Variational inequalities

Abstract

If $- \infty < \alpha < \beta < \infty $ and $f \in C^{3} \left( [ \alpha , \beta ] \times {\bf R}^{2} , {\bf R} \right) $ is bounded, while $y \in C^{2} \left( [ \alpha , \beta ] , {\bf R} \right) $ solves the typical one-dimensional problem of the calculus of variations to minimize the function $$F \left( y \right) = \int_{ \alpha }^{ \beta }f \left( x, y(x), y'(x) \right) dx,$$ then for any ${\phi } \in C^{2} \left( [ \alpha , \beta ] , {\bf R} \right) $ for which ${\phi }^{(k)}( \alpha ) = {\phi }^{(k)}( \beta ) = 0$ for every $k \in \{ 0, 1, 2 \} $, we prove that $\int_{\alpha }^{\beta } \left( \frac{ {\partial }^{2}f }{ \partial y^{2} } {\phi }^{2} - \frac{ {\partial }^{3}f }{ \partial y^{2} \partial y' } 2 {\phi }^{3} \right) dx$ $\geq \int_{\alpha }^{\beta } \left( \frac{ {\partial }^{2}f }{ \partial y \partial y' } 2 \phi \phi ' + \frac{ {\partial }^{3}f }{ \partial y {\partial y'}^{2} } 2 {\phi }^{2} \phi ' + \frac{ {\partial }^{2}f }{ {\partial y'}^{2} } \phi \phi " + \frac{ {\partial }^{3}f }{ \partial y {\partial y'}^{2} } \phi ' {\phi }^{2} + \frac{ {\partial }^{3}f }{ {\partial y'}^{3} } \phi {\phi '}^{2} \right) dx$, so either the above are variational inequalities of motion or the Lagrangian of motion is not $C^{3}$.