Perturbations of an abstract Euler-Poisson-Darboux equation

The stability of the uniform correctness of the Cauchy problem ,t>0,u(0)=u ₀,u′(0)=0 fork>0 with respect to perturbations of the operator is studied. Translated fromMatematickeskie Zameiki, Vol. 60, No. 3, pp. 363–369, September, 1996.     

Partial regularity for certain classes of polyconvex functionals related to nonlinear elasticity

Consider the variational integral where Ω⊂ℝ ⁿ andp≥n≥2. H: (0, ∞)→[0, ∞) is a smooth convex function such that . We approximateJ by a sequence of regularized functionalsJ _δ whose minimizers converge strongly to anJ-minimizing function and prove partial regularity results forJ _δ-minimizers.     

Structure of bounded solutions for a class of abstract differential equations

Summary The paper is concerned with bounded solutions of an equationu′(t)=Bu(t) in Hilbert spaces, . A representation formula is obtained depending on the zeros of Rez(θ).

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Riassunto è studiata la struttura delle soluzioni limitate di una equazione differenzialeu′(t)=Bu(t) in uno spazio di Hilbert, ove , in funzione dei zeri di Rez(θ).

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This research is supported through a grant of the National Research Council Canada.     

The first boundary-value problem for a singular nonlinear ordinary differential equation of fourth order

The solvability of the boundary-value problem

Solutions to Some Open Problems in Fluid Dynamics

Let u=u（x,t,uo）represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation ut-εuxxt＋δux＋γHuxx＋βuxxx＋f（u）x=αuxx,u（x,0）=uo（x）, whereα〉0,β〉0,γ〉0,δ〉0 andε〉0 are constants.This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations. The nonlinear function satisfies the conditions f（0）=0,｜f（u）｜→∞as ｜u｜→∞,and f∈C^1（R）,and there exist the following limits Lo=lim sup/u→o f（u）/u^3 and L∞=lim sup/u→∞ f（u）/u^5 Suppose that the initial function u0∈L^I（R）∩H^2（R）.By using energy estimates,Fourier transform,Plancherel＇s identity,upper limit estimate,lower limit estimate and the results of the linear problem vt-εv（xxt）＋δvx＋γHv（xx）＋βv（xxx）=αv（xx）,v（x,0）=vo（x）, the author justifies the following limits（with sharp rates of decay） lim t→∞[（1＋t）^（m＋1/2）∫｜uxm（x,t）｜^2dx]=1/2π（π/2α）^（1/2）m!!/（4α）^m[∫R uo（x）dx]^2, if∫R uo（x）dx≠0, where 0!!=1,1!!=1 and m!!=1·3…（2m-3）…（2m-1）.Moreover lim t→∞[（1＋t）^（m＋3/2）∫R｜uxm（x,t）｜^2dx]=1/2π（x/2α）^（1/2）（m＋1）!!/（4α）^（m＋1）[∫Rρo（x）dx]^2, if the initial function uo（x）=ρo′（x）,for some functionρo∈C^1（R）∩L^1（R）and∫Rρo（x）dx≠0.     

Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations

This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt ＋ α1 uδ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x,t）∈ E R^2, and δu/δt ＋ α2 δu/δx δ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x, t） ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0（x） ∈ H^s（R）, and s ≥ 5/4 for the first equation and s≥301/108 for the second equation.     

Existence of solutions with turning points for nonlinear singularly perturbed boundary-value problems

We consider the following singularly perturbed boundary-value problem:

Positive Solution of Multi-Point Boundary Value Problems for the One-Dimensional p-Laplacian with Singularities

In this paper, we obtain positive solution to the following multi-point singular boundary value problem with p-Laplacian operator,{（ φp（u＇））＇＋q（t）f（t,u,u＇）=0,0〈t〈1,u（0）=∑i=1^nαiu（ξi）,u＇（1）=∑i=1^nβiu＇（ξi）,whereφp（s）=｜s｜^p-2s,p≥2;ξi∈（0,1）（i=1,2,…,n）,0≤αi,βi〈1（i=1,2,…n）,0≤∑i=1^nαi,∑i=1^nβi〈1,and q（t） may be singular at t=0,1,f（t,u,u＇）may be singular at u＇=0     

Lower semicontinuity of some functionals under PDE constraints: An \mathcal{A}-quasiconvex pair

The problem of establishing necessary and sufficient conditions for l.s.c. under PDE constraints is studied for a special class of functionals:

Some existence and nonexistence theorems for compact graphs of constant mean curvature with boundary in parallel planes

Given H≥0 and bounded convex curves α₁, ...,⇌_n, α in the plane z=0 bounding domains D₁, …, D_n, D, respectively, with if i ∈ j and with D_i ⊂ D, we obtain several results proving the existence of a constanth depending only on H and on the geometry of the curves α_i, α such that the Dirichlet problem for the constant mean curvature H equation: where may accept or not a solution.     

A remark on the coefficients of bounded polynomials

Let be such that |p(e^iq)|≤1 for ϕ∈R and |p(1)|=a∈[0,1]. An inequality of Dewan and Govil for the sum |a_v|+|a_n|, 0≤u<v≤n is sharpened.     

