Elliptic residual and boundary stability from a maximum principle

Since $U$ is bounded, choose real numbers $a_1<b_1$ with $a_1\le x_1\le b_1$ on $\bar{U}$. Let $B=\mathrm{\parallel }{\tilde{b}}_1{\mathrm{\parallel }}_{C^0(U)}$ and $C_0=\mathrm{\parallel }\tilde{c}{\mathrm{\parallel }}_{C^0(U)}$. Choose $\kappa >0$ large enough that ${\kappa }^2{\lambda }_0-\kappa B-C_0\ge 1$. For $\psi (x)=e^{\kappa x_1}$, [F1] gives $L[\psi ](x)=e^{\kappa x_1}({\kappa }^2{a}_{11}(x)+\kappa {\tilde{b}}_1(x)+\tilde{c}(x))\ge e^{\kappa a_1}>0.$ Choose $M>e^{\kappa b_1}$ and set $\mathrm{\Phi }=M-\psi$. Then $\mathrm{\Phi }>0$ on $\bar{U}$ and $L[\mathrm{\Phi }]=M\tilde{c}-L[\psi ]\le -e^{\kappa a_1}$.

Put $R=\mathrm{\parallel }L[w]{\mathrm{\parallel }}_{C^0(U)}$, $Q=\mathrm{\parallel }w{\mathrm{\parallel }}_{C^0(\mathrm{\partial }U)}$, and $\eta =e^{\kappa a_1}$. For $z=w-Q-(R\mathrm{/}\eta )\mathrm{\Phi }$, the boundary values satisfy $z\le 0$. In $U$, $L[z]=L[w]-Q\tilde{c}-(R\mathrm{/}\eta )L[\mathrm{\Phi }]\ge -R+R=0,$ because $-Q\tilde{c}\ge 0$.

Suppose $z(x_0)>0$ at some $x_0\in U$. Since $z\le 0$ on $\mathrm{\partial }U$ and $z$ is continuous on $\bar{U}$, its positive maximum is attained at an interior point. For sufficiently small $\delta >0$, $z+\delta \psi$ has an interior positive maximum as well: its value at $x_0$ is at least $z(x_0)>0$, while its boundary values are at most $\delta e^{\kappa b_1}<z(x_0)$. At that interior maximum [F2] and $\tilde{c}\le 0$ give $L[z+\delta \psi ]\le 0$, contradicting $L[z+\delta \psi ]>0$. Hence $z\le 0$ on $U$, and therefore $w\le Q+(R\mathrm{/}\eta )\mathrm{\parallel }\mathrm{\Phi }{\mathrm{\parallel }}_{C^0(\bar{U})}\mathrm{.}$ Applying the same argument to $-w$ yields the corresponding lower bound. Taking $C_L=\max \{1,\mathrm{\parallel }\mathrm{\Phi }{\mathrm{\parallel }}_{C^0(\bar{U})}\mathrm{/}\eta \}$ proves the first estimate; applying it to $u^{\ast }-h$ proves the second.