$C^{2}$ regularity of a curve of solutions to a family of elliptic equations I have the following question, I apologize in advance if it looks classical, but I've not found any precise reference pointing to the solution so far. I have the solutions $u_s$ ($s>0$) to the family of elliptic equations $−Δu=f_s$ in a bounded domain $\Omega$, where $f_s$ are assumed to be smooth, with (say) zero Dirichlet boundary condition. Can I derive somehow the $C^{2}$ regularity of the map $s∈(0,+∞)→u_s∈X$, where $X$ is some functional space? I found this result cited in a paper, but no reference to such a result was given.
Thank you very much,
Bruno
 A: Sure, depending on how smooth $s\mapsto f_s$ is for some topology.
Assume for example that $s\mapsto f_s$ is $C^2(0,\infty;L^2)$. Then the map $s\mapsto u_s$ is $C^2(0,\infty;H^1_0(\Omega))$. To see this fix for example $s_0\in(0,\infty)$ and vary $s\to s_0$. By linearity you have first
$$
w_s:=\frac{u_s-u_{s_0}}{s-s_0},\,g_s=\frac{f_s-f_{s_0}}{s-s_0}:\qquad -\Delta w_s=g_s.
$$
Since $s\mapsto f_s\in L^2$ is differentiable you see that $g_s\to g_{s_0}=\partial_s f|_{s_0}$ when $s\to s_0$. By usual continuity of $(-\Delta)^{-1}:L^2\to H^{1}_0$ this implies that the sequence $w_s=(-\Delta)^{-1}g_s$ is Cauchy in $H^1_0$ hence converges to some $w_{s_0}$ in that space, with moreover $-\Delta w_{s_0}=g_{s_0}$. This simply means that the map $s\mapsto u_s\in H^1_0$ is Fréchet differentiable at $s=s_0$ with
$$
\partial_s u|_{s_0}=w_{s_0}=(-\Delta)^{-1}(\partial_s f|_{s_0}).
$$
Now if $s\mapsto f_s\in L^2$ is $C^1$ you see by continuity of $(-\Delta)^{-1}$ that $s\mapsto \partial_s u=(-\Delta)^{-1}(\partial_s f)\in H^1_0$ is continuous, hence $s\mapsto u_s\in H^1_0$ is in fact $C^1$. Repeating the same argument you immediately get that if $s\mapsto f_s\in L^2$ is $C^k$ then $s\mapsto u_s\in H^1_0$ is $C^k$ with
$$
\partial^{(k)}_s u=(-\Delta)^{-1}(\partial^{(k)}_s f).
$$
The same argument applies also for non-homogeneous boundary conditions. Since your problem is linear all you really need is continuity of the solution to the PDE with respect to the data (inhomogeneity + boundary conditions, as long as both are $C^k$ in $s$ for the relevant topologies).
