Gradient Estimates for the Perfect and Insulated Conductivity Problems with Multiple Inclusions

In this paper, we study the perfect and the insulated conductivity problems with multiple inclusions imbedded in a bounded domain in ? ⁿ , n ≥ 2. For these two extreme cases of the conductivity problems, the gradients of their solutions may blow up as two inclusions approach each other. We establish the gradient estimates for the perfect conductivity problems and an upper bound of the gradients for the insulated conductivity problems in terms of the distances between any two closely spaced inclusions.     

A Gradient Estimate for Solutions of the Heat Equation II

The author obtains an estimate for the spatial gradient of solutions of the heat equation, subject to a homogeneous Neumann boundary condition, in terms of the gradient of the initial data. The proof is accomplished via the maximum principle; the main assumption is that the sufficiently smooth boundary be convex.     

Hessian型方程Neumann边值问题的梯度估计

通过构造辅助函数,利用基本对称函数的性质以及函数在极大值点的性质,得到Hessian型方程S_k(D~2u-A(x,u,Du))=B(x,u)的梯度内估计,构造不同的辅助函数,分近边、边界和内部3种情形讨论该方程Neumann边值问题,进而得到全局梯度估计.     

带梯度项的椭圆型方程边界爆破解的高阶估计

In this paper,the higher order asymptotic behaviors of boundary blow-up solutions to the equation■in bounded smooth domain■are systematically investigated for p and q.The second and third order boundary behaviours of the equation are derived.The results show the role of the mean curvature of the boundary■and its gradient in the high order asymptotic expansions of the solutions.     

Gaussian Curvature Estimates for the Convex Level Sets of Solutions for Some Nonlinear Elliptic Partial Differential Equations

We give lower bound estimates for the Gaussian curvature of convex level sets of minimal surfaces and the solutions to semilinear elliptic equations in terms of the norm of boundary gradient and the Gaussian curvature of the boundary.     

The Convexity Estimates for the Solutions of Two Elliptic Equations

In this paper, for the solutions of two elliptic equations we find the auxiliary curvature functions which attain respective minimum on the boundary. These results are the generalization of the classical ones in Makar-Limanov [17 Makar-Limanov , L.G. ( 1971 ). Solution of Dirichlet's problem for the equation Δu = ?1 on a convex region . Math. Notes Acad. Sci. USSR 9 : 52 – 53 .[Crossref] , [Google Scholar]] for the torsion equation and Acker et al. [1 Acker , A. , Payne , L.E. , Philippin , G. ( 1981 ). On the convexity of level lines of the fundamental mode in the clamped membrane problem, and the existence of convex solutions in a related free boundary problem . Z. Angew. Math. Phys. 32 : 683 – 694 .[Crossref], [Web of Science ®] , [Google Scholar]] for the first eigenfunction of the Laplacian in convex domains of dimension 2. Then we get the new proof of the specific convexity of the solutions of the above two elliptic equations. As a consequence, for the elliptic equation vΔv = ? (1 + |?v|²) in a smooth, bounded and strictly convex domain Ω in ? ⁿ with homogeneous Dirichlet boundary value condition, we also get a sharply lower bound estimate of the Gaussian curvature for the solution surface by the curvature of the boundary of the domain.     

The Pointwise Estimates of Solutions to the Cauchy Problem of a Chemotaxis Model

This paper deals with an attraction-repulsion chemotaxis model(ARC) in multi-dimensions. By Duhamel's principle, the implicit expression of the solution to(ARC)is given. With the method of Green's function, the authors obtain the pointwise estimates of solutions to the Cauchy problem(ARC) for small initial data, which yield the W s,p(1 ≤p≤∞) decay properties of solutions.     

Quantum Dot and Antidot Infrared Photodetectors: Iterative Methods for Solving the Laplace Equation in Domains with Involved Geometry

We propose iteration methods for solving the Dirichlet problem in domains with involved geometry. Such problems arise in relation to the problem of optimizing quantum dot and antidot infrared detectors. We estimate the deviation of an approximate solution from the exact solution.     

