Boundary Conditions in Variational Problems

1984 ◽  
pp. 48-89
Author(s):  
Pierre Ninh Van Tu
Keyword(s):  
Boundary Conditions ◽  
Variational Problems

The equivalence of some variational problems for surfaces of prescribed mean curvature

1979 ◽  
Vol 20 (1) ◽  
pp. 87-104 ◽  
Author(s):  
Graham H. Williams
Keyword(s):  
Boundary Conditions ◽  
Minimization Problem ◽  
Mean Curvature ◽  
Variational Problems ◽  
Generalized Solution ◽  
Smooth Functions ◽  
Prescribed Mean Curvature ◽  
Class Of Functions ◽  
General Conditions ◽  
Non Parametric

One method of finding non-parametric hypersurfaces of prescribed mean curvature which span a given curve in Rn is to find a function which minimizes a particular integral amongst all smooth functions satisfying certain boundary conditions. A new problem can be considered by changing the integral slightly and then minimizing over a larger class of functions. It is possible to show that a solution to this new problem exists under very general conditions and it is usually known as the generalized solution. In this paper we show that the two problems are equivalent in the sense that the least value for the original minimization problem and the generalized problem are the same even though no solution may exist. The case where the surfaces are constrained to lie above an obstacle is also considered.

scholarly journals Natural Boundary Conditions in Geometric Calculus of Variations

2015 ◽  
Vol 65 (6) ◽  
Author(s):  
Giovanni Moreno ◽  
Monika Ewa Stypa
Keyword(s):  
Boundary Conditions ◽  
Variational Problems ◽  
Boundary Values ◽  
Natural Boundary ◽  
Natural Behavior ◽  
Geometric Calculus ◽  
Jet Bundles ◽  
Independent Variables ◽  
Natural Boundary Conditions ◽  
Dimensional Domain

AbstractIn this paper we obtain natural boundary conditions for a large class of variational problems with free boundary values. In comparison with the already existing examples, our framework displays complete freedom concerning the topology of Y - the manifold of dependent and independent variables underlying a given problem - as well as the order of its Lagrangian. Our result follows from the natural behavior, under boundary-friendly transformations, of an operator, similar to the Euler map, constructed in the context of relative horizontal forms on jet bundles (or Grassmann fibrations) over Y . Explicit examples of natural boundary conditions are obtained when Y is an (n + 1)-dimensional domain in ℝ

Numerical study of variational problems of moving or fixed boundary conditions by Muntz wavelets

2020 ◽  
pp. 107754632097479
Author(s):  
Ashish Rayal ◽  
Sag R Verma
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Approximation Method ◽  
Computational Algorithm ◽  
Numerical Study ◽  
Operational Matrix ◽  
Variational Problems ◽  
Algebraic Equations ◽  
Suggested Approach ◽  
Fixed Boundary

In this study, an approximation method with an integral operational matrix based on the Muntz wavelets basis is presented to solve the variational problems of moving or fixed boundary conditions and a computational algorithm is given for the suggested approach. First, the integral operational matrix is created through the Muntz wavelets. Then, by using this integral operational matrix with Lagrange multipliers, the present approach reduces the variational problem into the system of algebraic equations. This approach is examined by some illustrative examples, and the acquired results prove that the suggested approach can solve the variational problems effectively with higher accuracy. The proposed approach yields better and comparable results with some other existing schemes given in the literature. The approximate wavelet solutions derived by the suggested approach are very identical to the corresponding exact solution.

On a class of variational problems with nonlocal integrant

2017 ◽  
Vol 24 (1) ◽  
pp. 97-101
Author(s):  
Vladimir P. Maksimov
Keyword(s):  
Boundary Conditions ◽  
Variational Problem ◽  
Necessary Conditions ◽  
Integral Form ◽  
Variational Problems ◽  
Linear Constraints ◽  
Local Extremum ◽  
General Linear ◽  
Point Boundary ◽  
Deviating Argument

AbstractFor a class of functionals with nonlocal integrant, necessary conditions of local extremum under general linear constraints are obtained. An analog of Euler’s equation in an integral form is derived. An example of application to a variational problem for a functional with deviating argument and two-point boundary conditions is presented.

scholarly journals Discontinuous solutions of variational problems

1959 ◽  
Vol 1 (1) ◽  
pp. 27-37 ◽  
Author(s):  
D. F. Lawden
Keyword(s):  
Boundary Conditions ◽  
Finite Number ◽  
Calculus Of Variations ◽  
Variational Problems ◽  
Discontinuous Solutions ◽  
End Points ◽  
Elementary Problem ◽  
Image Position ◽  
Corner Conditions

The most elementary problem of the calculus of variations consists in finding a single-valued function y(x), defined over an interval [a, b] and taking given values at the end points, such that the integral is stationary relative to all small weak variations of the function y(x) consistent with the boundary conditions. Since y′ occurs in the integrand, it is clear that I is only defined when y(x) is differentiable and accordingly when y(x) is continuous. Usually y′(x) is also continuous. Occasionally, however, the boundary conditions can only be satisfied and a stationary value of I found, by permitting y′ (x) to be discontinuous at a finite number of points. The arc y = y(x) will then possess ‘corners’ and the well-known Weierstrass-Erdmann corner conditions [1]must be satisfied at all such points by any function y (x) for which I is stationary. Arcs y = y (x) for which y′(x) is continuous except at a finite number of points, are referred to as admissible arcs. In this paper, we shall extend the range of admissible arcs to include those for which y(x) is discontinuous at a finite number of points.

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