Subharmonic envelope

Question

I came across a more complicated version of the following problem. It is so elementary, I think that there had to be some research done on this in the past. If someone has any ideas please let me know.

Suppose $u: \bar D \to \Bbb R$ is a smooth function on the closure of the unit disc in $\Bbb C$. We introduce the following family: $$S_u = \lbrace v: D \to \Bbb R \mid v \leq u, v \text{ is continuous and subharmonic }\rbrace$$ Let $h$ be the upper envelope of the supremum of all elements in $S_u$. Then $h$ is an usc subharmonic function.

The question is as follows. Under what conditions on $u$ is the difference $\Delta = u(0)-h(0) \geq 0$ zero. This quantity is zero when $u$ is subharmonic, but its fate seems to be unclear in other situations.

Answer 1

$h$ is the solution of the obstacle problem: it minimizes the Dirichlet energy $$\int |\nabla v|^2$$ among all functions $v$ less or equal than $u$, and having the same boundary values as $u$. One can alo characterize $h$ as the solution of a free boundary problem, namely $h$ is harmonic on $\Omega \setminus \Gamma$, and $h=u$ on $\Gamma$ and on $\partial \Omega$. So the question is what $\Gamma$ looks like, and not much can be said in general (if anything at all, apart from various regularity results).