Natural boundary conditions for free-boundary extremals

Let \(f \in C ^ { 2 } ( \bar { D } )\) be an extremum of the functional \(S [ f ] = \int _ { { \cal D } } d x { \cal L } ( f , f ^ { \prime } )\) in a class of functions satisfying the boundary condition \(f | _ { \partial D } = f _ { 0 } \mathrm { ~ , ~ }\) What additional condition must be imposed on \(f\) to make it an extremum of \(S [ f ]\) in a class of \(C ^ { 2 } ( \bar { D } )\) functions with all possible boundary conditions?