I want to minimize the functional $I=\int_{-1}^1u^2(x)|2x-u'(x)|^2dx$
Here i applied and found the euler langrange equation and found the differential equation
$u'^2+2uu'-4u=4x^2$ given is $u\in C^1 $ and $u(-1)=0 , u(1)=1$
but the minimizer is given as $u(x)=0:x\in[-1,0], x^2:x\in[0,1]$
Can anyone help me how to go about with this problem .
I need some idea on this functional as well to see that if we choose approperiate $h\in (-1,1)$ i get a circle arc as a minimizer with radium $\frac{1}{h}$
Functional is $I=\int_0^1\sqrt{1+y'^2} +h y dx$
Thank you for your kind guidance .