The Cauchy Boundary Value Problems on Closed Piecewise Smooth Manifolds in <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Suppose that D is a bounded domain with a piecewise C^1 smooth boundary in C^n. Let ψ∈C^1 α(δD). By using the Hadamard principal value of the higher order singular integral and solid angle coefficient method of points on the boundary, we give the Plemelj formula of the higher order singular integral with the Boehner-Martinelli kernel, which has integral density ψ. Moreover, by means of the Plemelj formula and methods of complex partial differential equations, we discuss the corresponding Cauehy boundary value problem with the Boehner-Martinelli kernel on a closed piecewise smooth manifold and obtain its unique branch complex harmonic solution.     

Analysis of search-extension method for finding multiple solutions of nonlinear problem

For numerical computations of multiple solutions of the nonlinear elliptic problemΔu f(u)=0 inΩ, u=0 onΓ, a search-extension method (SEM) was proposed and systematically studied by the authors. This paper shall complete its theoretical analysis. It is assumed that the nonlinearity is non-convex and its solution is isolated, under some conditions the corresponding linearized problem has a unique solution. By use of the compactness of the solution family and the contradiction argument, in general conditions, the high order regularity of the solution u∈H~(1 α),α>0 is proved. Assume that some initial value searched by suitably many eigenbases is already fallen into the neighborhood of the isolated solution, then the optimal error estimates of its nonlinear finite element approximation are shown by the duality argument and continuation method.     

On Doubling Properties and the Unique Continuation at the Boundary in Dini Domains

Let u be a solution to second order elliptic equations in Dini domains, a direct and elementary proof of the doubling property for u ² over balls centered at points in the domain is presented. Moreover, under the zero Dirichlet boundary condition, the unique continuation at the boundary for Dini domains has been proved.     

A Decomposition Theorem for the Solutions to the Interface Problems of Quasi-Linear Elliptic Equations

Interface problems for second order quasi-linear elliptic partial differential equations in a two-dimensional space are studied. We prove that each weak solution can be decomposed into two parts near singular points, one of which is a finite sum of functions of the form cr^a log^m rφ(θ), where the coefficients c depend on the H^1-norm of the solution, the C^(0,δ) -norm of the solution, and the equation only; and the other one of which is a regular one, the norm of which is also estimated.     

Maximum Principles for a Class of Nonlinear Elliptic Boundary Value Problems

The Hopf＇s maximum principles are utilized to obtain maximum principles for functions defined on solutions of nonlinear elliptic equations in divergence form （g（u）u,i）,i ＋f（x,u,q）=0（q=｜△↓u｜^2）, subject The principles derived may be used to deduce bounds on the gradient q.     

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).     

‖A −1‖∞ and equidiagonal-dominance

In this paper, the matrix of equidiagonal-dominance is defined and several theorems about ‖A ⁻¹‖_∞ and its evaluation are established. Many interesting numerical examples are given. This work is supported by the National Natural Science Foundation of China and the Science Foundation of Academy of Engineering Physics of China     

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     

The Camassa–Holm Equation on the Half-Line: a Riemann–Hilbert Approach

We consider the initial-boundary value (IBV) problem for the Camassa–Holm (CH) equation u _t −u _txx +2u _x +3uu _x =2u _x u _xx +uu _xxx on the half-line x≥0. In this article, we aim to provide a characterization of the solution of the IBV problem in terms of the solution of a matrix Riemann–Hilbert (RH) factorization problem in the complex plane of the spectral parameter. The data of this RH problem are determined in terms of spectral functions associated to initial and boundary values of the solution. The construction requires more boundary data than those needed for a well-posed IBV problem. Their dependence is expressed in terms of an algebraic relation to be satisfied by the spectral functions. This RH formulation gives us the long-time asymptotics of a solution of the CH-equation. Dedicated to Gennadi Henkin in great admiration.     

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...     

A priori and universal estimates for global solutions of superlinear degenerate parabolic equations

We prove an a priori estimate and a universal bound for any global solution of the nonlinear degenerate reaction-diffusion equation u _t =Δu ^m +u ^p in a bounded domain with zero Dirichlet boundary conditions. Received: October 1, 2001?Published online: July 9, 2002     

Asymptotic Behaviors of the Solutions to Scalar Viscous Conservation Laws on Bounded Interval

Abstract This paper concerns the asymptotic behaviors of the solutions to the initial-boundary value prob-lem for scalar viscous conservations laws u_t+f(u)_x=u_(xx) on[0,1],with the boundary condition u(0,t) =u_,u(1,t)=u_+ and the initial data u(x,0)=u_0(x,0)=u_0(x),where u_≠u_+ and f is a given function satisfyingf'(u)>0 for u under consideration.By means of energy estimates method and under some more regular condi-tions on the initial data,both the global existence and the asymptotic behavior are obtained.When u_u_+, which corresponds to shock waves in inviscid conservation laws, it is established for weak shockwaves,which means that │u_-u_+│is small.Moreover,exponential decay rates are both given.     

