=u(s,Xxt,x),Zst,x= Dxu(s,Xst,x)σ(s,E[θ(Xs0,x0)],Xst,x).借助随机逆流、等价范数及测试函数,最终我们可以得到在假设3.1-(A2),(A4)下,u(t,x)=Ytt,x是平均场 PDE(3)的唯一 Sobolev 解。第二部分:第一,我们研究的是以下假设4.1条件成立的情况下,带全局单调系数的MFBSDE(2)解的存在唯一性定理。假设4.1:(H1)对任意固定的(ω,t),f(ω,t,.,.,.,.)连续;(H2)存在一过程ft∈HF2(0,T;R)和一个常数L0,使得|f(i,(?),(?),y,z)|≤ft + L(|(?)| + |(?)| + |y| + |z|).(H3)存在常数λ1,λ2 ∈ R,使得对任意的t∈[0,T],yi,(?)i ∈ Rn,z,(?) ∈ Rn×d(i = 1,2),(y1-y2)(f(t,(?),y1,(?),z)-f(t,(?),y2,(?),z))≤λ1(y1-y2)((?))+λ2|y1-y2|2.(H4)存在 L0,使得 P-a.s.对任意的t∈[0,T],y,y ∈ Rn,zi,zi ∈ Rn×d(i = 1,2),|f(t,y,y,z1,z1)-f(t,y,y,z2,z2)|2 ≤ L(|z1-z2|2 + |z1-z2|2).第二,我们研究的是带局部单调系数的MFBSDE(2)解的存在性和唯一性,假定如下条件成立,假设4.2:(H2,)存在L0和0 ≤ γ ≤ 1,使得|f(t,y,z,y,z)| ≤L(1 + |y|γ +|z|γ + |y|γ + |z|γ).(H3')对任意的N ∈ N,存在常数λN,λN∈R,使得对任意的t ∈[0,T],yi,yi∈Rn,z,z ∈ Rn×d 满足"yi|,|yi|,|z|,|z|≤N(i = 1,2),有(y1-y2)(f(t,y1,y1,z,z)-f(t,y2,y2,z,z))≤λN(y1-y2)(y1-y2)+ λN|y1-y2|2.(H4')对任意的N ∈N,存在LN0,使得P-a.s对任任的的f ∈[0,T],y,y∈Rn,zi,zi∈Rn×d满足|yi|,|yi|,|z|,|z|≤N(i=1,2),成立|f(t,y,y,z1,z1)-f(t,y,y,z2,z2)|2LN(|z1-z2|2 + |z1-z2|2).那么,我们可以得到在假设4.1-(H1)和假设4.2成立的情况下,且满足1 + exp(2L + 2|λN|+2λN-+ +2LNθ-1 + 2)→0,当N→∞时,(4)其中θ是一个任意固定的常数,使得0θ1-2α。带局部单调系数的MFBSDE(2)有唯一解(Y,Z)。第三:在前面的结论成立的情形下,我们可以开始研究相应平均场PDE(3)的Sobolev解的存在唯一性。首先,我们可以得到在以下假设下:假设4.3:(B1)6,σ 满足假设 3.1-(A1),(A4)。(B2)f,Φ 满足假设 3.1-(A2)-(i)(iii),以及假设 3.1-(A2)-(iv)成立,Φ ∈ L2(Rd,ρ(x)dx)。(B3)对任意的0≤t≤T,x1,x2,x1,x2∈Rn,y,y1,y2,y,y1,y2∈Rn,z,z1,z2,z,z1,z2 Rn×d,存在 C0,λ1,A2 ∈R,使得|Φ(x1,x1)-Φ(x2,x2)|2 +|f(t,x1,x1,y,y,z1,Z1)-f(t,x2,x2,y,y,z2,z2)|2C(|x1-x2|2 + |x1-x2|2 + |z1-z2|2 + |z1-z2|2).(y1-y2)(f(t,x1,x1,y1,y1,z,z)-f(t,x2,x2,y2,y2,z,z))≤ λ1(y1-y2)(y1-y2)+ λ2|y1-y2|2.(B4)|f(t,(?),x,(?),y,(?),z)| ≤ |f(t,(?),x,0,0,0,0)| + K(|y| + |y| + |z| + |z|),f(t,x,x,0,0,0,0)∈ L2(Rd,ρ(x)dx)且满足线性增长。值函数u(t,x):=Ytt,x是带全局单调系数的平均场PDE(3)的唯一Sobolev解。接着我们也可以得到在局部单调性的假设下平均场PDE的Sobolev解的存在唯一性定理的结论。相应的局部单调性假设,如下:假设4.4:(B3,)对任意的N ∈ N,存在LN0,λN,λN∈R,使得对x1,x1,x2,x2∈Rd,y1,y1,y2,y2∈Rn,z1,z1,z2,z2∈Rn×d,满足|y1|,|y1|,|y2|,|y2|,|z1|,|z1|,|z2|,|z2|≤N,成立|Φ(x1,x1)-Φ(x2,x2)|2 + |f(t,x1,x1,y1,y1,z1,z1)-f(t,x2,x2,y1,y1,z2,Z2)|2≤ LN(|x1-x2|2 + |x1-x2|2 + |z1-Z2|2 + |z1-z2|2),(y1-y2)(f(t,x1,x,x1,y1,y1,z1,z1)-f(t,x1,x1,y2,y2,z1,z1))≤ λN(y1-y2)(y1-y2)+ λN|y1-y2|2(B4,)存在K0和0≤γ≤1,使得|f(t,(?),x,(?),y,(?),z)|≤K(1 +|y|γ +|z|γ+ |y|γ +|z|γ),对任意的t,(?),x,(?),y,(?),z.于是,我们就可以得到在假设4.3-(B1),(B2),假设4.4和(4)式成立的条件下,带局部单调系数的平均场PDE(3)式存在唯一的Sobolev解。
[Abstract]:In 2009, Buckdahn, Djehiche, Li and Peng[1] took the lead in introducing the mean field backward stochastic differential equations (simple as, MFBSDEs). These equations are concerned. They have studied the relationship between MFBSDEs and the corresponding partial differential equation (Li, PDEs). This paper mainly deals with a new class of Weak Solutions of the mean field PDEs. The existence and uniqueness of the solution of Sobolev is not dependent on the result of the comparison theorem, so the coefficient of the equation can be dependent on (?). The main equations of this paper are as follows: the mean field SDE: mean field BSDE: and the new mean field PDE: first part: the main hypothesis conditions are: the 3.1: (A1) (I) function B and sigma (?), X satisfaction (II) B (., 0,0) and sigma (. 