Behavior of the Free Boundary Near Contact Points with the Fixed Boundary for Nonlinear Elliptic Equations

The aim of this paper is to study a free boundary problem for a uniformly elliptic fully non-linear operator. Under certain assumptions we show that free and fixed boundaries meet tangentially at contact points.     

Behavior of the Free Boundary Near Contact Points with the Fixed Boundary for Nonlinear Elliptic Equations

The aim of this paper is to study a free boundary problem for a uniformly elliptic fully non-linear operator. Under certain assumptions we show that free and fixed boundaries meet tangentially at contact points.     

具临界指数和临界非线性边界条件的拟线性椭圆型方程Neumann问题

本文给出ＲＮ（Ｎ３）中有界光滑区域Ω上的拟线性椭圆型方程：－∑Ｎｉ＝１ｘｉ·｜Ｄｕ｜ｐ－２ｕｘｉ＝λ｜ｕ｜ｐ－２ｕ＋ａ（ｘ）｜ｕ｜ｐ－２ｕ＋ｆ（ｘ，ｕ），ｘ∈Ω（λ＞０，ｐ＝Ｎｐ／（Ｎ－ｐ），２ｐ＜Ｎ）在边界条件：－｜Ｄｕ｜ｐ－２Ｄνｕ｜Ω＝ψ（ｘ）｜ｕ｜ｑ－２ｕ（ｑ＝（Ｎ－１）ｐ／（Ｎ－ｐ））下的多解性结果．     

Boundary Homogenization of an Inverse Non-local Elliptic Boundary-Value Problem

In this paper, a mathematical analysis is given of an inversenon–local elliptic boundary–value problem for conductivityprospecting with a patched–electrode well-log instrument.As the electrode cells of the patched electrode system becomesmaller and smaller, the microscopic structure of the electrodesystem becomes increasingly complicated and themathematicalanalysis becomes much more difficult. Nevertheless, it is shownthat the electrode system can be homogenized macroscopically;that is, the solution of the inverse problem for the patchedelectrode system converges to the solution of the correspondinginverse problem for the original electrode system.     

Uniqueness in a Free Boundary Problem

We study overdetermined boundary conditions for positive solutions to the p Laplacian in a bounded domain D. We show these conditions imply uniqueness in certain free boundary problems.     

关于一个拟线性椭圆边值问题的研究

本文研究一个主部为N维P-Laplace微分算子的非线性偏微分方程的边值问题.方程中的非线性项关于其变量可以具有奇性和间断性．这类方程的一种典型情形是大家熟知的具负指数的Emden-Fowler方程．本文利用摄动技巧，Schauder不动点定理，经对解的性质的精细分析．获得了解的存在性和唯一性结果．     

On a Free Boundary Problem

This paper considers a discontinuous semilinear elliptic problem: \[ -\Delta u=g(u)H(u-\mu )\quad \text{in }\Omega,\qquad u=h\quad \text{on }% \partial \Omega, \] −Δu=g(u)H(u−μ) in Ω, u=h on ∂Ω, where H is the Heaviside function, μ a real parameter and Ω the unit ball in ℝ². We deal with the existence of solutions under suitable conditions on g, h, and μ. It is shown that the free boundary, i.e. the set where u=μ, is sufficiently smooth.     

带有临界指数的椭圆型方程的正解的存在性

本文主要采用变分方法来研究一类带有临界指数的椭圆型方程的正解的存在性问题.并且,在Ω领域(有界或无界)中的许多条件下,可以证明其基态解的存在性.     

Infinitely Many Solutions for an Elliptic Problem with Critical Exponent in Exterior Domain

We consider the following nonlinear problem {-△u=uN＋2/N-2,u〉0 in R^N/Ω,u（x）→0 as｜x｜→＋∞,δu/δn=0 on δΩ,where Ω belong to RN,N ≥ 4 is a smooth and bounded domain and n denotes inward normal vector of δΩ. We prove that the above problem has infinitely many solutions whose energy can be made arbitrarily large when Ω is convex seen from inside （with some symmetries）.     

