STABILIZED NUMERICAL APPROXIMATIONS OF THE BACKWARD PROBLEM OF A PARABOLIC EQUATION

1　IntroductionLetΩ be a bounded domain in Rn and Ω be its boundary.ThenΣ =Ω× ( 0 ,1 ) is abounded domain in Rn+1 .We consider the following backwad problem of a prabolic equa-tion: u t= ni,j=1 xiaij( x) u xj -c( x) u,　　 ( x,t)∈Σ,( 1 )u| Ω× [0 ,1 ] =0 , ( 2 )u| t=1 =g( x) . ( 3 )　　 Where { aij( x) } are smooth functions given onΩ satisfyingaij( x) =aji( x) ,　　 1≤ i,j≤ n, ( 4)α0 ni=1ζ2i ≤ ni,j=1aij( x)ζiζj≤α1 ni=1ζ2i,　　 ζ∈ Rn,x∈Ω. ( 5)　　Where0 <α…     

一类椭圆型方程Dirichlet外问题的奇摄动

§ 1 .　IntroductionSingularperturbationofDirichletproblemsforellipticequationswerediscussedbysomeauthors[1 ] -[4] ,butmostofwhathavebeenconsideredareboundeddomain .InthispapertheauthorconsiderDirichletexteriorproblemsasfollow :εL1 [u]+L2 [u]=f(x ,u ,ε) ,x∈Rn -Ω ,　　 ( 1)u(x) =g(x ,ε) ,x∈ Ω ,( 2 )whereL1 issecondorderellipticoperator:L1 [u]=∑ni,j=1aij(x) 2 u xi xj+∑ni=1ai(x) u xi +a(x ,u) ,∑ni,j=1aijζiζj ≥δ0 >0 ,x∈Rn -Ω , ζ∈Rn ,ζ≠ 0 ,L2 isfirstorderdifferentialopera…     

Potential space estimates in local Hardy spaces for Green potentials in convex domains

Let Ω be a bounded co.nvex domain in Rn(n≥3) and G(x,y) be the Green function of the Laplace operator -△ on Ω. Let hrp(Ω) = {f ∈ D'(Ω) :(E)F∈hp(Rn), s.t. F|Ω = f}, by the atom characterization of Local Hardy spaces in a bounded Lipschitz domain, the bound of f→(△)2(Gf) for every f ∈ hrp(Ω) is obtained, where n/(n 1)＜p≤1.     

The Regularity of a Class of Degenerate Elliptic Monge-Ampere Equations

In the present paper the regularity of solutions to Dirichlet problem of degenerate elliptic Monge-Ampere equations is studied. Let Ω R^2 be smooth and convex. Suppose that u ∈ C^2（^-Ω） is a solution to the following problem： det（uij） = K（x）f（x,u, Du） in Ω with u = 0 on аΩ. Then u ∈ C^∞（f）） provided that f（x,u,p） is smooth and positive in ^-Ω × R × R^2, K〉0 in Ω and near αΩ, K = d^m ^-K, where d is the distance to αΩ, m some integer bigger than 1 and ^-K smooth and positive on ^-Ω.     

p阶平均曲率算子Dirichlet问题的无穷多个解

§ 1　 IntroductionSince the Mountain Pass Theorem came out,the existence of nontrivial solutions,pos-sibly multiple,ofnonlinear elliptic equations has been extensively studied.In this paper,weconsider the following Dirichlet problem for p-(generalized) mean curvature operator:-div((1 +| u|2 ) p- 22 u) =f(x,u) ,　 x∈Ω,u∈ W1 ,p0 (Ω ) , (1 .1 )whereΩ is a bounded domain in Rn(n>p>1 ) with smooth boundary Ω.First let us recall the following Dirichletproblem for p-Laplacian:-Δpu≡ -div…     

EXISTENCE OF SOLUTIONS TO THE PARABOLIC EQUATION WITH A SINGULAR POTENTIAL OF THE SOBOLEV-HARDY TYPE

We study the existence of solutions to the following parabolic equation{ut-△pu=λ/|x|s|u|q-2u,(x,t)∈Ω×(0,∞),u(x,0)=f(x),x∈Ω,u(x,t)=0,(x,t)∈Ω×(0,∞),(P)}where-△pu ≡-div(|▽u|p-2▽u),1相似文献   

NEW PROOFS OF THE DECAY ESTIMATE WITH SHARP RATE OF THE GLOBAL WEAK SOLUTION OF THE n-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

Consider the Cauchy problem for the n-dimensional incompressible NavierStokes equations:??tu-α△u+(u·?)u+?p = f(x, t), with the initial condition u(x, 0) = u0(x) and with the incompressible conditions ? · u = 0, ? · f = 0 and ? · u0= 0. The spatial dimension n ≥ 2.Suppose that the initial function u0∈ L1(Rn) ∩ L2(Rn) and the external force f ∈ L1(Rn× R+) ∩ L1(R+, L2(Rn)). It is well known that there holds the decay estimate with sharp rate:(1 + t)1+n/2∫Rn|u(x, t)|2 dx ≤ C, for all time t 0, where the dimension n ≥ 2, C 0 is a positive constant, independent of u and(x, t).The main purpose of this paper is to provide two independent proofs of the decay estimate with sharp rate, both are complete, systematic, simplified proofs, under a weaker condition on the external force. The ideas and methods introduced in this paper may have strong influence on the decay estimates with sharp rates of the global weak solutions or the global smooth solutions of similar equations, such as the n-dimensional magnetohydrodynamics equations, where the dimension n ≥ 2.     

