I have a complex optimization problem. I would like to explain a simple version. Consider the Laplace equation in 1D. Our space is the unit interval $[0,1]$. The problem is to solve the following boundary value problem: $$- \partial _x^2u(x) = 0,~~~ x \in [0,1]~~~(1)$$ $$u(0) = 0.5~~~(2)$$ $$u(1) = 1.5~~~(3)$$ while the solution must obey an integral constraint: $$1. ~~~~~C_1= \int\limits_0^1 {dx} u(x) = 2~~~(4)$$ or $$2. ~~~~~C_2= \int\limits_0^{0.5} {dx} u(x) = 2~~~(5)$$ Beginning with the first case, as far as I understand, the Integral Constraint is global so the Lagrange Multiplier, $\lambda$, in the penalty term is a constant. So: $$L[u(x),{\partial _x}u(x)] = \frac{1}{2}{\left( {{\partial _x}u(x)} \right)^2} - \lambda \left( {\int\limits_0^1 {dy} u(y) - 2} \right),~~~S = \int\limits_0^1 {dx} L ~~~(6)$$ Variation towards $\lambda$ is trivial and gives back the constraint (4). Variation towards $u(x)$ seems a bit trickier. From the Euler-Lagrange equations we have: $$\frac{{\partial L}}{{\partial u(x)}} = \lambda \frac{\delta }{{\delta u(x)}}\int\limits_0^1 {dy} u(y) = \lambda \int_0^1 {dy\delta (x - y) = \lambda ,}~~~~ {\partial _x}\frac{{\partial L}}{{\partial \left( {{\partial _x}u(x)} \right)}} = \partial _x^2u(x) (7)$$ This gives the equation of motion $- \partial _x^2u(x) = \lambda$. Direct integration and imposing of (2), (3), (4) to find $\lambda$ and the two integration constants gives the answer $u(x)=-6x^2+7x+0.5$.
So what puzzles me: are the above correct? The way I change the functional derivatives in (7) is not something I understand clearly. Also, how can I do the same with variations, i.e. $$S[u(x) + \delta u(x),{\partial _x}\left( {u(x) + \delta u(x)} \right)] = \int\limits_0^1 {dx} \left[ {\frac{1}{2}{{\left( {{\partial _x}\left( {u(x) + \delta u(x)} \right)} \right)}^2} + \left( {\lambda + \delta \lambda } \right)\left( {\int\limits_0^1 {dy\delta (x - y)} \left( {u(x) + \delta u(x)} \right) - 2} \right)} \right] ?$$
Finally, can I use the above for the constraint from eq. (5)? This is a local constraint and I think that $\lambda$ should be a function of $x$. However, functional derivatives like $\frac{\delta }{{\delta u(x)}}\int\limits_0^1 {dz} \lambda (z)\int\limits_0^{0.5} {dy} u(y)$ arise that I am not sure are correct.
First post here, my apologies if I am not clear or for the lenght of the post.