0,0) are F- sequential measurable continuous functions and exist constant l0, making any 0 less t less than T, (?), X (?) R~d (A2) is a measurable random variable. Constructs f importantly Marxism importantly Marxist Marxist Marxist Marxist Marxist Marxist Marxist Marxist societies traditions Marxism traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Gamma: The Lipschitz continuous function of n * D. (A3) Rd * Rn x R x D, gas to any s [0, T]. Under assumption 3.1, the average field PDE (3) has a unique solution and satisfies the following relational expression Yst, x=u (s, Xxt, x), Zst, x= Dxu (s, Xst, x)). With the aid of random currents, equivalent norms and test functions, we can finally get the only solution of average field (3). Second: First, we study the existence and uniqueness theorem of MFBSDE (2) solution with global monotone coefficients under the condition of the following hypothesis 4.1 conditions. 4.1: (H1) is assumed to be arbitrarily fixed (omega, t), f (omega, t,,,.) continuous; (H2) there exists a process FT HF2 (? 0, T; R) and a constant L0. 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Z2|2 + |z1-z2|2). Second, we study the existence and uniqueness of the MFBSDE (2) solution with local monotone coefficient, assuming the following conditions are established, assuming that 4.2: (H2,) exists L0 and 0 < < < 1 >, and makes |f (T, y, Z, y, z) less than equal. 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Then, we can get the assumption that 4.1- (H1) and hypothesis 4.2 are set up and satisfy the 1 + exp (2L + 2| lambda [lambda] [[lambda], theta 2) - 0. (4) theta is a arbitrarily fixed constant, which makes 0 theta 1-2 alpha (2) with local monotone coefficient. Third: Third: Third: Third: Third: before third: Third: before third: Third: Third: before third: Third: Third: before We can begin to study the existence and uniqueness of the Sobolev solution of the corresponding mean field PDE (3). First, we can get the following hypothesis: assuming 4.3: (B1) 6, sigma satisfies the hypothesis 3.1- (A1), (B2) f, and the hypothesis is assumed to be 3.1- (A2) - (I). , x1, X2, x1, Rn, Rn, y, Y1, Rn, Rn, Y1, Rn, Rn, Rn, Y1, Rn, Y1 less than 1, 1 and 2 |y1-y2|2. (B4) |f (T, (?), x, y, z), z) and K (x, 0,0,0,0) and satisfy linear growth. The corresponding local monotonicity hypothesis, as follows: assuming 4.4: (B3,) for arbitrary N N, LN0, N, N R. -f (T, X2, X2, X2, X2, X2, Y1, Y1, Y1, Y1, Y1) they are (?). So, we can get the only Sobolev solution of the mean field PDE (3) with local monotone coefficient under the assumption that 4.3- (B1), (B2), hypothesis 4.4 and (4) are established.
