Energy Functionals and Complex Monge–Ampère Equations

2010 ◽  
Vol 9 (3) ◽  
pp. 463-476 ◽  
Author(s):  
Zuoliang Hou ◽  
Qi Li
Keyword(s):  
Boundary Conditions ◽  
Bounded Domains ◽  
Energy Functionals ◽  
Convergence Of Solutions ◽  
Monge Ampere

AbstractWe introduce certain energy functionals to complex Monge–Ampère equations over bounded domains with inhomogeneous boundary conditions, and use these functionals to show the convergence of solutions to certain parabolic Monge–Ampère equations.

Natural Frequencies of Parallelogrammic Plates Using Characteristic Orthogonal Polynomials in Rayleigh-Ritz Method

1992 ◽  
Author(s):  
L. T. Lee ◽  
W. F. Pon
Keyword(s):  
Boundary Conditions ◽  
Coordinate System ◽  
Orthogonal Polynomials ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Energy Functional ◽  
Comprehensive Treatment ◽  
Energy Functionals ◽  
Simply Supported ◽  
Cartesian Coordinate

Abstract Natural frequencies of parallelogrammic plates are obtained by employing a set of beam characteristic orthogonal polynomials in the Rayleigh-Ritz method. The orthogonal polynomials are generalted by using a Gram-Schmidt process, after the first member is constructed so as to satisfy all the boundary conditions of the corresponding beam problems accompanying the plate problems. The strain energy functional and kinetic energy functionals are transformed from Cartesian coordinate system to a skew coordinate system. The natural frequencies obtained by using the orthogonal polynomial functions are compared with those obtained by other methods with all four edges clamped boundary conditions and greet agreements are found between them. The natural frequencies for parallelogrammic plates with other boundary conditions, such as four edges simply supported, clamped-free and simply supported-free, are also obtained. This method is considered as a better and accurate comprehensive treatment for this type of problems.

scholarly journals Eigenvalues of the negative Laplacian for arbitrary multiply connected domains

1996 ◽  
Vol 19 (3) ◽  
pp. 581-586
Author(s):  
E. M. E. Zayed
Keyword(s):  
Boundary Conditions ◽  
Spectral Function ◽  
Three Dimensions ◽  
Bounded Domains ◽  
Multiply Connected Domains ◽  
Multiply Connected ◽  
Asymptotic Formulae ◽  
Negative Laplacian

The purpose of this paper is to derive some interesting asymptotic formulae for spectra of arbitrary multiply connected bounded domains in two or three dimensions, linked with variation of positive distinct functions entering the boundary conditions, using the spectral function∑k=1∞{μk(σ1,…,σn)+P}−2asP→∞. Further results may be obtained.

scholarly journals A reaction infiltration problem: classical solutions

1997 ◽  
Vol 40 (2) ◽  
pp. 275-291 ◽  
Author(s):  
John Chadam ◽  
Xinfu Chen ◽  
Roberto Gianni ◽  
Riccardo Ricci
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Parabolic Equation ◽  
Elliptic Equation ◽  
Boundary Conditions ◽  
Classical Solution ◽  
Existence And Uniqueness ◽  
Classical Solutions ◽  
Bounded Domains ◽  
Global Classical Solution

In this paper, we consider a reaction infiltration problem consisting of a parabolic equation for the concentration, an elliptic equation for the pressure, and an ordinary differential equation for the porosity. Existence and uniqueness of a global classical solution is proved for bounded domains Ω⊂RN, under suitable boundary conditions.

scholarly journals Convergence analysis of time-discretisation schemes for rate-independent systems

2019 ◽  
Vol 25 ◽  
pp. 65 ◽  
Author(s):  
Dorothee Knees
Keyword(s):  
Convergence Analysis ◽  
Numerical Schemes ◽  
Solution Concepts ◽  
Continuous Solutions ◽  
Energy Functionals ◽  
Convergence Of Solutions ◽  
Time Discretisation ◽  
Energetic Solutions ◽  
Nonconvex Energy ◽  
The Given

It is well known that rate-independent systems involving nonconvex energy functionals in general do not allow for time-continuous solutions even if the given data are smooth. In the last years, several solution concepts were proposed that include discontinuities in the notion of solution, among them the class of global energetic solutions and the class of BV-solutions. In general, these solution concepts are not equivalent and numerical schemes are needed that reliably approximate that type of solutions one is interested in. In this paper, we analyse the convergence of solutions of three time-discretisation schemes, namely an approach based on local minimisation, a relaxed version of it and an alternate minimisation scheme. For all three cases, we show that under suitable conditions on the discretisation parameters discrete solutions converge to limit functions that belong to the class of BV-solutions. The proofs rely on a reparametrisation argument. We illustrate the different schemes with a toy example.

scholarly journals Homogenization limit for the Diffusion Equation in a domain Perforated along (n - 1)-Dimensional Manifold with Dynamic Conditions on the Boundary of the Perforations: Critical Case

2019 ◽  
Vol 486 (1) ◽  
pp. 12-19
Author(s):  
M. N. Zubova ◽  
T. A. Shaposhnikova
Keyword(s):  
Diffusion Equation ◽  
Boundary Conditions ◽  
Critical Case ◽  
Original Problem ◽  
Dimensional Manifold ◽  
Dynamic Boundary Conditions ◽  
Dynamic Conditions ◽  
Convergence Of Solutions ◽  
Homogenization Model ◽  
With Memory

The problem of homogenization the diffusion equation in a domain perforated along an (n - 1)-dimensional manifold with dynamic boundary conditions on the boundary of the perforations is studied. A homogenization model is constructed that is a transmission problem for the diffusion equation with the transmission conditions containing a term with memory. A theorem on the convergence of solutions of the original problem to the solution of the homogenized one is proved.

scholarly journals On the spectrum of the negative Laplacian for general doubly-connected bounded domains

1995 ◽  
Vol 18 (2) ◽  
pp. 245-254
Author(s):  
E. M. E. Zayed ◽  
A. I. Younis
Keyword(s):  
Boundary Conditions ◽  
Laplace Operator ◽  
Three Dimensional ◽  
Asymptotic Formulas ◽  
Bounded Domains ◽  
Impedance Boundary ◽  
Asymptotic Expressions ◽  
The Difference ◽  
Negative Laplacian ◽  
Positive Functions

This paper is devoted to asymptotic formulas for functions related with the spectrum of the standard Laplace operator in two and three dimensional bounded doubly connected domains with impedance boundary conditions, where the impedances are assumed to be positive functions. Moreover, asymptotic expressions for the difference of eigenvalues related to impedance boundary value problems with different impedances are derived. Further results may be obtained.

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