Variational principle for Poisson's equation

Consider the functional \(J [ f ] = \iint d x d y \left[ \left( { \frac { \partial f } { \partial x } } \right) ^ { 2 } + \left( { \frac { \partial f } { \partial y } } \right) ^ { 2 } + 2 f g \right]\) , where D is a domain in (xy)-plane with the boundary \(\partial D\), and \(g = g ( x , y )\) is a continuous function in \(\bar { D }\) Write the equation on \(f ,\) whose solution In a class of functions \(f \in\) \(C ^ { 2 } ( \bar { D } )\) , satisfying the boundary condition \(f ( x , y ) | _ { \partial D } = f _ { 0 } ( x , y )\) , is an extremum of \(J [ f ]\)