A microscopic convexity principle for nonlinear partial differential equations

We study microscopic convexity property of fully nonlinear elliptic and parabolic partial differential equations. Under certain general structure condition, we establish that the rank of Hessian ∇ ² u is of constant rank for any convex solution u of equation F(∇ ² u,∇ u,u,x)=0. The similar result is also proved for parabolic equations. Some of geometric applications are also discussed. Research of the first author was supported in part by NSFC No.10671144 and National Basic Research Program of China (2007CB814903). Research of the second author was supported in part by an NSERC Discovery Grant.     

A solution of the finite-dimensional homogeneous Banach space problem

It is proved that if a finite-dimensional Banach space X has the property that all its αn-dimensional subspaces are K-isomorphic, for some 0 < α < 1 and K ≥ 1, then X is f(α, K)-isomorphic to a Hilbert space, where f(α, K) is cK^3/2, if 0 < α < 2/3 and cK², if 2/3 < α < 1, and where c = c(α) depends on α only. Part of the work on this paper was done while this author held a visiting position at Odense University sponsored by the Odense University and Danish Natural Sciences Council Grant No. 11-7639. Supported in part by National Sciences and Engineering Research Council Grant.     

Incompleteness and minimality of complex exponential system

A necessary and sufficient condition is obtained for the incompleteness of a complex exponential system E(A,M)in C_α,where C_αis the weighted Banach space consisting of all complex continuous functions f on the real axis R with f(t)exp(-α(t))vanishing at infinity,in the uniform norm‖f‖_α=sup{|f(t)e~(-α(t))|:t∈R}with respect to the weightα(t).If the incompleteness holds, then the complex exponential system E(?)is minimal and each function in the closure of the linear span of complex exponential system E(?)can be extended to an entire function represented by a Taylor-Dirichlet series.     

The weak density of properly d-r.e. branching degree in the d-r.e. degrees

It is proved that for any r.e. degreesv <u, there exists a properly d-r.e. degreec such thatc is a branching degree andv <c <u. Project supported by the National Natural Science Foundation of China (Grant No. 19771045) and National 863 Hi-Tech R&D Programme.     

Global Existence and Nonexistence for a Strongly Coupled Parabolic System with Nonlinear Boundary Conditions

This paper deals with the strongly coupled parabolic system ut = v^m△u, vt = u^n△v, （x, t） ∈Ω × （0,T） subject to nonlinear boundary conditions 偏du/偏dη = u^αv^p, 偏du/偏dη= u^qv^β, （x, t） ∈ 偏dΩ × （0, T）, where Ω 包含 RN is a bounded domain, m, n are positive constants and α,β, p, q are nonnegative constants. Global existence and nonexistence of the positive solution of the above problem are studied and a new criterion is established. It is proved that the positive solution of the above problem exists globally if and only if α 〈 1,β 〈 1 and （m ＋p）（n ＋ q） ≤ （1 - α）（1 -β）.     

Approximation bounds for quadratic maximization and max-cut problems with semidefinite programming relaxation

In this paper,we consider a class of quadratic maximization problems.For a subclass of the problems,we show that the SDP relaxation approach yields an approximation solution with the ratio is dependent on the data of the problem with α being a uniform lower bound.In light of this new bound,we show that the actual worst-case performance ratio of the SDP relaxation approach (with the triangle inequalities added) is at least α δd if every weight is strictly positive,where δd ＞ 0 is a constant depending on the problem dimension and data.     

Analysis of a Free Boundary Problem Modeling Tumor Growth

In this paper, we study a free boundary problem arising from the modeling of tumor growth. The problem comprises two unknown functions： R = R（t）, the radius of the tumor, and u = u（r, t）, the concentration of nutrient in the tumor. The function u satisfies a nonlinear reaction diffusion equation in the region 0 〈 r 〈 R（t）, t 〉 0, and the function R satisfies a nonlinear integrodifferential equation containing u. Under some general conditions, we establish global existence of transient solutions, unique existence of a stationary solution, and convergence of transient solutions toward the stationary solution as t →∞.     

Scale limit of a variational inequality modeling diffusive flux in a domain with small holes and strong adsorption in case of a critical scaling

In this paper we study the asymptotic behavior of solutions u _ɛ of the elliptic variational inequality for the Laplace operator in domains periodically perforated by balls with radius of size C ₀ɛ_α, C ₀ > 0, α = n/n−2, and distributed with period ɛ. On the boundary of balls, we have the following nonlinear restrictions u _ɛ ≥ 0, ∂_ν u _ɛ ≥ −ɛ^−ασ(x, u _ɛ), u _ɛ(∂_ν u _ɛ + ɛ^−ασ(x, u _ɛ)) = 0. The weak convergence of the solutions u _ɛ to the solution of an effective variational equality is proved. In this case, the effective equation contains a nonlinear term which has to be determined as solution of a functional equation. Furthermore, a corrector result with respect to the energy norm is given.     

Regularity of viscosity solutions near KAM torus

In this paper, we give weak regularity theorems on P of u~ε(x, P), where u~ε(x, P)is the viscosity solution of the cell problem H_ε(P D_xu~ε, x)=H_ε(P).     

