Complementary variational principles for biharmonic boundary-value problems

1971 ◽  
Vol 2 (12) ◽  
pp. 593-597 ◽  
Author(s):  
N. Anderson ◽  
A. M. Arthurs
Keyword(s):  
Boundary Value Problems ◽  
Variational Principles ◽  
Boundary Value

On dual approximation principles and optimization in continuum mechanics

1969 ◽  
Vol 265 (1162) ◽  
pp. 319-351 ◽  
Keyword(s):  
Mathematical Programming ◽  
Boundary Value Problems ◽  
Continuum Mechanics ◽  
Variational Principles ◽  
Diffusion Theory ◽  
Boundary Value ◽  
Uniqueness Theorems ◽  
Systematic Method ◽  
And Diffusion ◽  
Dual Functions

A unified expression of some of the boundary value problems of continuum mechanics is developed. A central role is given to the notion of a Legendre dual transformation in displaying the simple analytical structure of each problem considered. A systematic method of deriving reciprocal variational principles is described. General boundary value problems governed by inequalities as well as equations are then considered. Convexity of the dual functions related by the Legendre transformation is shown to be the basis of uniqueness theorems and extremum principles. Attention is drawn to the relevance of the literature on mathematical programming theory. M any examples are given, involving new or recent results in elasticity, plasticity, fluid mechanics and diffusion theory.

scholarly journals On the Travelling Waves for the Generalized Nonlinear Schrödinger Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Mahir Hasanov
Keyword(s):  
Quantitative Analysis ◽  
Boundary Value Problems ◽  
Eigenvalue Problems ◽  
Variational Principles ◽  
Travelling Waves ◽  
Boundary Value ◽  
Dispersion Relations ◽  
Duality Principle ◽  
Nonlinear Schrödinger ◽  
Generalized Nonlinear Schrödinger Equation

This paper is devoted to the analysis of the travelling waves for a class of generalized nonlinear Schrödinger equations in a cylindric domain. Searching for travelling waves reduces the problem to the multiparameter eigenvalue problems for a class of perturbedp-Laplacians. We study dispersion relations between the eigenparameters, quantitative analysis of eigenfunctions and discuss some variational principles for eigenvalues of perturbedp-Laplacians. In this paper we analyze the Dirichlet, Neumann, No-flux, Robin and Steklov boundary value problems. Particularly, a “duality principle” between the Robin and the Steklov problems is presented.

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