On the removability of isolated singular points for degenerating nonlinear elliptic equations

The well known results on the removable singularity of elliptic equations are generalized to the class of degenerating nonlinear elliptic equations. A sufficient condition for the isolated singular point to be removable has been found. In the absence of degeneration, this condition coincides with already known results.     

Univalent Interpolation in Besov Spaces and Superposition into Bergman Spaces

We establish the best possible condition for point singularities to be removable for nonlinear elliptic equations in divergent form with lower order terms from the non-linear Kato classes.      

Removable sets for Hölder continuous solutions of quasilinear elliptic equations with lower order terms

We consider quasi-linear second order elliptic differential equations with lower order terms and study removable sets for Hölder continuous solutions of the equation.     

On qualitative properties of the solutions of semilinear elliptic equations

Semilinear elliptic equations of an arbitrary order 2m are considered. A theorem on the removable singularities of the solutions and a Liouville type theorem are proved.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 16, pp. 186–190, 1992.     

分布参数系统能控能观性问题的统一处理及应用

本文的主要目的是介绍一个统一的处理分布参数系统能控能观性问题的方法,并给出该方法在分布参数控制理论中的应用。为此,本文将从一类“类抛物” 偏微分算子（即没有椭圆性条件）的带权恒等式出发,给出所有已知的关于抛物型方程、双曲型方程、Schrödinger 方程和板方程的基于整体Carleman 估计的能控能观性结果。同时,基于该带权的恒等式,本文还给出它在双曲型系统的稳定性问题和在拟线性复Ginzburg-Landau 方程能控能观性等问题中的应用。     

A class of quasilinear degenerate elliptic problems

We establish the existence of solutions for a class of quasilinear degenerate elliptic equations. The equations in this class satisfy a structure condition which provides ellipticity in the interior of the domain, and degeneracy only on the boundary. Equations of transonic gas dynamics, for example, satisfy this property in the region of subsonic flow and are degenerate across the sonic surface. We prove that the solution is smooth in the interior of the domain but may exhibit singular behavior at the degenerate boundary. The maximal rate of blow-up at the degenerate boundary is bounded by the “degree of degeneracy” in the principal coefficients of the quasilinear elliptic operator. Our methods and results apply to the problems recently studied by several authors which include the unsteady transonic small disturbance equation, the pressure-gradient equations of the compressible Euler equations, and the singular quasilinear anisotropic elliptic problems, and extend to the class of equations which satisfy the structure condition, such as the shallow water equation, compressible isentropic two-dimensional Euler equations, and general two-dimensional nonlinear wave equations. Our study provides a general framework to analyze degenerate elliptic problems arising in the self-similar reduction of a broad class of two-dimensional Cauchy problems.     

A variational theory of the Hessian equation

By studying a negative gradient flow of certain Hessian functionals we establish the existence of critical points of the functionals and consequently the existence of ground states to a class of nonhomogenous Hessian equations. To achieve this we derive uniform, first‐ and second‐order a priori estimates for the elliptic and parabolic Hessian equations. Our results generalize well‐known results for semilinear elliptic equations and the Monge‐Ampère equation. © 2001 John Wiley & Sons, Inc.     

Estimates for Solutions to Model Problems Connected with Quasistationary Approximation for the Stefan Problem

Model problems for elliptic equations are considered. The time-derivative of a solution can be contained in the boundary condition or in the conjugation condition. Suich problems appear, for example, in the study of free boundary problems for elliptic equations that can be considered as quasistationary approximations to free boundary problems for parabolic equations. Estimates for solutions to the model problems are obtained. Bibliography: 11 titles.     

Existence and uniqueness of positive solutions of semilinear elliptic equations

This paper is devoted to the study of existence,uniqueness and non-degeneracy of positive solutions of semi-linear elliptic equations.A necessary and sufficient condition for the existence of positive solutions to problems is given.We prove that if the uniqueness and non-degeneracy results are valid for positive solutions of a class of semi-linear elliptic equations,then they are still valid when one perturbs the differential operator a little bit.As consequences,some uniqueness results of positive solutions under the domain perturbation are also obtained.     

