Consider for example, the Black Schole's equation
$\partial_tu+0.5\sigma^2s^2\partial_{ss}u+rs\partial_su-ru=0$
Subject to boundary conditions
$u(s,T)=f(s)$ and $u(0,t)=g(t)$
The solution depends on the parameter $\sigma$, which is not a variable in the PDE, but it is possible to show the solution is differentiable with respect to $\sigma$ for certain f and g. Do there exists analysis tools and results which deal with smoothness of solutions with respect to perturbation of coefficients of PDEs?