= Parameter-dependent boundary data in a regular perturbation hierarchy
{title2=$L_0u_n=f_n-L_1u_{n-1},\quad Bu_n=g_n$}
For $(L_0+\epsilon L_1)u=f(\epsilon)$ and $Bu=g(\epsilon)$, expand all three quantities in powers of $\epsilon$. The order-$n$ problem has $L_0u_n=f_n-L_1u_{n-1}$ and $Bu_n=g_n$, with $u_{-1}=0$. In particular, <Taylor series> boundary coefficients are generally nonzero even when the differential operator depends linearly on the parameter. If the homogeneous realization of $L_0$ is invertible and boundary data admit smooth lifts, the same inverse solves each successive inhomogeneous <boundary value problem>. Induction establishes unique coefficients at every order. This formal conclusion does not alone assert convergence of the full <power series>; convergence needs additional bounds on the inverse and the perturbation.
Back to article page
