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On a boundary value problem for improperly elliptic equations in multiply connected domain

Authors:

Armenak Babayan

NATIONAL POLYTECHNIC UNIVERSITY OF ARMENIA

The paper considers a Dirichlet-type boundary value problem for the elliptic equation ∂ m+n u ∂z ¯ m ∂z n =0,∂u ∂z ¯≡1 2∂u ∂x+i∂u ∂y,∂u ∂z≡1 2∂u ∂x-i∂u ∂y in a multiply connected domain. The problem is reduced to a Dirichlet problem for the n-harmonic equation. An existence theorem is proved.

A Riemann-Hilbert type problem for second order non-regular elliptic equation in weighted spaces

Article

Apr 2010

The paper investigates a Riemann-Hilbert type problem for second order nonregular elliptic equation in weighted spaces. It is established that the number of linearly independent solutions of the homogeneous problem and the number of conditions on the boundary functions depend not only on the order of singularity of the weight function and coefficient indices of the considered problem, but also on the behavior of these functions at singular points.

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The Stabilization Theorems For Parabolic Systems With Analytic Nonlinearity And Ljapunov Functional

July 2004 · Boletim da Sociedade Paranaense de Matematica

Vishnevskii Mikhail

Let u(x,t) denote the solution of a boundary value problem forparabolic system . We say the solution u(x; t) stabilizes as t tends to plus infinity (minus infinity) if the set of all partial limits as t tends to plus infinity (menus infinity) of the solution u(x,t) consists of a single stationary solution. In this communication we consider the nonlinear parabolic system with analytic dependence ... [Show full abstract] ofu(x,t) and gradient of u(x,t) on the space variable and with Liapunov functional.It is shown that any solution of the problem uniformly bounded for positive t (or fornegative t) stabilizes. In particular the global attractor of this kind of problem con-sists of stationary solution and connected orbits. The °ow on global attractor is agradient-like °ow. The similar result obtained also for the Canh - Hilliard equation.

Chapter

Concentrating Solutions for a Two-dimensional Elliptic Problem with Large Exponent in Nonlinearity

January 2007

Angela Pistoia

We study the existence of positive and sign-changing solutions to the boundary value problem − Δ u = |u| p−1 u in a bounded smooth domain Ω in ℝ2, with homogeneous Dirichlet boundary condition, when p is a large exponent. We find topological conditions on Ω which ensure the existence of a positive solution concentrating at exactly m points as p→∞. In particular, for a non-simply connected domain ... [Show full abstract] such a solution exists for any given m ≥ 1. Moreover, for p large enough, we prove the existence of two pairs of solutions which change sign exactly once and whose nodal lines intersect the boundary of Ω.

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September 1990 · Computing

Panayot S. Vassilevski

A unified approach of deriving band approximate inverses of band symmetric positive definite matrices is considered. Such band approximations to the inverses of successive Schur complements are required throughout incomplete block factorizations of block-tridiagonal matrices. Such block-tridiagonal matrices arise, for example, in finite element solution of second order elliptic differential ... [Show full abstract] equations. A sharp decay rate estimate for inverses of blocktridiagonal symmetric positive definite matrices is given in addition. Numerical tests on a number of model elliptic boundary value problems are presented comparing thus derived preconditioning matrices.Es wird ein einheitlicher Zugang fr die Ableitung von approximierten Inversen von symmetrischen, positiv definiten Bandmatrizen beschrieben. Solche Band-Approximationen fr die Inversen von aufeinanderfolgenden Schur-Komplementen werden bentigt bei der unvollstndigen Blockfaktorisierung von blocktridiagonalen Matrizen, die bei der Finite-Elemente-Lsung von elliptischen Differentialgleichungen zweiter Ordnung entstehen.Eine scharfe Abschtzung fr die Abklingrate der Blcke der Inversen von blocktridiagonalen symmetrischen positiv definiten Matrizen wird zustzlich gegeben.Numerische Tests fr einige elliptische Modelldifferentialgleichungen werden prsentiert, um die abgeleiteten Prkonditionierungsmatrizen miteinander zu vergleichen.

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The Kernel Bundle of a Holomorphic Fredholm Family

January 2013 · Communications in Partial Differential Equations

Let $\Y$ be a smooth connected manifold, $\Sigma\subset\C$ an open set and $(\sigma,y)\to\scrP_y(\sigma)$ a family of unbounded Fredholm operators $D\subset H_1\to H_2$ of index 0 depending smoothly on $(y,\sigma)\in \Y\times \Sigma$ and holomorphically on $\sigma$. We show how to associate to $\scrP$, under mild hypotheses, a smooth vector bundle $\kerb\to\Y$ whose fiber over a given $y\in \Y$ ... [Show full abstract] consists of classes, modulo holomorphic elements, of meromorphic elements $\phi$ with $\scrP_y\phi$ holomorphic. As applications we give two examples relevant in the general theory of boundary value problems for elliptic wedge operators.

