Limit Behaviour of Solutions to a Class of Equivalued Surface Boundary Value Problems for Parabolic Equations

In this paper, we discuss the limit behaviour of solutions for a class of equivalued surface boundary value problems for parabolic equations. When the equivalued surface boundary \overline{\Gamma}^\varepsilon_1 shrinks to a fixed point on boundary \Gamma_1, only homogeneous Neumann boundary conditions or Neumann boundary conditions with Dirac function appear on \Gamma_1.     

Insensitizing controls for the parabolic equation with equivalued surface boundary conditions

This paper is devoted to the study of the existence of insensitizing controls for the parabolic equation with equivalued surface boundary conditions. The insensitizing problem consists in finding a control function such that some energy functional of the equation is locally insensitive to a perturbation of the initial data. As usual, this problem can be reduced to a partially null controllability problem for a cascade system of two parabolic equations with equivalued surface boundary conditions. Compared the problems with usual boundary conditions, in the present case we need to derive a new global Carleman estimate, for which, in particular one needs to construct a new weight function to match the equivalued surface boundary conditions.     

椭圆问题的非线性边界条件均匀化

在本文我们讨论了在等值面边值问题中的非线性边界条件的均匀化,推广了相应的边界条件均匀化结果,而且可应用到用于处理热敏电阻问题中的一类非线性非局部边值问题的边界条件均匀化问题。     

Limit behaviour of solutions to boundary value problem with equivalued surface

In this paper, we discuss the limit behaviour of solutions to boundary value problem with equivalued surface with m inner holes and give a different proof from that of Li Ta-tsien et al. (1998).     

ASYMPTOTIC BEHAVIOR FOR A CLASS OF ELLIPTIC EQUIVALUED SURFACE BOUNDARY VALUE PROBLEM WITH DISCONTINUOUS INTERFACE CONDITIONS

Spontaneous potential well-logging is one of the important techniques in petroleum exploitation. A spontaneous potential satisfies an elliptic equivalued surface boundary value problem with diseontinuous interface conditlons. In practice, the measuring electrode is so small that we can simplify the corresponding equivalued surface to a point. In this paper, we give a positive answer to this approximation process: when the equivalued surface shrinks to a point, the solution of the original equivalued surface boundary value problem converges to the solution of the corresponding limit boundary value problem.     

Limit behaviour of solutions to equivalued surface boundary value problem for p‐Laplacian equations

In this paper, we discuss the limit behaviour of solutions to boundary value problem with equivalued surface for p‐Laplacian equations when the equivalued surface boundary shrinks to a point in certain way. Copyright © 2000 John Wiley & Sons, Ltd.     

The Comparison of Asymptotio Behavior of the Solution of Equivalued Boundary Value Problems for Nonlinear and Linear Elliptic Equations

This paper shows the different asymptotic behavior of the solution of equivalued boundary value problems for nonlinear equation from the solution to linear one, while the boundary, on which the equivalued boundary value is carried, shrinks to afixed point.     

Limit behaviour of solutions to equivalued surface boundary value problem for p‐Laplacian equations: II

In this paper (which is a continuation of Part‐I), we discuss the limit behaviour of solutions to boundary value problem with equivalued surface for p‐Laplacian equations in the case of 1<p?2?1/N when the equivalued surface boundary shrinks to a point in certain way. Copyright © 2001 John Wiley & Sons, Ltd.     

Hyperbolic boundary value problem with equivalued surface on a domain with thin layer

This paper deals with a kind of hyperbolic boundary value problems with equivalued surface on a domain with thin layer. Existence and uniqueness of solutions are given, and the limit behavior of solutions is studied in this paper. Project partially supported by NSFC (No:10401009) and NCET of China (No:060275).     

Nonlinear elliptic boundary value problem with equivalued surface in a domain with thin layer

This paper deals with certain kinds of boundary value problems with equivalued surface of nonlinear elliptic equations on a domain with thin layer. We introduce the concept of renormalized solution to this problem. Existence and uniqueness of renormalized solutions are given, and the limit behaviour of solutions is studied in this paper.     

