ASYMPTOTIC SOLUTION OF SINGULAR PERTURBATION PROBLEMS FOR THE FOURTH-ORDER ELLIPTIC DIFFERENTIAL EQUATIONS

In this paper we consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the energy estimates of the solutionand its derivatives and construct the formal asymptotic solution by Lyuternik- Vishik 's method. Finally, by means of the energy estimates we obtain the bound of the remainder of the asymptotic solution.

Symmetry Reductions of Cauchy Problems for Fourth-Order Quasi-Linear Parabolic Equations

This paper is devoted to studying symmetry reduction of Cauchy problems for the fourth-order quasi-linear parabolic equations that admit certain generalized conditional symmetries （GCSs）. Complete group classification results are presented, and some examples are given to show the main reduction procedure.

A PRIORI ESTIMATE FOR MAXIMUM MODULUS OF GENERALIZED SOLUTIONS OF QUASI-LINEAR ELLIPTIC EQUATIONS

Let G he a hounded domain in E Consider the following quasi-linear elliptic equationAlthough the houndedness of generalized solutions of the equation is proved for very general structural conditions, it does not supply a priori estimate for maximum modulus of solutions. In this paper an estimate to the maximum modulus is made firstly for a special case of quasi-linear elliptic equations, i.e. the A and B satisfy the following structural conditions

Existence of Infinitely Many High Energy Solutions for a Fourth-Order Kirchhoff Type Elliptic Equation in R³

In this paper, we consider the following fourth-order equation of Kirchhoff type where a, b > 0 are constants, 3 < p < 5, V ∈ C (R3, R);Δ2: = Δ (Δ) is the biharmonic operator. By using Symmetric Mountain Pass Theorem and variational methods, we prove that the above equation admits infinitely many high energy solutions under some sufficient assumptions on V (x). We make some assumptions on the potential V (x) to solve the difficulty of lack of compactness of the Sobolev embedding. Our results improve some related results in the literature.

Superconvergence Analysis of C^(m)Finite Element Methods for Fourth-Order Elliptic Equations I:One Dimensional Case

In this paper,we study three families of C^(m)(m=0,1,2)finite element methods for one dimensional fourth-order equations.They include C^(0)and C1 Galerkin methods and a C^(2)-C^(0)Petrov-Galerkin method.Existence,uniqueness and optimal error estimates of the numerical solution are established.A unified approach is proposed to study the superconvergence property of these methods.We prove that,for kth-order elements,the C^(0)and C1 finite element solutions and their derivative are superconvergent with rate h2k−2(k≥3)at all mesh nodes;while the solution of the C^(2)-C^(0)Petrov-Galerkin method and its first-and second-order derivatives are superconvergent with rate h^(2k−4)(k≥5)at all mesh nodes.Furthermore,interior superconvergence points for the l-th(0≤l≤m+1)derivate approximations are also discovered,which are identified as roots of special Jacobi polynomials,Lobatto points,and Gauss points.As a by-product,we prove that the C^(m)finite element solution is superconvergent towards a particular Jacobi projection of the exact solution in the Hl(0≤l≤m+1)norms.All theoretical findings are confirmed by numerical experiments.

Infinitely Many Solutions for a Class of Quasi-linear Elliptic Problem

In this paper,we study the following quasi-linear elliptic equation:■where Ω?R^(N) is a bounded domain,λ>0 is a parameter.The function ψ(|t|)t is the subcritical term,and φ(|t|)t is the critical Orlicz-Sobolev growth term with respect to φ.Under appropriate conditions on φ,ψ and φ,we prove the existence of infinitely many weak solutions for quasi-linear elliptic equation,for λ∈(0,λ_(0)),where λ_(0)> 0 is a fixed constant.

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~α log^m r(?)(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.

Existence Theorem for a Class of Nonlinear Fourth-order Schrodinger-Kirchhoff-Type Equations

This paper is concerned with the existence of nontrivial solutions for the following fourth-order equations of Kirchhoff type {Δ^2u-（a＋b∫R^N｜▽u｜^2dx）Δu＋λv（x）u=f（x,u）,x∈R^N,u∈H^2（R^N）,where a,b are positive constants, λ≥ 1 is a parameter, and the nonlinearity f is either superlinear or sublinear at infinity in u. With the help of the variational methods, we obtain the existence and multiplicity results in the working spaces.

Existence and uniqueness for variational problem from progressive lens design

We study a functional modelling the progressive lens design,which is a combination of Willmore functional and total Gauss curvature.First,we prove the existence for the minimizers of this class of functionals among the class of revolution surfaces rotated by the curves y=f(x)about the x-axis.Then,choosing such a minimiser as background surfaces to approximate the functional by a quadratic functional,we prove the existence and uniqueness of the solution to the Euler-Lagrange equation for the quadratic functionals.Our results not only provide a strictly mathematical proof for numerical methods,but also give a more reasonable and more extensive choice for the background surfaces.