Consider the following problem \[ \begin{cases} F_f - \dfrac{\partial}{\partial x} F_{f_x} - \dfrac{\partial}{\partial y} F_{f_y} = 0, & (x, y) \in D \\ F_{f_x} n_x + F_{f_y} n_y + = g(s), & x \in \partial D, \end{cases} \tag{16} \] where \(f = f(x, y) \in C^2(\bar{D})\), \(F = F[f, f_x, f_y](x, y) \in C^2(\mathbb{R}^3 \times \bar{D})\), \(g \in C^1(\partial D)\), \(\partial/\partial n\) denotes a normal derivative on \(\partial D\), \(f_x \equiv \frac{\partial f}{\partial x}\) and \(F_f \equiv \frac{\partial F}{\partial f}\).
Show that the solutions to this equation are given by extrema of the functional \[ J[f] = \iint_D dx\,dy\,F - \int_{\partial D} ds\,f g. \tag{17} \]
How must the functional above be modified to give the mixed boundary conditions on the function \(f\): \[ F_{f_x} n_x + F_{f_y} n_y + h(s) f = g(s), \quad h \in C^1(\partial D) \tag{18} \]
Erratum: the boundary condition in (16) is printed in the source as “\(F_{f_x} n_x + F_{f_y} n_y + = g(s)\), \(x \in \partial D\)”, with a dangling plus sign. The condition that makes part (a) true, and that (18) reduces to for \(h = 0\), is \(F_{f_x} n_x + F_{f_y} n_y = g(s)\) on \(\partial D\), where \((n_x, n_y)\) is the outward unit normal and \(s\) is arc length along \(\partial D\).