Overlapping Schwarz methods for Maxwell''s equations in three dimensions

Summary. A two-level overlapping Schwarz method is considered for a Nédélec finite element approximation of 3D Maxwell's equations. For a fixed relative overlap, the condition number of the method is bounded, independently of the mesh size of the triangulation and the number of subregions. Our results are obtained with the assumption that the coarse triangulation is quasi-uniform and, for the Dirichlet problem, that the domain is convex. Our work generalizes well–known results for conforming finite elements for second order elliptic scalar equations. Numerical results for one and two-level algorithms are also presented. Received November 11, 1997 / Revised version received May 26, 1999 / Published online June 21, 2000     

A family of potentials for elliptic equations with one singular coefficient and their applications

Potentials play an important role in solving boundary value problems for elliptic equations. In the middle of the last century, a potential theory was constructed for a two-dimensional elliptic equation with one singular coefficient. In the study of potentials, the properties of the fundamental solutions of the given equation are essentially and fruitfully used. At the present time, fundamental solutions of a multidimensional elliptic equation with one degeneration line are already known. In this paper, we investigate the double- and simple-layer potentials for this kind of elliptic equations. Results from potential theory allow us to represent the solution of the boundary value problems in the form of an integral equation. By using some properties of the Gaussian hypergeometric function, we first prove limiting theorems and derive integral equations concerning the densities of the double- and simple-layer potentials. The obtained results are then applied in order to find an explicit solution of the Holmgren problem for the multidimensional singular elliptic equation in the half of the ball.     

Multiplicity results for some fourth-order elliptic equations

This paper deals with some fourth-order elliptic equations with Navier boundary condition. By using the variational method, some existence and multiplicity results are established.     

Linear second-order divergence equations in Lipschitz domains

In this paper we generalize gradient estimates in Lp spaces to Orlicz spaces for weak solutions of second-order divergence elliptic equations with small BMO coefficients in Lipschitz domains. Our results improve the known results for such equations using the harmonic analysis method.     

二阶椭圆偏微分方程的Pinching-估计

Pinching-估计是研究解的凸性的一种重要方法,主要给出了半线性二阶椭圆偏微分方程的Pinching-估计,并将其推广到一类完全非线性二阶椭圆偏微分方程.     

Jacobi elliptic function solutions for two variant Boussinesq equations

A general Jacobi elliptic function expansion method is proposed to construct abundant Jacobi elliptic function (doubly periodic) solutions for two variant Boussinesq equations. These Jacobi elliptic function solutions degenerate to the soliton wave solutions and trigonometric function solutions at a certain limit condition.     

含距离位势的拟线性椭圆方程解的存在性

变分原理证明了一类含距离位势的拟线性椭圆方程齐次Dirichlet边界条件下第一特征值问题的可解性.进一步,利用临界点理论得到了一类含距离位势的非线性椭圆方程非平凡解的存在性.     

On the asymptotic integration of nonlinear differential equations

The aim of the present paper is twofold. Firstly, the paper surveys the literature concerning a specific topic in asymptotic integration theory of ordinary differential equations: the class of second order equations with Bihari-like nonlinearity. Secondly, some general existence results are established with regard to a condition that has been found recently to be of significant use in the theory of elliptic partial differential equations.     

A note on Chudnovsky?s Fuchsian equations

We show that four exceptional Fuchsian equations, each determined by the four parabolic singularities, known as the Chudnovsky equations, are transformed into each other by algebraic transformations. We describe equivalence of these equations and their counterparts on tori. The latters are the Fuchsian equations on elliptic curves and their equivalence is characterized by transcendental transformations which are represented explicitly in terms of elliptic and theta functions.     

Determinants of elliptic hypergeometric integrals

We start from an interpretation of the BC ₂-symmetric “Type I” (elliptic Dixon) elliptic hypergeometric integral evaluation as a formula for a Casoratian of the elliptic hypergeometric equation and then generalize this construction to higher-dimensional integrals and higher-order hypergeometric functions. This allows us to prove the corresponding formulas for the elliptic beta integral and symmetry transformation in a new way, by proving that both sides satisfy the same difference equations and that these difference equations satisfy a needed Galois-theoretic condition ensuring the uniqueness of the simultaneous solution.     

Boundary subspaces for the finite element method with Lagrange multipliers

Summary The paper is concerned with the problem of constructing compatible interior and boundary subspaces for finite element methods with Lagrange multipliers to approximately solve Dirichlet problems for secondorder elliptic equations. A new stability condition relating the interior and boundary subspaces is first derived, which is easier to check in practice than the condition known so far. Using the new condition, compatible boundary subspaces are constructed for quasiuniform triangular and rectangular interior meshes in two dimensions. The stability and optimal-order convergence of the finite element methods based on the constructed subspaces are proved.This work was supported by the Finnish National Research Council for Technical Sciences and by the Finnish-American ASLA-Fulbright Foundation