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

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

Zbl 0533.00030