= Monotone iteration for a semilinear elliptic equation
{title2=$\Delta u_{k+1}=V(u_k)$}
For smooth non-increasing $V$ and an <ordered subsolution and supersolution>, begin with $u_0=\varphi^-$ and solve successive <Poisson equations> with forcing $V(u_k)$ and the fixed boundary data. The <weak maximum principle for elliptic operators> applied to the negatives of consecutive differences gives
$$
\varphi^-\leq u_1\leq u_2\leq\cdots\leq\varphi^+.
$$
The <global Schauder estimate> and the <Hölder interpolation inequality> bound the iterates uniformly in $C^{2,\alpha}$. Their bounded <monotone sequence> converges, and <compact embedding of Hölder spaces> permits passing to the equation. The uniform <Hölder seminorm> bounds retain $C^{2,\alpha}$ regularity of the limit.
Back to article page
