Consider the problem of minimizing $$J(u) = \int_0^1 -\log u(x) - \log(1-u(x)) \, dx,$$ where the logarithm is defined as $\log(\xi) = -\infty$ when $\xi\leq 0$. Note that the function $\xi \to -\log(\xi)-\log(1-\xi)$ is convex with a minimum at $\xi = 1/2$ so the natural domain of definition of $J(u)$ is $$D = \{u(x) \; : \; u(x) \text{ is measurable and } 0 < u(x) < 1 \}.$$ I'm interested in doing functional analysis on sets like $D$. (For example, $J(u)$ could have $u'(x)$ in its definition instead of $u(x)$, and then one would pair this with boundary conditions to study certain nonlinear PDEs.)
Is there a "functional analysis" for such domains? For example, it would be natural to define a metric from $J(u)$ (since it is convex) and obtain a unique minimizer, if not in $D$, in some completion of $D$.