Some Parabolic Monge-Ampère Equations From Theoretics And Application

This paper includes the Theory and Application two parts.In theory, for the general form Monge-Amp`ere operator that be raisedby Ca?arelli-Nirenberg-Spruck and the Hessian operator which being devel-oped in the sequel from itthe corresponding two types parabolic Monge-Amp`ere equations were in-vestigated. Which extended the results of [4], [5] and [20] respectively.In application, we revised the model that has been constructed by YongJiongmin in optimization investment and investigated the following cauchyproblem of one dimension parabolic Monge-Amp`ere equationIn chapter 1, we consider the first initial-boundary value problem of thefollowing general parabolic Monge-Amp`ere equation Where is a bounded convex domain in n-dimensional Euclidean spaceRn, T > 0 is a given constant.ψ(x,t),(x,t) andσ= (σij(x,t)) are givenfunctions and symmetric matrix defined on Q respectively.The main result of this chapter is the following theorem:Theorem 1 Suppose that is strictly convex, for some 0 <α< 1,∈C4+α,ψ(x,t)∈C2+α,1+α/2(Q), (x,t)∈C4+α,2+α/2(Q),ψ(x,t) and(x,t) satisfy the compatibility conditions up to the second order. Thenproblem (1) has a unique admissible solution u(x,t)∈C4+α,2+α/2(Q), if thecondition (H1) and one of the conditions (H2) and (H2) is fulfilled.(H1) (Dij(x,0) +σij(x,0)) > 0, for all x∈.(H2)σ=σ(x)∈C2+α(), mQinψ+ mpiQn Dt 21ad2≡ν1 > 0 (d is theradius of the minimal ball Bd(x0) containing , a = max{0,mQa xDtψ}.(H2)σ=σ(x,t)∈C2+α,1+α/2(Q), mpiQn (Dt +ψ) =ν1 > 0 and thematrix (DijψDtσij) is nonpositive in Q.In chapter 2, we investigate the first initial-boundary value problem ofthe following general parabolic Hessian equation:Where is a bounded domain in n dimensional Euclidean space Rn, T > 0is a given constant.ψ(x,t),(x,t) andσ= (σij(x,t)) are given func-tions and symmetric matrix defined on QT respectively. f(λ) is a givensmooth and symmetric function ofλ= (λ1,···,λn) with a special case thatf(λ) = Sk1 /k(λ) for Sk(λ) being elementary symmetrical polynomials(k =2,3,···,n). D2u = (Diju) is the Hessian of u with respect to the variablex,λ(D2u +σ) is the eigenvalue vector of the matrix D2u +σ.This parabolic type equation is corresponding to the elliptic Hessian equa- tions in [2] and [3]. Where in [2] and [3], assumptions about f(λ) and isthat:f(λ) is assumed a smooth function defined in an open convex coneΓRn,which is dierent from Rn, with vertex at the origin, containing the positivecone: {λ∈Rn, each componentλi > 0} and satisfying the following inΓ:Γand f are assumed to be invariant under interchange of any twoλi,namely, they are symmetric inλi. It follows thatFor every C > 0 and every compact set K inΓthere is a number R =R(C,K) such thatAs to the shape of , they supposed that there exists a number R su-ciently large such that at every point x∈, ifκ1,···,κn1 represent theprincipal curvatures of (relative to the interior normal), then(κ1,···,κn1,R)∈Γ. (8)In this chapter we assume that f(λ),Γand satisfied (3)-(8) as [2]and[3].In addition we also assume that there exist some positive constantsν0andν1 such that n Our main result of this chapter is the following theorem.Theorem 2 Suppose that, for someα∈(0,1),ψ(x,t)∈C2+α,1+α/2(Q),(x,t)∈C4+α,2+α/2(Q),ψ(x,t) and (x,t) satisfy the compatibility con-ditions up to the second order,∈C4+αand ,f,Γsatisfy (3)-(10).Then problem (2) has a unique admissible solution u∈C4+α,2+α/2(Q) ifthe condition (H1) and forν1 comes from (10), one of the assumptions(H2) and (H2) is fulfilled:(H1)λ(Dij(x,0) +σij(x,0))∈Γ,for all x∈.