On a lemma in the direct method of the calculus of variations

1957 ◽  
Vol 6 (1) ◽  
pp. 109-113 ◽  
Author(s):  
L. Cesari ◽  
L. H. Turner
Keyword(s):  
Calculus Of Variations ◽  
Direct Method

Isoperimetric Problems in the Calculus of Variations

1952 ◽  
Vol 4 ◽  
pp. 257-280 ◽  
Author(s):  
William Karush
Keyword(s):  
Calculus Of Variations ◽  
Classical Theory ◽  
Indirect Method ◽  
Direct Method ◽  
Isoperimetric Problems ◽  
End Points ◽  
Sufficiency Theorems ◽  
Theory Of Fields ◽  
A Direct Method ◽  
Problem Of Bolza

We are concerned with establishing sufficiency theorems for minima of simple integrals of the parametric type in a class of curves with variable end points and satisfying isoperimetric side conditions. The results which are obtained involve no explicit assumptions of normality. Such results can be derived by transforming our problem to a problem of Bolza and using the latest developments in the theory of that problem. More recently [6] an indirect method of proof has been published. Our object is to present a direct method of proof without transformation of the problem which is based upon a generalization of the classical theory of fields.

scholarly journals Direct Method of Calculus of Variations in Elasticity

2018 ◽  
pp. 1-7
Author(s):  
Alessandro Della Corte ◽  
Francesco dell’Isola
Keyword(s):  
Calculus Of Variations ◽  
Direct Method

Single-Term Walsh Series Direct Method for the Solution of Nonlinear Problems in the Calculus of Variations

2004 ◽  
Vol 10 (7) ◽  
pp. 1071-1081 ◽  
Author(s):  
M. Razzaghi ◽  
B. Sepehrian
Keyword(s):  
Calculus Of Variations ◽  
Nonlinear Problem ◽  
Direct Method ◽  
Nonlinear Problems ◽  
Variational Problems ◽  
Walsh Series ◽  
Algebraic Equations ◽  
Brachistochrone Problem ◽  
Single Term ◽  
A Direct Method

A direct method for solving variational problems using single-term Walsh series is presented. Two nonlinear examples are considered. In the first example the classical brachistochrone problem is examined. and in the second example a higher-order nonlinear problem is considered. The properties of single-term Walsh series are given and are utilized to reduce the calculus of variations problems to the solution of algebraic equations. The method is general, easy to implement and yields accurate results.

OPTIMALLY IMMERSED PLANAR CURVES UNDER MÖBIUS ENERGY

2011 ◽  
Vol 20 (10) ◽  
pp. 1381-1390
Author(s):  
RYAN P. DUNNING
Keyword(s):  
Calculus Of Variations ◽  
Direct Method ◽  
Energy Parameter ◽  
Energy Estimates ◽  
Singular Behavior ◽  
Planar Curves ◽  
Intersecting Curve ◽  
Knot Energy ◽  
Parameter Dependent ◽  
Selection Of

This paper investigates the existence of optimally immersed planar self-intersecting curves. Because any self-intersecting curve will have infinite knot energy, parameter-dependent renormalizations of the Möbius energy remove the singular behavior of the curve. The direct method of the calculus of variations allows for the selection of optimal immersions in various restricted classes of curves. Careful energy estimates allow subconvergence of these optimal curves as restrictions are relaxed.

ON MINIMIZATION OF A NON-CONVEX FUNCTIONAL IN VARIABLE EXPONENT SPACES

2014 ◽  
Vol 25 (01) ◽  
pp. 1450011 ◽  
Author(s):  
GERARDO R. CHACÓN ◽  
RENATO COLUCCI ◽  
HUMBERTO RAFEIRO ◽  
ANDRÉS VARGAS
Keyword(s):  
Calculus Of Variations ◽  
Sobolev Spaces ◽  
Variable Exponent ◽  
Direct Method ◽  
Convex Functional ◽  
Lebesgue Spaces ◽  
Variable Exponent Spaces ◽  
Existence Of Minimizers ◽  
Variable Exponent Sobolev Spaces

We study the existence of minimizers of a regularized non-convex functional in the context of variable exponent Sobolev spaces by application of the direct method in the calculus of variations. The results are new even in the framework of classical Lebesgue spaces.

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