The isometric extension of “into” mappings on unit spheres of AL -spaces

In this paper, we show that if V ₀ is an isometric mapping from the unit sphere of an AL-space onto the unit sphere of a Banach space E, then V ₀ can be extended to a linear isometry defined on the whole space. This work was supported by the Research Foundation for Doctor Programme (Grant No. 20060055010) and the National Natural Science Foundation of China (Grant No. 10571090)     

Four types of bounded wave solutions of CH-γ equation

Recently, many authors have studied the following CH-γ equationut c0ux 3uux - α2(uxxt uuxxx 2uxuxx) γuxxx =0,where α2, c0 and γ are paramters. Its bounded wave solutions have been investigated mainly for the case α2 ＞ 0. For the case α2 ＜ 0, the existence of three bounded waves (regular solitary waves,compactons, periodic peakons) was pointed out by Dullin et al. But the proof has not been given.In this paper, not only the existence of four types of bounded waves periodic waves, compacton-like waves, kink-like waves, regular solitary waves, is shown, but also their explicit expressions or implicit expressions are given for the case α2 ＜ 0. Some planar graphs of the bounded wave solutions and their numerical simulations are given to show the correctness of our results.     

An integral estimate of Bessel function and its application

In this paper the authors give a new integral estimate of the Bessel function,which is an extension of Calder(?)n-Zygmund's result.As an application of this result,we prove that the parameterized Marcinkiewicz integralμ_Ω~p with variable kernels is of type (2,2),where the kernel functionΩdoes not have any smoothness on the unit sphere in R~n.     

A mathematical analysis for a diffusive epidemic model with criss-cross dynamics

Abstract. In this paper, an initial boundary value problem with homogeneous Neumann bound-ary condition is studied for a reaction diffusion system which models the spread of infectious dis-eases within two population groups by means of serf and criss-cross infection mechanism, Exis-tence, uniqueness and houndedness of the nonnegative global solution     

Superconvergence of tricubic block finite elements

In this paper, we first introduce interpolation operator of projection type in three dimen- sions, from which we derive weak estimates for tricubic block finite elements. Then using the estimate for the W 2, 1-seminorm of the discrete derivative Green’s function and the weak estimates, we show that the tricubic block finite element solution uh and the tricubic interpolant of projection type Πh3u have superclose gradient in the pointwise sense of the L∞-norm. Finally, this supercloseness is applied to superc...     

Precise asymptotic formulas for variational eigencurves of semilinear two-parameter elliptic eigenvalue problems

We consider the two-parameter nonlinear eigenvalue problem?−Δu = μu − λ(u + u ^p + f(u)), u > 0 in Ω, u = 0 on ∂Ω,?where p>1 is a constant and μ,λ>0 are parameters. We establish the asymptotic formulas for the variational eigencurves λ=λ(μ,α) as μ→∞, where α>0 is a normalizing parameter. We emphasize that the critical case from a viewpoint of the two-term asymptotics of the eigencurve is p=3. Moreover, it is shown that p=5/3 is also a critical exponent from a view point of the three-term asymptotics when Ω is a ball or an annulus. This sort of criticality for two-parameter problems seems to be new. Received: February 9, 2002; in final form: April 3, 2002?Published online: April 14, 2003     

Oscillation criteria for second order nonlinear differential equations

Summary Various sufficient conditions are obtained which guarantee that all continuable solutions of (1.1) y″+q(t)y′+p(t)f(y)=0 are oscillatory. No explicit sign assumptions are made on p(t) although certain integral conditions are assumed to hold with regard to f(y), p(t) and q(t). Examples are given of the form p(t) = λ/t^μ + (βsint)/t^α, λ, β, μ, α>0. This research was supported by NRC Grant A-7673 and CMC Summer Research Grant. Entrata in Redazione il 4 settembre 1971.     

Multiple Solutions and Its Morse Index for One-Dimensional Nonlinear Problem

The multiple solutions for one-dimensional cubic nonlinear problem u" u~3=0,u(0)=u(π)=0are computed,on the basis of the eigenpairs of-φ"_k=λ_(kφk),k=1,2,3....There exist two nonzero solutions±u_k corresponding to each k,and their Morse index MI(k) for 1(?)k(?)20 is to be exactly determined.It isshown by the numerical results that MI(k)(?)k.     

Optimal Three Cylinder Inequality at the Boundary for Solutions to Parabolic Equations and Unique Continuation Properties

Let Γ be a portion of a C ^1,α boundary of an n-dimensional domain D. Let u be a solution to a second order parabolic equation in D × (–T, T) and assume that u = 0 on Γ × (–T, T), 0 ∈ Γ. We prove that u satis.es a three cylinder inequality near Γ × (–T, T) . As a consequence of the previous result we prove that if u (x, t) = O (|x|^k) for every t ∈ (–T, T) and every k ∈ ℕ, then u is identically equal to zero. This work is partially supported by MURST, Grant No. MM01111258     

The Feynman-Stratonovich semigroup and stratonovich integral expansions in nonlinear filtering

Representations for the solution of the Zakai equation in terms of multiple Stratonovich integrals are derived. A new semigroup (the Feynman-Stratonovich semigroup) associated with the Zakai equation is introduced and using the relationship between multiple Stratonovich integrals and iterated Stratonovich integrals, a representation for the unnormalized conditional density,u(t,x), solely in terms of the initial density and the semigroup, is obtained. In addition, a Fourier seriestype representation foru(t,x) is given, where the coefficients in this representation uniquely solve an infinite system of partial differential equations. This representation is then used to obtain approximations foru(t,x). An explicit error bound for this approximation, which is of the same order as for the case of multiple Wiener integral representations, is obtained. Research supported by the National Science Foundation and the Air Force Office of Scientific Research Grant No. F49620 92 J 0154 and the Army Research Office Grant No. DAAL03-92-G0008.     

Decompositions of the Hardy space Hz^1（Ω）

We give a decomposition of the Hardy space Hz^1（Ω） into ＂div-curl＂ quantities for Lipschitz domains in R^n. We also prove a decomposition of Hz^1（Ω） into Jacobians det Du, u ∈ W0^1,2 （Ω,R^2） for Ω in R^2. This partially answers a well-known open problem.