I have to find a minimum, if exists, for the functional $$F[f]=\int_0^1 f'(x)^2dx-\log \left(f\left(\frac{1}{2}\right)\right)$$ On the space $$f\in H^1((0,1)):\ f\ge 0,\ \int_0^1 f(x)\ dx=1$$ (this may not be the best choice, if you have a better ides for the functional space be free to suggest).
At first I tried with the Euler-Lagragne equation with the lagrange multipliers, which gave me the differential equation $$2f''(x)=\lambda+\frac{\delta_{1/2}(x)}{f(1/2)}\qquad \text{ in } D'((0,1))$$ with Neumann boundary conditions: $$f'(0)=0,\qquad f'(1)=0$$ Is there a way to solve it (analitically or numerically)? Does it make sense that the solution becomes a kind of spline?
This $\delta$ on the rhs can be translated to a jump in $f'$ at $1/2$: $$ f'(x) = f'(0)+ \int_0^x f''(t) = \lambda x+ \begin{cases} 0 & \text{ if } x<1/2\\ \frac1{f(1/2)} & \text{ if } x > 1/2. \end{cases} $$ This can be made rigorous using smooth test function and integration by parts on $(0,1/2)$ and $(1/2,1)$.
Then $f$ is a quadratic polynomial on $(0,1/2)$ and $(1/2,1)$, continuous, with prescribed jump of $f'$ at $1/2$. This should make it possible to solve for $f$.