Suppose $D \subset \Bbb C$ with smooth boundary. Let $f \in C^{1,1}(D)$. Let $\varphi$ be the supremum of all members in the set

$$\lbrace g \in C^{\infty}(\overline{D})| g \ is \ subharmonic \ and \ g \leq f\rbrace$$

It is known that $\varphi$ is subharmonic in $D$ and harmonic on $E =\lbrace \varphi < f\rbrace$, the complement of the contact set. It is also known that $\varphi$ is C^{1,1}.

Suppose now that $\lbrace f_t \rbrace$, $t \in(0,1)$ is a smooth family of smooth obstacles. Will the variation of the family $\lbrace \varphi_t \rbrace$ be at least $C^1$?

If this is too much to ask, then let $\tilde E \subset (0,1)\times D$, be the open? domain for which each $t-$slice $E_t$ is the set where $\varphi_t$ will be harmonic. Will the variation of $\lbrace \varphi_t\rbrace$ be $C^1$ on $\tilde E$?

Any reference or chunk of information will be appreciated.

I don't know almost anything about obstacle problems (but you know this by now :D). There seems to be no obvious reference out there. Can someone recommend something to a grad student who is faimiliar with the Gilbarg and Trudinger stuff but not much more?