单侧障碍问题解梯度的Hlder连续性

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Stability Estimates for the Inverse Conductivity Problem for Less Regular Conductivities

A log-type stability estimate for the inverse conductivity problem in space dimension n ≥ 3, if the conductivity has C ^3/2+ε regularity is proven.     

Global Existence and Pointwise Estimates of Solutions to Generalized Benjamin-Bona-Mahony Equations in Multi Dimensions

This paper is concerned with the global existence and pointwise estimates of solutions to the generalized Benjamin-Bona-Mahony equations in all space dimensions. By using the energy method, Fourier analysis and pseudo-differential operators, the global existence and pointwise convergence rates of the solution are obtained. The decay rate is the same as that of the heat equation and one can see that the solution propagates along the characteristic line.     

Computing Approximate Solutions of the Maximum Covering Problem with GRASP

We consider the maximum covering problem, a combinatorial optimization problem that arises in many facility location problems. In this problem, a potential facility site covers a set of demand points. With each demand point, we associate a nonnegative weight. The task is to select a subset of p > 0 sites from the set of potential facility sites, such that the sum of weights of the covered demand points is maximized. We describe a greedy randomized adaptive search procedure (GRASP) for the maximum covering problem that finds good, though not necessarily optimum, placement configurations. We describe a well-known upper bound on the maximum coverage which can be computed by solving a linear program and show that on large instances, the GRASP can produce facility placements that are nearly optimal.     

Error Estimates for Barycentric Finite Volumes Combined with Nonconforming Finite Elements Applied to Nonlinear Convection-Diffusion Problems

The subject of the paper is the derivation of error estimates for the combined finite volume-finite element method used for the numerical solution of nonstationary nonlinear convection-diffusion problems. Here we analyze the combination of barycentric finite volumes associated with sides of triangulation with the piecewise linear nonconforming Crouzeix-Raviart finite elements. Under some assumptions on the regularity of the exact solution, the L ²(L ²) and L ²(H ¹) error estimates are established. At the end of the paper, some computational results are presented demonstrating the application of the method to the solution of viscous gas flow.     

Decay Estimates of Solutions to Wave Equations in Conical Sets

We consider the wave equation in an unbounded conical domain, with initial conditions and boundary conditions of Dirichlet or Neumann type. We give a uniform decay estimate of the solution in terms of weighted Sobolev norms of the initial data. The decay rate is the same as in the full space case.     

Viscosity Solutions for the Two-Phase Stefan Problem

We introduce a notion of viscosity solutions for the two-phase Stefan problem, which incorporates possible existence of a mushy region generated by the initial data. We show that a comparison principle holds between viscosity solutions, and investigate the coincidence of the viscosity solutions and the weak solutions defined via integration by parts. In particular, in the absence of initial mushy region, viscosity solution is the unique weak solution with the same boundary data.     

一类退化非线性抛物型方程整体解的衰减估计

J.L.L ions用紧致性方法证明了一类退化非线性抛物型方程初边值问题整体解的存在唯一性,但解的衰减性很少有人考虑.应用M.N akao建立的差分不等式研究了整体解的衰减估计.     

放缩,反演和障碍问题

代替移动平面法和滑动区域法，作者利用放缩和反演变换研究某些半线性椭圆型分分下等方程解的单调性和对称性，这些变换依赖某一参数，在关于数据的单调性的适当假定下，当参数近某一初始值最大值时，解和其变换的差在某一“窄区域”满足一个椭圆型不等式，利用Ｖａｒａｄｈａｎ的极值原理，即得这个差是正的并得到解的单调性。     

电报方程双周期解的极大值原理与强正性估计及应用

本文讨论非线性电报方程u_(tt)-u_(xx)+cu_t=F(t,x,u),(t,x)∈R~2时空双2π周期解的存在性。改进了Ortega与Robles-Perez关于线性电报方程双周期解的极大值原理,应用新获得的极大值原理,推广了相应的上下解定理,并且加强了极大值原理的结论,建立了线性方程解的强正性估计,利用这个强正性估计及锥上的不动点定理获得了超线性电报方程及奇异电报方程正双周期解的存在性。