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Necessary and sufficient conditions for one-dimensional variational problems with applications to elasticity

Pavol Quittner

Abstract

This paper deals with necessary and sufficient conditions for weak and strong minimizers of functionals $Φ(u)=\int_a^b f(x,u(x),u'(x))\,dx$, where $u\in C^1([a,b],{\mathbb R}^N)$. We first derive conditions which are simpler than the known ones, and then apply them to several particular problems, including stability problems in the elasticity theory. In particular, we solve some open problems in [A. Majumdar, A. Raisch: Stability of twisted rods, helices and buckling solutions in three dimensions, Nonlinearity 27 (2014), 2841--2867] by finding optimal conditions for the stability of a naturally straight Kirchhoff rod under various types of endpoint constraints.

Necessary and sufficient conditions for one-dimensional variational problems with applications to elasticity

Abstract

This paper deals with necessary and sufficient conditions for weak and strong minimizers of functionals , where . We first derive conditions which are simpler than the known ones, and then apply them to several particular problems, including stability problems in the elasticity theory. In particular, we solve some open problems in [A. Majumdar, A. Raisch: Stability of twisted rods, helices and buckling solutions in three dimensions, Nonlinearity 27 (2014), 2841--2867] by finding optimal conditions for the stability of a naturally straight Kirchhoff rod under various types of endpoint constraints.
Paper Structure (7 sections, 14 theorems, 151 equations, 9 figures, 1 table)

This paper contains 7 sections, 14 theorems, 151 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Jacobi theory
  4. Stability of a twisted rod
  5. Field of extremals
  6. Scalar examples with variable endpoints
  7. Appendix

Key Result

Proposition 1

Let $f\in C^1$ and let $u^0$ be a critical point of $\Phi$ in $u^0+C^1_{\mathcal{D}}$. Then $u^0$ is an extremal (i.e. it satisfies the Euler equations $\frac{d}{dx}(f^0_{p_i})=f^0_{u_i}$, $i=1,2,\dots,N$), and $u^0$ also has to satisfy the natural boundary conditions If $f_{p_i}\in C^1$ for $i=1,2,\dots,N$, and the strengthened Legendre condition is true, then $u^0\in C^2$.

Figures (9)

  • Figure 1: The case $I^{{\mathcal{D}}}_0\cap\{1,2\}\ne\emptyset\ne I^{{\mathcal{D}}}_1\cap\{1,2\}$.
  • Figure 2: The case $I^{{\mathcal{D}}}_0\cap\{1,2\}=\emptyset$.
  • Figure 3: Phase plane and extremals for Example \ref{['ex-LG']} and $0\leq u\leq4\pi$; $C^-<2M<C^+$, $Z_i=(\varphi(i,\alpha),\varphi_x(i,\alpha))$, $i=0,1$, $Y_1=(A_1+\alpha,K)$, $X_i=(A_i,K)=(u^0(i),(u^0)'(i))$, $i=0,1$.
  • Figure 4: Graphs of $g$ in the symmetric and non-symetric cases.
  • Figure 5: Graph of $h$ in the symmetric case.
  • ...and 4 more figures

Theorems & Definitions (37)

  • Proposition 1
  • Definition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Proposition 9
  • proof
  • ...and 27 more