Minimizers and the Euler-Lagrange equations. (English) Zbl 0547.73013

Ball, J. M.

Trends and applications of pure mathematics to mechanics, Symp., Palaiseau/France 1983, Lect. Notes Phys. 195, 1-4 (1984).

[For the entire collection see Zbl 0533.00030.]
The author exposes some results developed in other papers, in preparation or to appear, concerning the minimization of \(I(u)=\int_{\Omega}f(x,u(x),\nabla u(x))dx\), with \(\Omega\subset {\mathbb{R}}^ m\). It is known that the problem is equivalent to the condition that u(x) satisfies the Euler-Lagrange equations in the weak form. For the existence of the solution often stronger growth conditions on f are assumed.
In two examples the author shows: 1. there can also exist solutions of the minimizing problem for smooth and regular functions f; 2. for a nonlinear problem of elastostatics which requires the minimization of \(I(u)=\int_{\Omega}W(\nabla u(x))dx\) there can exist a solution such that \(\nabla u(x)\) or \(\nabla u(x)^{-1}\) have infinite values, according to the occurrence of fractures, in a mechanical interpretation.

Reviewer: St.Zanfir

MSC:

74S30	Other numerical methods in solid mechanics (MSC2010)
74B99	Elastic materials
74H99	Dynamical problems in solid mechanics
70H03	Lagrange’s equations

Keywords:

minimization; Euler-Lagrange equations

Citations:

Zbl 0533.00030