Elliptic Functions With Critical Points Eventually Mapped Onto Infinity

We consider the class of elliptic functions whose critical points in the Julia set are eventually mapped onto ∞. This paper is a continuation of our previous papers, namely [11] and [12]. We study the geometry and ergodic properties of this class of elliptic functions. In particular, we obtain a lower bound on the Hausdorff dimension of the Julia set that is bigger than the estimate proved in [11]. Let h be the Hausdorff dimension of the Julia set of f. We construct an atomless h-conformal measure m and prove the existence of a (unique up to a multiplicative constant) σ-finite f-invariant measure μ equivalent to m. The measure μ is ergodic and conservative.     

On the Interaction of Boundary Singular Points in the Dirichlet Problem for an Elliptic Equation with Piecewise Constant Coefficients in a Plane Domain

Computational Mathematics and Mathematical Physics - For an elliptic equation in divergent form with a discontinuous scalar piecewise constant coefficient in an unbounded domain $$Omega subset...     

A Free Boundary Problem for Some Modified Predator-Prey Model in a Higher Dimensional Environment

Applications of Mathematics - We focus on the free boundary problems for a Leslie-Gower predator-prey model with radial symmetry in a higher dimensional environment that is initially well populated...     

Elliptic Functions With Critical Points Eventually Mapped Onto Infinity

We consider the class of elliptic functions whose critical points in the Julia set are eventually mapped onto ∞. This paper is a continuation of our previous papers, namely [11] and [12]. We study the geometry and ergodic properties of this class of elliptic functions. In particular, we obtain a lower bound on the Hausdorff dimension of the Julia set that is bigger than the estimate proved in [11]. Let h be the Hausdorff dimension of the Julia set of f. We construct an atomless h-conformal measure m and prove the existence of a (unique up to a multiplicative constant) σ-finite f-invariant measure μ equivalent to m. The measure μ is ergodic and conservative.     

A Free Boundary Problem with Curvature

Abstract

In this paper we are interested in a free boundary problem with a motion law involving the mean curvature term of the free boundary. Viscosity solutions are introduced as a notion of global-time solutions past singularities. We show the comparison principle for viscosity solutions, which yields the existence of minimal and maximal solutions for given initial data. We also prove uniqueness of the solution for several classes of initial data and discuss the possibility of nonunique solutions.     

球外部区域上一类椭圆边值问题的正径向解

研究了一类椭圆边值问题在球外部区域上正径向解的存在性,当非线性项f（u）关于u超线性或次线性增长的情形,获得了该问题正径向解的存在性.     

一个反应扩散方程的自由边界问题

该文研究了一个描述原细胞生长的反应扩散方程的自由边界问题.利用非线性分析中的线性化思想和偏微分方程的估计理论,证明了该自由边界问题局部古典解的存在唯一性.     

A Free Boundary Problem Arising in Smoulder Combustion

The smouldering combustion is modeled as a free boundary problem here. By using the Duvaut's transform, the problem is reduced to a variational inequality. Existence and uniqueness are established. Tbe properties of the free boundary are studied in various cases. The asymptotic behavior of the free boundary with respect to the parameter is rigorously proved, which confirms the result of a previous work by J. Adler and D. M. Herbert, obtained by using asymptotic expansion. Furthermore, we show that the time dependent problem will actually converge to a travelling wave solution if the boundary data converge to the corresponding travelling wave solution.     

A Time-Dependent Strange Term Arising in Homogenization of an Elliptic Problem with Rapidly Alternating Neumann and Dynamic Boundary Conditions Specified at the Domain Boundary: The Critical Case

Doklady Mathematics - A strange term arising in the homogenization of elliptic (and parabolic) equations with dynamic boundary conditions given on some boundary parts of critical size is...