scholarly journals Necessary and sufficient conditions for the strong local minimality of C1 extremals on a class of non-smooth domains

2020 ◽  
Vol 26 ◽  
pp. 49
Author(s):  
Judith Campos Cordero ◽  
Konstantinos Koumatos
Keyword(s):  
Boundary Conditions ◽  
Calculus Of Variations ◽  
Materials Science ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Local Minimality ◽  
Sufficiency Theorem ◽  
Necessary And Sufficient ◽  
Quasiconvexity At The Boundary

Motivated by applications in materials science, a set of quasiconvexity at the boundary conditions is introduced for domains that are locally diffeomorphic to cones. These conditions are shown to be necessary for strong local minimisers in the vectorial Calculus of Variations and a quasiconvexity-based sufficiency theorem is established for C1 extremals defined on this class of non-smooth domains. The sufficiency result presented here thus extends the seminal theorem by Grabovsky and Mengesha (2009), where smoothness assumptions are made on the boundary.

Analysis and Design of Fixed–Fixed Bistable Arch-Profiles Using a Bilateral Relationship

2019 ◽  
Vol 11 (3) ◽  
Author(s):  
Safvan Palathingal ◽  
G. K. Ananthasuresh
Keyword(s):  
Boundary Conditions ◽  
Closed Form ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Single Variable ◽  
Fixed Boundary ◽  
Analysis And Design ◽  
Bilateral Relationship ◽  
Variable Equation ◽  
Necessary And Sufficient

Arch-profiles of bistable arches, in their two force-free equilibrium states, are related to each other. This bilateral relationship is derived for arches with fixed–fixed boundary conditions in two forms: a nonlinear single-variable equation for analysis and a closed-form analytical expression for design. Some symmetrical features of shape as well as necessary and sufficient conditions for bistability are presented as corollaries. Analysis and design of arch-profiles using the bilateral relationship are illustrated through examples.

The displacement problem for elastic crystals

1989 ◽  
Vol 113 (1-2) ◽  
pp. 159-180 ◽  
Author(s):  
I. Fonseca ◽  
L. Tartar
Keyword(s):  
Boundary Conditions ◽  
Convex Function ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Displacement Boundary ◽  
Displacement Problem ◽  
Body Forces ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Elastic Crystals

SynopsisIn this paper we obtain necessary and sufficient conditions for the existence of Lipschitz minimisers of a functional of the typewhere h is a convex function converging to infinity at zero and u is subjected to displacement boundary conditions. We provide examples of body forces f for which the infimum of J(.) is not attained.

scholarly journals Inverse problems for the Sturm–Liouville equation with a spectral parameter in the boundary condition

2017 ◽  
Author(s):  
Namig J. Guliyev
Keyword(s):  
Boundary Condition ◽  
Inverse Problems ◽  
Boundary Conditions ◽  
Spectral Parameter ◽  
Liouville Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Eigenvalue Parameter ◽  
Sturm Liouville ◽  
Necessary And Sufficient

Inverse problems of recovering the coefficients of Sturm--Liouville problems with the eigenvalue parameter linearly contained in one of the boundary conditions are studied: (1) from the sequences of eigenvalues and norming constants; (2) from two spectra. Necessary and sufficient conditions for the solvability of these inverse problems are obtained.

Sufficiency and the Jacobi Condition in the Calculus of Variations

1986 ◽  
Vol 38 (5) ◽  
pp. 1199-1209 ◽  
Author(s):  
Frank H. Clarke ◽  
Vera Zeidan
Keyword(s):  
Calculus Of Variations ◽  
Basic Problem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Jacobi Condition ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Differentiable Mappings

Besides stating the problem and the results, we shall give in this section a brief overview of the classical necessary and sufficient conditions in the calculus of variations, in order to clearly situate the contribution of this article.1.1 The problem. We are given an interval [a, b], two points xa, xb in Rn, and a function L (the Lagrangian) mapping [a, b] × Rn × Rn to R. The basic problem in the calculus of variations, labeled (P), is that of minimizing the functionalover some class X of functions x and subject to the constraints x(a) = xa, x(b) = xb. Let us take for now the class X of functions to be the continuously differentiable mappings from [a, b] to Rn; we call such functions smooth arcs.

scholarly journals Convexity of certain integrals of the calculus of variations

1987 ◽  
Vol 107 (1-2) ◽  
pp. 15-26 ◽  
Author(s):  
B. Dacorogna
Keyword(s):  
Calculus Of Variations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary Condition ◽  
Necessary And Sufficient

SynopsisIn this paper we study the convexity of the integral over the space . We isolate a necessary condition on f and we find necessary and sufficient conditions in the case where f(x, u, u′) = a(u)u′2n or g(u) + h(u′).

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