## Method of continuity and second order elliptic equation

$\textbf{Thm:}$ Suppose $L_t: \mathfrak{B}\to \mathfrak{V}$ is a family of linear operators for $t\in[0,1]$ between two banach spaces. And $L_t$ is continuous for $t\in[0,1]$. If

(1) (closedness)$\displaystyle ||L_tx||_{\mathfrak{V}}\leq C||x||_{\mathfrak{B}}$, $x\in \mathfrak{B}$, Or namely $L_t$ is uniformly bounded.

(2)(openness) $\displaystyle C||L_tx||_{\mathfrak{V}}\geq ||x||_{\mathfrak{B}}$. Or $L_t$ is invertible.

Here $C$ is independent of $t,x$. Then $L_0$ is onto $\mathfrak{B}$ to $\mathfrak{V}$ if and only if $L_1$ is onto.

How can we use this theorem? Well, consider the Dirichlet problem of Laplacian equation

$\displaystyle \begin{cases}\Delta u=f \text{ in }\Omega\\ u=0 \text{ on }\partial \Omega\end{cases}$

is solvable for under some condition of $f$ and $\Omega$. For the general second order elliptic equation

$Lu=\displaystyle \begin{cases}a^{ij}(x) u_{ij}(x)+b^i(x)u_i(x)+c(x)u=f \text{ in }\Omega\\ u=0 \text{ on }\partial \Omega\end{cases}$

is related to the Dirichlet problem by defining $L_t=(1-t)\Delta+tL$.

Let us assume $\partial \Omega\in C^{2,\alpha}$. $\mathfrak{B}=\{u\in C^{2,\alpha}(\overline{\Omega})|u=0 \text{ on }\partial \Omega\}$, $\mathfrak{V}=\{u\in C^{\alpha}(\overline{\Omega})\}$ and

(I) all the coefficients in $L$ are in $\mathfrak{V}$

(II) $L$ is strictly elliptic

(III) $c\leq 0$

Apparently, (1) is satisfied by $L_t$. For (2), $||x||_{\mathfrak{B}}\leq C||L_tx||_{\mathfrak{V}}$ is equivalent to the solution of $L_tu_t=f$ satisfies

$\displaystyle ||u||_{2,\alpha}\leq C||f||_{\alpha}$

this is guaranteed by the schauder estimate.

So we have the following theorem

$\textbf{Thm:}$ $L$ and $\Omega$ satisfies (I)-(III). Then the problem

$Lu=f$ in $\Omega$ and  $u=0$ on $\partial \Omega$

having a solution in $\mathfrak{B}$ is equivalent to

$\Delta u=f$ in $\Omega$ and  $u=0$ on $\partial \Omega$

having a solution in $\mathfrak{B}$.