Variational inequalities

Abstract

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