(H2)σ=σ(x)∈C2+α(),mQinψ+ mpiQn Dt 21ad2ν1 (d is the radiusof the minimal ball Bd(x0) containing , a = max{0,ν10 mQa xDtψ}).(H2)σ=σ(x,t)∈C2+α,1+α/2(Q),mpiQn (Dt +ψ)ν1 and the matrix(DiiψDtσii) is nonpositive (i.e. its eigenvalues are nonpositive) in Q.In chapter 3, we try to extend the investigation of parabolic Monge-Amp`ere equation from Euclidean space Rn to Riemann manifolds, andconsider the classical solvability of the problemDtug1(x)det( iju) =ψ(x,t), (x,t)∈Q =×(0,T],u(x,t) = (x,t), (x,t)∈pQ. (11)Where Mn is a smooth Riemann manifold of dimension n 2 and Mnis a compact domain with smooth boundary . gij denotes the metric ofMn, g = det(gij) > 0.ψand are given functions defined on Q and pQrespectively.The main result of this chapter is:Theorem 3 Suppose the following conditions are fulfilled: (H1) There exists a functionΨ(x) defined in such thatΨ(x)∈C4(),with 2Ψbeing positive definite everywhere in Q andΨ| = 0.(H2)ψ(x,t)∈C2,1(Q) andψ(x,t) > 0 on Q.(H3) For someα∈(0,1),(x,t)∈C4+α,2+α/2(Q), 2(x,0) is positivedefinite everywhere in , Dt(x,t) < 0 for x∈.(H4) Compatibility conditions up to the first order of problem (11) aresatisfied.Then the problem (11) has a unique admissible solution u(x,t)∈C4,2(Q).In chapter 4, we extend the operator above from Monge-Amp`ere to Hes-sian and investigate the first initial-boundary value problem of the followingparabolic Hessian equation in Riemann manifoldsDtug1(x)f(λ( 2u)) =ψ(x,t), (x,t)∈Q =×(0,T],u(x,t) = (x,t), (x,t)∈pQ. (12)Where gij denotes the metric of Mn, g = det(gij) > 0. 2u denotes theHessian of a function u on Mn and, for a (0,2) tensor h on Mn,λ(h) =(λ1,···,λn) denotes the eigenvalues of h with respect to the metric g.ψand are given functions defined on Q and pQ respectively.As in [2] and [3], we assume that f and satisfied (3)-(8).In addition, as in [12], we also assume that there exist some positiveμ0andν0 such that and for anyψ1 >ψ0 > 0,Our main result of this chapter is the following theorem.Theorem 4 Suppose that, for someα∈(0,1), (x,t)∈C4+α,2+α/2(Q)and satisfies the necessary conditionλ( 2(x,0))∈Γfor all x∈.Dt(x,t) < 0 on Q,ψ(x,t)∈C2,1(Q) andψ(x,t) > 0 on Q,ψ(x,t) and(x,t) satisfy the compatibility conditions up to the first order.∈C4and ,f,Γsatisfy the conditions (3)-(8),(13)-(16). Then problem (12) hasa unique admissible solution u∈C4,2(Q).In chapter 5, we discuss the model that has been constructed by YongJiongmin in optimization investment [40]. In order to approach practise,we investigate the first initial-boundary value problem of the followingparabolic Monge-Amp`ere equation instead of the initial value problem one.We get the existence of solution in C4+α,2+α/2(Q) to the problem (17) forr = 0 and r > 0 two cases.The necessary conditions we need is that:(H1) For some constantα∈(0,1),μ> 0, u0(t),u1(t)∈C2+α([0,T]),g(x)∈C4+α([0,1]) satisfies u0(t)μ,u1(t) ru0(t) + ru1(t)μon [0,T],g (x) < 0,g (x)μon [0,X].(H2) Problem (17) satisfy the compatibility conditions up to the secondorder.The sucient condition we need is: (H3) There exist constantsα∈(0,1),v > 0 and strong convex monotonyfunctions u0(x,t),u(x,t)∈C4+α,2+α/2(Q) such thatTheorem 5 Suppose the conditions (H1)-(H3) are fulfilled, then theproblem (17) has a unique strong convex monotony function u = u(x,t)∈C4+α,2+α/2(Q).