A reverse Denjoy theorem Ⅲ

Suppose that C 1 and C 2 are two simple curves joining 0 to ∞, non-intersecting in the finite plane except at 0 and enclosing a domain D which is such that, for all large r, the set {θ : re iθ∈ D} has measure at most 2α, where 0 α π. Suppose also that u is a non-constant subharmonic function in the plane such that u(z) = Φ(|z|) for all large z ∈ C 1 ∪ C 2 ∪～D, where Φ(|z|) is a convex, non-decreasing function of |z| and ～D is the complement of D. Let A D (r, u) = inf{u(z) : z ∈ D and |z| = r}. It is shown that if A D (r, u) = O(1) then lim inf r→∞ B(r, u)/r π/(2α) 0.     

NUMERICALLY SOLVING PERIODICALLY PERTURBED CONSERVATIVE SYSTEMS BY PARAMETER EMBEDDING METHODS

1　IntroductionConsider the system of differential equationsTu≡ u"+ F( t,u) =0 ( 1 )where F:R×Rn→Rnis a continuos function of2 π-period with respect to tand F( t,· )∈ C1( Rn,Rn) has a symmetric derivative for all t∈R and allξ∈Rn.When the system is of the formu"+ grad G( u) =e( t) ( 2 )where G∈C2 ( Rn,R) ,e:R→Rncontinuousand2 π-periodic.Equation( 2 ) can be interpreted asthe Newtonian equation ofmotion ofa mechanicalsystem subjectto conservative internal forcesand periodic e…     

Jacobian-determinant equation in the critical Besov spaces

Let Ω be a bounded domain in R~n with smooth boundary. Here we consider the following Jacobian-determinant equation det u(x)=f(x),x∈Ω;u(x)=x,x∈?Ω where f is a function on Ω with min_Ω f = δ 0 and Ωf(x)dx = |Ω|. We prove that if f ∈B_(p1)~(np)(Ω) for some p∈(n,∞), then there exists a solution u ∈ B_(p1)~(np+1)(Ω)C~1(Ω) to this equation. On the other hand, we give a simple example such that u ∈ C_0~1(R~2, R~2) while detu does not lie in B_(p1)~(2p)(R~2) for any p∞.     

Extension of Floquet''''s theory to nonlinear quasiperiodic differential equations

In this paper, we consider the following autonomous system of differential equations: x = Ax f(x,θ), θ = ω, where θ∈Rm, ω = (ω1,…,ωm) ∈ Rm, x ∈ Rn, A ∈ Rn×n is a constant matrix and is hyperbolic, f is a C∞ function in both variables and 2π-periodic in each component of the vector e which satisfies f = O(||x||2) as x → 0. We study the normal form of this system and prove that under some proper conditions this system can be transformed to an autonomous system: x = Ax g(x), θ = ω. Additionally, the proof of this paper naturally implies the extension of Chen's theory in the quasi-periodic case.     

A Unified Boundary Behavior of Large Solutions to Hessian Equations

This paper is concerned with strictly k-convex large solutions to Hessian equations S_k(D²u(x)) = b(x)f(u(x)), x ∈ ?, where ? is a strictly(k-1)-convex and bounded smooth domain in Rn, b ∈ C^∞(?) is positive in ?, but may be vanishing on the boundary.Under a new structure condition on f at infinity, the author studies the refined boundary behavior of such solutions. The results are obtained in a more general setting than those in [Huang, Y., Boundary asymptotical b...     

The eigenvalue problem for the Laplacian equations

This article studies the Dirichlet eigenvalue problem for the Laplacian equations △u = -λu, x ∈Ω, u = 0, x ∈ (δ)Ω, where Ω (∩) Rn is a smooth bounded convex domain. By using the method of appropriate barrier function combined with the maximum principle, authors obtain a sharp lower bound of the difference of the first two eigenvalues for the Dirichlet eigenvalue problem. This study improves the result of S.T.Yau et al.     

