Help to obtain Euler Lagrange equation

Question

I am pretty new with the Euler Lagrange equation. Set $$ F(u,\nabla u)=\frac{1}{2}\int_\Omega |v-u|^2\ dx +\frac{1}{2}\int_\Omega|\nabla u|^2\ dx \qquad u\in H^1(\Omega)$$ Where $v\in H^1(\Omega)$ is fixed. A straight computation shows that there exists $u_0\in H^1(\Omega)$ s.t. $F(u_0)\leq F(u)$ for all $u$. And so, this $u_0$ must satisfy the Euler lagrange eq. My question is how can be this equation for this concrete example obtained? What is the common procedure?

Answer 1

Let's take a 1D example. We can combine the integrals and simplify the notation a bit: $$F\left(u,u'\right)=\int_{\Omega}\frac{(u-v)^2+(u')^2}{2}\,dx.$$ Then $L=\dfrac{(u-v)^2+(u')^2}{2}$ and the EL equation says \begin{align*} \frac{\partial L}{\partial u}-\frac{d}{dx}\,\frac{\partial L}{\partial u'}&=0 \\ u-v-\frac{d}{dx}u'&=0 \\ u-v-u''&=0. \end{align*} Without knowing $v,$ this is as far as we can go, but getting this far is an important first step.