Consider the biharmonic equation
\[ { \frac { \partial ^ { 4 } f } { \partial x ^ { 4 } } } + 2 { \frac { \partial ^ { 4 } f } { \partial x ^ { 2 } \partial y ^ { 2 } } } + { \frac { \partial ^ { 4 } f } { \partial y ^ { 4 } } } = 0 .\tag{15} \]
Find a functional \(J [ f ]\) whose extrema in a class of \(C ^ { 4 } ( \bar { D } )\) functions \(f = f ( x , y )\) with fixed boundary conditions, \(f ( x , y ) | _ { \partial D } = f _ { 0 } ( x , y )\) , satisfy this equation.