I am looking for some references that I can look into the stability of PDEs (or ODEs) with respect to boundary data. Suppose $$ \mathcal{L}u_i(x) = f(x),\qquad x \in \Omega $$ and $u_i(x) = g_i(x)$ in $\partial \Omega$, for $i=1,2$.
For time-dependent problem, the linear stability is the notion I am looking for. However, I couldn't find any reference that shows the stability with respect to the boundary data for time-independent problem.
I am looking for some statements saying if $u_i^*$ are the solutions to the above PDEs, $$ \|u_1^* - u_2^*\| \le C \|g_1 - g_2\|, $$ with appropriate norms. If $\mathcal{L}$ is linear, the problem is equivalent to show that the solution to $$ \mathcal{L}u = 0, \quad x \in \Omega, \qquad u(x) = \epsilon(x) \quad x \in \partial \Omega $$ satisfies $\|u^*\| \le C\|\epsilon\|$.
Any answers/suggestions/comments will be very appreciated.