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 $-\Delta u=f_s$ 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 $s\in (0,+\infty)\rightarrow u_s\in X$, 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
Yes, if the solution operator is continuous from (whatever space $f_s$ live in) to $X$.
Let's say $f_s$ belong to some function space $Y$. Assume $s\to f_s$ is $C^2$ smooth, in the sense that
$$f_{s+h} = f_s+hp_s+h^2q_s+o(h^2)\tag1 $$ where $f_s$, $p_s$, $q_s$ are all norm-continuous with respect to $s$, and $o$ is with respect to the norm in $Y$. The Poisson equation is linear, and its solution is given by the convolution with Green's function $G$. Therefore, we convolve (1) with $G$ to obtain $$ u_{s+h} = G*f_{s+h} = u_s+h G*p_s+h^2 G*q_s+G*o(h^2) \tag2 $$ By assumption, convolution with $G$ is continuous from $Y$ to $X$. Hence, $$ u_{s+h} = u_s+h P_s+h^2 Q_s+ o(h^2) \tag3 $$ where $P_s$ and $Q_s$ are norm-continuous functions into $X$, and $o$ is with respect to the norm in $X$.