Method of continuity

The method of continuity transfers invertibility along a continuous family of bounded linear operators between fixed Banach spaces. Let $L_t:X\to Y$ be continuous in operator norm for $0\le t\le 1$, let $L_0$ be invertible, and assume the uniform estimateThen every $L_t$ is invertible. The estimate gives injectivity and bounds any inverse by the same $C$.

For openness, if $L_s$ is invertible and $\Vert L_s^{-1}(L_t-L_s)\Vert <1$, a Neumann series inverts $I+L_s^{-1}(L_t-L_s)$. For closedness, suppose invertible $L_{t_j}$ approach $L_t$. For fixed $f\in Y$, put $u_j=L_{t_j}^{-1}f$. The uniform bound gives $\Vert u_j{\Vert }_X\le C\Vert f{\Vert }_Y$, and applying the estimate to $u_j-u_k$ givesThus $u_j$ converges in $X$ and its limit solves $L_{t}u=f$. The nonempty invertible parameter set is both open and closed in the connected interval.

For an elliptic boundary value problem with Dirichlet boundary conditions, one often uses $L_t=(1-t)\Delta +tL$ on the fixed space $C^{2,\alpha }$ with zero boundary values and target $C^{0,\alpha }$. A uniform global Schauder estimate together with a uniqueness/supremum estimate supplies the required bound. An estimate that still contains an uncontrolled solution norm on its right side is insufficient.^[ref]