Euler Lagrange equation

Question

I am trying to minimize the following expression $\int^b_a|u'|dx+\int^b_a \psi(|u'|)|u''|dx + \lambda \int^b_a |u-g|^2$. ($\psi$ can be any bounded borel function) I want to derivate the Euler Lagrange equation but I am stuck. Can someone help me out?

Answer 1

ByOur Lagrangian is $L=|u^\prime|+\psi(|u^\prime|)u^{\prime\prime}+\lambda(u-g)^2$. The ELE is$$0=\frac{\partial L}{\partial u}-\frac{d}{dx}\frac{\partial L}{\partial u^\prime}+\frac{d^2}{dx^2}\frac{\partial L}{\partial u^{\prime\prime}}=2\lambda(u-g)-\frac{d}{dx}(\operatorname{sgn}u^\prime+\psi_1(|u^\prime|)|u^{\prime\prime}|\operatorname{sgn}u)+\frac{d^2}{dx^2}\left(\psi(|u^\prime|)\operatorname{sgn}u^{\prime\prime}\right),$$where $f_1$ is the derivative of a unary function $f$ with respect to its argument (I'll use this notation again in a moment for a function $f$ other than $\psi$) and $\operatorname{sgn}y$ is the sign $\pm1$ of y. I'll leave it to you to evaluate further, using $\operatorname{sgn}_1y=2\delta(y)$ ($\delta$ being the Dirac delta).