The First and Second Variational Formulas of the Energy

2018 ◽  
pp. 21-45
Keyword(s):  
Variational Formulas

Variational Formulas for Higher Affine Mean Curvatures

1988 ◽  
Vol 13 (3-4) ◽  
pp. 318-326 ◽  
Author(s):  
Li An-Min
Keyword(s):  
Variational Formulas

scholarly journals Extremal Problems for Functions Starlike in the Exterior of the Unit Circle

1962 ◽  
Vol 14 ◽  
pp. 540-551 ◽  
Author(s):  
W. C. Royster
Keyword(s):  
Unit Circle ◽  
Analytic Functions ◽  
Upper Bounds ◽  
Extremal Problems ◽  
Simple Pole ◽  
Image Position ◽  
Inverse Functions ◽  
Variational Formulas

Let Σ represent the class of analytic functions(1)which are regular, except for a simple pole at infinity, and univalent in |z| > 1 and map |z| > 1 onto a domain whose complement is starlike with respect to the origin. Further let Σ- 1 be the class of inverse functions of Σ which at w = ∞ have the expansion(2).In this paper we develop variational formulas for functions of the classes Σ and Σ- 1 and obtain certain properties of functions that extremalize some rather general functionals pertaining to these classes. In particular, we obtain precise upper bounds for |b2| and |b3|. Precise upper bounds for |b1|, |b2| and |b3| are given by Springer (8) for the general univalent case, provided b0 =0.

The Numerical Simulation for Vibroacoustic Phenomena of a Slender Elastic Shell

2012 ◽  
Vol 479-481 ◽  
pp. 1365-1370
Author(s):  
Zhi Xi Yang ◽  
Sheng Hua Qiu
Keyword(s):  
Numerical Simulation ◽  
Finite Element ◽  
Thin Shell ◽  
Acoustic Pressure ◽  
Pressure Field ◽  
Elastic Shell ◽  
Numerical Examples ◽  
Longitudinal Acoustic ◽  
3 Dimensional ◽  
Variational Formulas

The vibroacoustic phenomena for the slender elastic thin shell filled with water by finite element method is introduced in this paper. The unsymmetric (u, p) variational formulas and finite element procedures are implemented for 3 dimensional structures of vibroacoustic environment based on the displacement field u and the fluid acoustic pressure field p. As illustrated by numerical examples, the longitudinal acoustic pressure eigenmodes will be occurred besides the transverse bendable eigenmodes of the slender shell, nonetheless the eigenvalues and the order of eigenmodes for the fluid acoustic pressure field can only be determined by the flexibility and geometry stiffness of the slender shell.

MINIMALITY OF AFFINE LAGRANGIAN SUBMANIFOLDS IN COMPLEX EQUIAFFINE SPACES

2011 ◽  
Vol 22 (07) ◽  
pp. 887-906 ◽  
Author(s):  
BARBARA OPOZDA
Keyword(s):  
Lagrangian Submanifolds ◽  
Riemannian Case ◽  
Phase Functions ◽  
Variational Formulas ◽  
Calabi Yau Manifolds

The notion of complex equiaffine manifolds is an affine generalization of Calabi–Yau manifolds. Similarly as in the Riemannian case the minimality of affine Lagrangian submanifolds in complex equiaffine spaces can be studied via calibrations, phase functions and variational formulas.

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