I need to tackle a variational optimization problem in two variables with an integral-constraint with respect to one variable and a point-wise constraint with respect to the other variable. One may want to call it a 'line-wise' constraint. Specifically, seeking for $f\colon [a, b]^2 \to \mathbb{R}$:
\begin{align} \tag{1} \text{minimize} & \quad \int_a^b \int_a^b \; K(x, y, f, f_x, f_y) \; dx \; dy \\ \text{subject to} & \quad \int_a^b \; G(x, y, f) \; dx \; = \; 0 \quad \forall \; y \in (a, b) \tag{2} \end{align} where $f_x$, $f_y$ are the derivatives of $f$ by $x$ and $y$ respectively and assuming natural boundary conditions, i.e. resulting in Neumann boundary conditions for the final PDE.
I would expect that one can formulate this by using a Lagrange multiplier that is a function in $y$, yielding a Lagrangian like so
\begin{equation} \tag{3} L \;=\; K + \lambda(y) G \end{equation}
then apply Euler-Lagrange equations for two independent variables on $L$ as found in Arfken, 'Mathematical Methods for Physicists' or here.
There is plenty of literature that treats the case of one or several integral constraints in one or several variables or of a point-wise constraint in one variable. I did not find the case of 'line-wise' constraints explicitly treated in literature and my knowledge of calculus of variations is too fragile to confidently derive this case. So I would kindly ask for
- confirmation of my intuition that (3) is the proper way to tackle problem (1, 2) or a correction otherwise
- pointers to literature where this case is handled
- hints how to derive (3) by variation or by reducing it to known cases
Edit: I read that Lagrange multipliers can only handle holonomic constraints, i.e. $G$ must not depend on $f_y$. It appears to be fine if functionals in integral constraints depend on derivatives with respect to the variable of integration.
Additional question: Do I understand correctly that $G$ in (2) may well depend on $f_x$ but not on $f_y$ to count as holonomic, as it is integrated over $x$? I.e. (2) could also be written with $G(x, y, f, f_x)$.
Sorry if I overlooked a duplicate of this question. I searched for them and there does not seem to be one.