【学位授予单位】：山东大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O211.63
本文编号：2171795
[Abstract]:In 2009, Buckdahn, Djehiche, Li and Peng[1] took the lead in introducing the mean field backward stochastic differential equations (simple as, MFBSDEs). These equations are concerned. They have studied the relationship between MFBSDEs and the corresponding partial differential equation (Li, PDEs). This paper mainly deals with a new class of Weak Solutions of the mean field PDEs. The existence and uniqueness of the solution of Sobolev is not dependent on the result of the comparison theorem, so the coefficient of the equation can be dependent on (?). The main equations of this paper are as follows: the mean field SDE: mean field BSDE: and the new mean field PDE: first part: the main hypothesis conditions are: the 3.1: (A1) (I) function B and sigma (?), X satisfaction (II) B (., 0,0) and sigma (. 0,0) are F- sequential measurable continuous functions and exist constant l0, making any 0 less t less than T, (?), X (?) R~d (A2) is a measurable random variable. Constructs f importantly Marxism importantly Marxist Marxist Marxist Marxist Marxist Marxist Marxist Marxist societies traditions Marxism traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist traditions Marxist Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Marxist traditions Gamma: The Lipschitz continuous function of n * D. (A3) Rd * Rn x R x D, gas to any s [0, T]. Under assumption 3.1, the average field PDE (3) has a unique solution and satisfies the following relational expression Yst, x=u (s, Xxt, x), Zst, x= Dxu (s, Xst, x)). With the aid of random currents, equivalent norms and test functions, we can finally get the only solution of average field (3). Second: First, we study the existence and uniqueness theorem of MFBSDE (2) solution with global monotone coefficients under the condition of the following hypothesis 4.1 conditions. 4.1: (H1) is assumed to be arbitrarily fixed (omega, t), f (omega, t,,,.) continuous; (H2) there exists a process FT HF2 (? 0, T; R) and a constant L0. Beings Marxist Marxist traditions Marxist Marxist societies veins veins occasions veins veins veins occasions veins veins veins veins occasions veins veins veins veins veins veins occasions veins traditions souls veins veins veins occasions veins traditions souls veins veins veins veins veins occasions veins veins veins veins occasions veins veins veins veins veins occasions veins veins veins occasions veins veins veins veins veins occasions veins veins veins veins veins occasions veins veins veins occasions veins veins veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins traditions souls veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins traditions souls veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins traditions souls veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins it is the same. Z2|2 + |z1-z2|2). Second, we study the existence and uniqueness of the MFBSDE (2) solution with local monotone coefficient, assuming the following conditions are established, assuming that 4.2: (H2,) exists L0 and 0 < < < 1 >, and makes |f (T, y, Z, y, z) less than equal. Derive Rn clauses importantly Marxist traditions Marxist Marxist souls societies veins veins veins occasions veins veins veins veins veins veins occasions veins veins veins veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins occasions veins veins veins veins veins veins veins veins veins veins occasions veins veins veins veins veins veins veins veins occasions veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins occasions veins veins veins veins veins veins veins veins veins veins veins veins veins occasions veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins occasions veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins occasions veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins occasions veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins veins Y, y, Z2, Z2) |2LN (|z1-z2|2 + |z1-z2|2). Then, we can get the assumption that 4.1- (H1) and hypothesis 4.2 are set up and satisfy the 1 + exp (2L + 2| lambda [lambda] [[lambda], theta 2) - 0. (4) theta is a arbitrarily fixed constant, which makes 0 theta 1-2 alpha (2) with local monotone coefficient. Third: Third: Third: Third: Third: before third: Third: before third: Third: Third: before third: Third: Third: before We can begin to study the existence and uniqueness of the Sobolev solution of the corresponding mean field PDE (3). First, we can get the following hypothesis: assuming 4.3: (B1) 6, sigma satisfies the hypothesis 3.1- (A1), (B2) f, and the hypothesis is assumed to be 3.1- (A2) - (I). , x1, X2, x1, Rn, Rn, y, Y1, Rn, Rn, Y1, Rn, Rn, Rn, Y1, Rn, Y1 less than 1, 1 and 2 |y1-y2|2. (B4) |f (T, (?), x, y, z), z) and K (x, 0,0,0,0) and satisfy linear growth. The corresponding local monotonicity hypothesis, as follows: assuming 4.4: (B3,) for arbitrary N N, LN0, N, N R. -f (T, X2, X2, X2, X2, X2, Y1, Y1, Y1, Y1, Y1) they are (?). So, we can get the only Sobolev solution of the mean field PDE (3) with local monotone coefficient under the assumption that 4.3- (B1), (B2), hypothesis 4.4 and (4) are established.
【学位授予单位】：山东大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O211.63
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