Regularity of Solutions for the Evolution p Laplacian Equations

In this note we consider the following Cauchy problem: u t=div(| u| p- 2 u) ,(x,t)∈ QT=Rn × (0 ,T) ,u(x,0 ) =u0 (x)∈ L∞ (RN) ,x∈ RN,(1 )where p>2 is a constant.It is well known that there exists a solution u∈ Cαloc(QT)∩L∞ (QT) to (1 ) ,with uxj∈ Cβ,β/ 2loc (QT) ,j=1 ,2 ,… ,N (see [1 ] ,[2 ] ) .The proofs of uxj∈Cβ,β/ 2loc (QT) are very complicated and difficult.In this note we use another approach toprove the Holder continuity.We proveuxj∈ Cβ,β/ (1 +β)loc (QT…     

The Gradient Theory of the Phase Transitions with Dirichlet Boundary Condition

§ 1.Introduction　 In this paper,we study the asymptotic behavior of minimizers{ uε} ε>0 (ε→ 0 ) of thevariational probleminf∫Ω[εp- 1 | Du| p + 1εW(x,u) ] dx:u∈ W1 ,p(Ω ) ,u =g,x∈ Ω (Pε)where p>1 ,N≥ 2 ,andΩ RN is a bounded open domain with Ω∈C2 .　 This kind of problems comes from the theory of phase transitions and are widelystudied in recent years.In [1 ]— [1 4] ,the authors investigated these problems underthe integral constraint∫Ωudx =constant,In [1 5]— [1 6] ,…     

AN IMPROVEMENT OF A RESULT OF IVOCHKINA AND LADYZHENSKAYA ON A TYPE OF PARABOLIC MONGE-AMP\`ERE EQUATION

For the initial-hotmdary value problem about a type of parabolic Monge-Ampere equation of the form (IBVP): {-Dtu + (deD^2xu)^1/n = f(x,t), (x,t) ∈ Q = Ω&#215;(0,T)}, u(x,t) =Ф(x,t)(x,t) ∈δpQ}, where Ω is a bounded convex domain in R^n, the result in [4] by Ivochkina and Ladyzheokaya is improved in the sense that, under assumptions that the data of the problempossess lower regularity and satisfy lower order compatibility conditions than than in [4], the existence of classical solution to (IBVP) is still established (see Theorem 1.1 below). This cannot be reallzed by only using the method in [4]. The main additional effort the authors have done is a kind of nonlinear perturbation.     

DIRICHLET FORMS AND SYMMETRIC DIFFUSIONS ON A BOUNDED DOMAIN IN R~d

Let D be a bounded C~3-domain in R~d and(a_(ij))be a bounded symmetric matrixdefined on D.Consider the symmetric form(u,v)=1/2∫_D a_(ij)(x)(u(x))/(x_i) (v(x))/(x_j)dx,u,v∈H~1(D).Under some assumptions it is shown that the diffusion process associated with the regularDirichlet space(,(H~1(D))on L~2(D)can be characterized as a unique solution of acertain stochastic differential equation.     

Quasilinear elliptic equations with Neumann boundary and singularity

Let Ω be a bounded domain with a smooth C2 boundary in RN（N ≥ 3）, 0 ∈Ω, and n denote the unit outward normal to ЭΩ.We are concerned with the Neumann boundary problems： -div（｜x｜α｜△u｜p-2△u）=｜x｜βup（α,β）-1-λ｜x｜γup-1,u（x）〉0,x∈Ω,Эu/Эn=0 on ЭΩ,where 1〈p〈N and α〈0,β〈0 such that p（α,β）△=p（N＋β）/N-p＋α〉p,y〉α-p.For various parameters α,βorγ,we establish certain existence results of the solutions in the case 0∈Ω or 0∈ЭΩ.     

Existence of Solutions to Generalized Vector Quasi-Equilibrium Problems with Discontinuous Mappings

Let X, Y be two finite-dimensional topological vector spaces, Z a Hausdorff topological vector space, K C X and D C Z be two nonempty sets, C be a pointed, closed, and convex cone in Y with int C ≠θ Let S ： K → 2^K and T ： K → 2^D be two multivalued mappings, and φ ： K × D × K → Y be a trifunction. In this paper, we consider the generalized vector quasi-equilibrium problem, which is formulated by finding X∈ K and y∈ T（x） such that x∈ E S（x） and φ（x,y, u） （∈/） -int C for all u ∈ S（x）. We establish an existence result in which T is not supposed to have any continuity property. Our results extend and improve the corresponding results of Cubiotti, Yao and Guo.     

Ergodic Retraction Theorem and Weak Convergence Theorem for Reversible Semigroups of Non-Lipschitzian Mappings

Let G be a semitopological semigroup. Let C be a closed convex subset of a uniformly convex Banaeh space E with a Frechet differentiable norm, and T = {Tt ： t ∈ G} be a continuous representation of G as nearly asymptotically nonexpansive type mappings of C into itself such that the common fixed point set F（T） of T in C is nonempty. It is shown that if G is right reversible, then for each almost-orbit u（.） of T, ∩s∈G ^-CO{u（t） ： t ≥ s} ∩ F（T） consists of at most one point. Furthermore, ∩s∈G ^-CO{Ttx ： t ≥ s} ∩ F（T） is nonempty for each x ∈ C if and only if there exists a nonlinear ergodic retraction P of C onto F（T） such that PTs - TsP = P for all s ∈ G and Px ∈^-CO{Ttx ： s ∈ G} for each x ∈ C. This result is applied to study the problem of weak convergence of the net {u（t） ： t ∈ G} to a common fixed point of T.