**Question 1:** It appears that when studying an elliptic equation $Lu=f$ in $\Omega$ with $u = g$ on $\partial \Omega$ we need to have $g=0$ in order that the inverse operator, $K=L^{-1}$ is linear. Otherwise $K(f_1+f_2) \neq Kf_1 + Kf_2$.

Is this inideed the case? Does it make any sense at all to speak of the "spectrum" of $L$ on $\Omega$ with respect to the boundary condition $g$?

I'm trying to understand when a maximum principle holds for $-\Delta u - \epsilon u$ on a domain $\Omega$. If I fix Dirichlet boundary conditions then for $\epsilon$ small enough I will have only the trivial $u \equiv 0$ solution. In some sense then, a maximum principle holds for the operator $-\Delta - \epsilon I$. However I can't seem to say anything about a general maximum principle here since my $\epsilon$ depended on my Dirichlet boundary conditions.

**Question 2:** Does a maximum principle still hold for $-\Delta u - \epsilon u$ on some arbitrary domain $\Omega$ when $\epsilon$ is small enough?