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A Discontinuous Petrov--Galerkin Method with Lagrangian Multipliers for Second Order Elliptic Proble...

January 2005 · SIAM Journal on Numerical Analysis

We present a Discontinuous Petrov-Galerkin method (DPG) for nite element discretization scheme of second order elliptic boundary value problems. The novel approach emanates from a one-element weak for- mulation of the differential problem (that is typical of Discontinuous Galerkin methods (DG)) which is based on introducing variables dened in the interior and on the boundary of the element. The ... [Show full abstract] interface variables are suitable Lagrangian multipliers that enforce interelement continuity of the solution and of its normal derivative, thus provid- ing the proper connection between neighboring elements. The internal variables can be eliminated in favor of the interface variables using static condensation to end up with a system of reduced size having as unknowns the La- grangian multipliers. A stability and convergence analysis of the novel formulation is carried out and its connection with mixed-hybrid and DG methods is explored. Numerical tests on several benchmark problems are included to validate the convergence performance and the ux-conserv ation properties of the DPG method.

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Viscosity‐splitting scheme for the Navier‐Stokes equations

June 2005 · Numerical Methods for Partial Differential Equations

Ying Lung‐an

A viscosity-splitting scheme for the initial boundary value problems of the Navier-Stokes equations is considered. In the scheme, the Stokes equation is solved in conjunction with a nonhomogeneous boundary condition which connects the tangent flow with a no-slip flow. Convergence is proved.

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Numerical determination of time transfer in general relativity

December 2007 · General Relativity and Gravitation

Ángel San Miguel Hernández

This paper determines the travel time of a light ray connecting two points in the space-time solving numerically a two-point boundary value problem by means of the shooting method. For the resolution of this problem multiple precision floating-point arithmetic is required. We have studied distinct implementations of the shooting method attending to the way in which the Jacobian appearing at each ... [Show full abstract] iteration is approximated. We have shown that by using Magnus expansions to solve the differential equation for the Jacobian, one obtains a better efficiency of the shooting method. Also, a comparison between the numerical results obtained through the application of the shooting method and those derived by Le Poncin-Lafitte etal. (Class Quantum Grav, 21:4463-4483, 2004) from perturbative expansions of the world function is made.

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Impedance boundary condition and extensions of the Maxwell operator

January 2005

The von Neumann theory of selfadjoint extension of symmetric operators used here for constructing a nonselfadjoint in general closed extension, corresponding to impedance boundary value problem for the Maxwell equation. Deficiency spaces of corresponding operator A<sup>S </sup> <sub>0</sub> are presented as a space Lfr of pairs of linear differential forms distributions on the boundary surface S. ... [Show full abstract] So the problem of constructing the operator connecting deficiency spaces and defining the closed extensions is reduced to the inversion problem for a pseudodifferential operator acting in the space Lfr and depending on the impedance operator on a space of linear differential forms distributions on the boundary surface S

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Critical semilinear equations on the Heisenberg group: The effect of the topology of the domain

October 2001 · Nonlinear Analysis

The effect of domain topology on critical semilinear equations on the Heisenberg group was considered. A critical semilinear boundary value problem was considered in a connected bounded domain. It was proved that the problem had a solution if at least a nontrivial homology group existed for the smooth connected bounded domain of the Heisenberg group.

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On integral-algebraic systems of equations with Cauchy operators

January 2006 · Doklady Mathematics

P. A. Krutitskii

The study results of the general properties of mixed boundary value problems for the Laplace equation on exterior two-dimensional multiply connected domains, are presented. The exterior two-dimensional multiply connected domains are bounded by closed and on closed curves. On the nonclosed curve, the Dirichlet condition is imposed, while on the closed curves, the von Neumann or skew derivative ... [Show full abstract] condition is applied. The mixed boundary value problem for the Laplace equation outside nonclosed curve on the plane is investigated. Integral representations in the form of potentials with densities satisfying integral-algebraic systems of equations with Cauchy operators are obtained for the solution of all the problems. The inhomogeneous system generally has no solution in he space, as the solution has singularities at the endpoint of Γ.

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Difference equations on weighted graphs

December 2003 · Journal of Convex Analysis

In discrete systems graphs setting, we mimic the variational formulation of boundary value problems. Working on with unnormalized weights rather than normalized weights, discrete Dirichlet and Neumann problems, and their probabilistic interpretations are discussed. Moreover, nonsymmetric forms, nonvariational setting, and an identification problem are also considered.

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Modified Schwarz alternating method to the solution of the Riemann--Hilbert problem on doubly connec...

May 2007 · Complex Variables and Elliptic Equations

Tatsiana Vaitsiakhovich

The article is devoted to the solution of a Riemann--Hilbert boundary value problem on a doubly connected unbounded domain by a modification of the Schwarz alternating method. The obtained solution is given by the series. Each element of this series represents the solution of the corresponding Riemann--Hilbert problem on a simply connected circular domain. The operator corresponding to the ... [Show full abstract] formula of the solution is also proposed. Convergence of the method is shown.