Estimates of the rate of decay of solutions to the impedance mixed problem for the wave equation in a region with noncompact boundary

The paper is devoted to the study of the behavior of the following mixed problem for large values of time:

Some remarks concerning the consistency of LS and LAD estimates of multiple regression

1.IntroductionConsideralinearregressionmodelwhereK,xi,doandeiaretheobservationofthetargetvariable,aknownHvector,theunknownparametervector,andtherandomerror,respectively.LetpbeaconvexfunctiondefinedonRI.TheM-estimateofpo,tobedenotedbyac,isdefinedasaminimizingpointofthefunctionH(P)=ZP(K--x:g).Denotetheleftandrightderivativesofpbyop--andi=1op .Regardingtheweakconsistencyofac,Zhao,RaoandChenll]establishedthefollowingresult:TheoremA.Lete15eZt'belid.Supposethatthereexistsfunctionop,satisfy…     

On the weighted elliptic problems involving multi-singular potentials and multi-critical exponents

Suppose Ω belong to R^N（N≥3） is a smooth bounded domain,ξi∈Ω,0〈ai〈√μ,μ：=（（N-1）/2）^2,0≤μi〈（√μ-ai）^2,ai〈bi〈ai＋1 and pi：=2N/N-2（1＋ai-bi）are the weighted critical Hardy-Sobolev exponents, i = 1, 2,..., k, k ≥ 2. We deal with the conditions that ensure the existence of positive solutions to the multi-singular and multi-critical elliptic problem ∑i=1^k（-div（｜x-ξi｜^-2ai△↓u）-μiu/｜x-ξi｜^2（1＋ai）-u^pi-1/｜x-ξi｜^bipi）=0with Dirichlet boundary condition, which involves the weighted Hardy inequality and the weighted Hardy-Sobolev inequality. The results depend crucially on the parameters ai, bi and #i, i -- 1, 2,..., k.     

ON A CLASS OF BESICOVITCHFUNCTIONS TO HAVE EXACT BOX DIMENSION： A NECESSARY AND SUFFICIENT CONDITION

This paper summarized recent achievements obtained by the authors about the box dimensions of the Besicovitch functions given byB(t) := ∞∑k=1 λs-2k sin(λkt),where 1 ＜ s ＜ 2, λk ＞ 0 tends to infinity as k →∞ and λk satisfies λk 1/λk ≥λ＞ 1. The results show thatlimk→∞ log λk 1/log λk = 1is a necessary and sufficient condition for Graph(B(t)) to have same upper and lower box dimensions.For the fractional Riemann-Liouville differential operator Du and the fractional integral operator D-v,the results show that if λ is sufficiently large, then a necessary and sufficient condition for box dimension of Graph(D-v(B)),0 ＜ v ＜ s - 1, to be s - v and box dimension of Graph(Du(B)),0 ＜ u ＜ 2 - s, to be s uis also lim k→∞logλk 1/log λk = 1.     

一阶时滞微分方程解的零点分布

Abstract. The paper gives two estimates of the distance between adjacent zeros of solutions     

A mean ergodic theorem for resolvent operators

Let {R(t)} _t≥0 be a uniformly bounded strongly continuous resolvent operator for the Volterra equation of convolution typeu=g+k*Au, whereA is a closed and densely defined operator on a Banach spaceX andk is a scalar kernel. We show that whenX is reflexive and that the average given by {R(t)} _t≥0 andk converges on the closed subspace to a bounded projection. This work was partially supported by DICYT 92-33LY and FONDECYT 91-0471     

A necessary and sufficient condition for singular nonlinear second-order boundary value problems to have positive solutions

In this paper the following result is obtained: Suppose f(g,u,v) is nonnegative, continuous in (a, 6) ×R⁺ ×R ⁺ ; f may be singular at κ = a(and/or κ = b) and υ = 0; f is nondecreasing on u for each κ,υ,nonincreasing on υ for each κ,u; there exists a constant q ε (0,1) such that

The Cauchy problem for nonstrictly second order hyperbolic equations with nonregular coefficients

In this paper we consider the Cauchy problem for the equation , where the matrix {a _jk(x)} is non-negative, and the first derivatives of the coefficients have a singularity of orderq≥3 att=T>0; under these assumptions, the Cauchy problem is well-posed in all Gevrey classes of indexs<q/(q−1).     

Higher integrability for minimizers of variational integrals with nonstandard growth

We consider a (possibly) vector-valued function u: Ω→R ^N, Ω⊂R ⁿ, minimizing the integral , whereD _iu=∂u/∂x _i, or some more general functional retaining the same behaviour; we prove higher integrability forDu:D ₁u,…,D_n−1u∈L^q, under suitable assumptions ona _i(x).

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Sunto Consideriamo una funzione u: Ω→R ^N, Ω⊂R ⁿ che minimizzi l'integrale , doveD _iu=∂u/∂x_i, o un funzionale con un comportamento simile; sotto opportune ipotesi sua _i(x), dimostriamo la seguente maggiore integrabilità perDu:D ₁u,…,D_n−1uεL^q.