非线性椭圆方程等值面边值问题熵解的存在性和正则性

本文研究了非线性椭圆方程等值面过值问题摘解的存在性和正则性,并改进了以前的结果。     

A reliable treatment of singular Emden-Fowler initial value problems and boundary value problems

In this paper, the variational iteration method (VIM) is used to study the singular Emden-Fowler initial value problems and boundary value problems arising in physics and astrophysics. The VIM overcomes the singularity at the origin. The Lagrange multipliers for all cases of the equations are determined. The work is supported by analyzing few initial value problems and boundary value problems where the convergence of the results is emphasized.     

Solvability of non‐linear parabolic boundary value problem with equivalued surface

In this paper, by means of the method of implicit discretization in time, we obtain the existence of weak solution for a class of non‐linear parabolic boundary value problem with equivalued surface. Copyright © 1999 John Wiley & Sons, Ltd.     

Limit behaviour of the solution of an evolution boundary‐value problem involving the p‐Laplacian operator for the case of an equivalued condition on a shrinking surface

In this paper, we discuss the limit behaviour of the solution of an evolution boundary‐value problem involving the p‐Laplacian operator for the case of an equivalued condition on a shrinking surface. Copyright © 2004 John Wiley & Sons, Ltd.     

Boundary Homogenization in the Spontaneous Potential Well-Logging

Motivated by the study on the spontaneous potential well-logging, this paper deals with the homogenization of boundary conditions for a class of elliptic problems with jump interface conditions.     

Towards computability of elliptic boundary value problems in variational formulation

We present computable versions of the Fréchet–Riesz Representation Theorem and the Lax–Milgram Theorem. The classical versions of these theorems play important roles in various problems of mathematical analysis, including boundary value problems of elliptic equations. We demonstrate how their computable versions yield computable solutions of the Neumann and Dirichlet boundary value problems for a simple non-symmetric elliptic differential equation in the one-dimensional case. For the discussion of these elementary boundary value problems, we also provide a computable version of the Theorem of Schauder, which shows that the adjoint of a computably compact operator on Hilbert spaces is computably compact again.     

Boundary point lemmas and overdetermined problems

Two elliptic boundary value problems are considered: a problem of mixed type in a cylindrical domain, and a Dirichlet problem in an annular domain. Under some overdetermined conditions on the boundary gradient, symmetry results for domain and solution are proved. The method of proof involves the classical boundary point lemma by Hopf, as well as a suitable adaptation of it that works well at certain corners.     

Fredholm alternative and boundary value problems

This note presents a simple proof of A. Lasota's application of the nonlinear Fredholm alternative to the existence proofs of the boundary value problems involving ordinary differential equations. It then uses Lasota's result to get a stronger version of the theorem of Herzog and Lemmert on the Dirichlet boundary value problem for the second-order systems of ordinary differential equations.

     

Overdetermined partial boundary value problems on finite networks

In this study, we define a class of non-self-adjoint boundary value problems on finite networks associated with Schrödinger operators. The novel feature of this study is that no data are prescribed on part of the boundary, whereas both the values of the function and of its normal derivative are given on another part of the boundary. We show that overdetermined partial boundary value problems are crucial for solving inverse boundary value problems on finite networks since they provide the theoretical foundations for the recovery algorithm. We analyze the uniqueness and the existence of solution for overdetermined partial boundary value problems based on the nonsingularity of partial Dirichlet-to-Neumann maps. These maps allow us to determine the value of the solution in the part of the boundary where no data was prescribed. We also execute full conductance recovery for spider networks.     

Asymptotic behavior for differences of eigenvalues of two Sturm-Liouville problems with smooth potentials

The asymptotics for the differences of the eigenvalues of two Sturm-Liouville problems defined on [0,π] with the same boundary conditions and different smooth potentials is considered. Under the assumptions of that both problems with a suite of boundary conditions have the same one full spectrum and both potential functions and their derivatives are the same at the endpoint x=π, the asymptotic expressions associated with other boundary conditions are provided.