Solutions and Multiple Solutions for p（x）-Laplacian Equations with Nonlinear Boundary Condition

The authors study the p（x）-Laplacian equations with nonlinear boundary condition. By using the variational method, under appropriate assumptions on the perturbation terms f1 （x, u）, f2（x, u） and h1（x）, h2（x）, such that the associated functional satisfies the ＂mountain pass lemma＂ and ＂fountain theorem＂ respectively, the existence and multiplicity of solutions are obtained. The discussion is based on the theory of variable exponent Lebesgue and Sobolev spaces.     

Remarks on bifurcations ofu + μu − u k=0 (4≤kεZ +)

In this paper, we investigate the bifurcations of one class of steady-state reaction-diffusion equations of the formu″ + μu − u ^k=0, subjectu(0)=u(π)=0, where μ is a parameter, 4≤kεZ ⁺. Using the singularity theory based on the Liapunov-Schmidt reduction, some satisfactory results are obtained. This work is supported by the National Natural Science Foundation of China (No.19971057) and the Youth Science Foundation of Shanghai Municipal Commission of Education (No.99QA66).     

An elliptic continuation result for harmonic functions in two dimensions

Let u be harmonic in a simply connected domainG ⊂ ℝ² and letK be a compact subset of G. In this note, it is proved there exists an “elliptic continuation” of u, namely there exist a smooth functionu ₁ and a second order uniformly elliptic operatorL with smooth coefficients in ℝ², satisfying:u ₁=u inK, Lu ₁=0 in ℝ². A similar continuation theorem, with u itself a solution to an elliptic second order equation inG, is also proved.     

Interval Valued Intuitionistic （S, T）-fuzzy Hv-submodules

On the basis of the concept of the interval valued intuitionistic fuzzy sets introduced by K. Atanassov, the notion of interval valued intuitionistic fuzzy Hv-submodules of an Hv-module with respect to a t-norm T and an s-norm S is given and the characteristic properties are described. The homomorphic image and the inverse image are investigated. In particular, the connections between interval valued intuitionistic （S, T）-fuzzy Hv-submodules and interval valued intuitionistic （S, T）-fuzzy submodules are discussed.     

Existence and nonexistence of global solutions for nonlinear evolution equation of fourth order

Abstract. The existence and uniqueness of classical global solutions and the nonexistence of global solutions to the first boundary value problem and the second boundary value problem for the equation ua-a1uxx-a2uxxl-a3uxxu=ψ(ux)x are proved     

On the Cauchy Problem of Evolution P-Laplacian Equations with Strongly Non-linear Sources When 1 < p < 2

We study the solvability of the Cauchy problem (1.1)–(1.2) for the largest possible class of initial values, for which (1.1)–(1.2) has a local solution. Moreover, we also study the critical case related to the initial value u ₀, for 1 < p < ∞. Project supported by the National Natural Science Foundation of China (19971070)     

Definition of Measure-theoretic Pressure Using Spanning Sets

We introduce a new definition of measure–theoretic pressure for ergodic measures of continuous maps on a compact metric space. This definition is similar to those of topological pressure involving spanning sets. As an application, for C ^{1+ α} (α > 0) diffeomorphisms of a compact manifold, we study the relationship between the measure–theoretic pressure and the periodic points. Project Supported by National Natural Science Foundation of China     

Superlinear／Quadratic One-step Smoothing Newton Methodf or P0-NCP

We propose a one-step smoothing Newton method for solving the non-linear complementarity problem with P0-function (P0-NCP) based on the smoothing symmetric perturbed Fisher function(for short, denoted as the SSPF-function). The proposed algorithm has to solve only one linear system of equations and performs only one line search per iteration. Without requiring any strict complementarity assumption at the P0-NCP solution, we show that the proposed algorithm converges globally and superlinearly under mild conditions. Furthermore, the algorithm has local quadratic convergence under suitable conditions. The main feature of our global convergence results is that we do not assume a priori the existence of an accumulation point. Compared to the previous literatures, our algorithm has stronger convergence results under weaker conditions.