CoV22

Find the Euler equation for the functional

$J(u)=\iint\limits_G\left[u_x^2+u_y^2+2f(x,y)u(x,y)\right]dxdy\,$

where $G\,$ is a closed region in the $xy\,$ plane and $u\,$ has continuous second partial derivatives.

$L_u - \frac{\partial}{\partial x}L_{u_x} - \frac{\partial}{\partial y}L_{u_y}$	$= 0\,$
$2f(x, y) - \frac{\partial}{\partial x}(2u_x) - \frac{\partial}{\partial y}(2u_y)$	$= 0\,$
$2f(x, y) - 2u_{xx} - 2u_{yy}\,$	$= 0\,$
$f(x, y)\,$	$= u_{xx} + u_{yy}\,$