= Solution
The preceding estimate gives constants $A,B>0$ such that, whenever $\|w\|_{H^2}\leq R$,
$$
\|\Phi(w)\|_{H^2}\leq A\|f\|_2+BR^3.
$$
Choose $R>0$ so that $BR^2\leq1/2$, and then choose $r>0$ so that $Ar\leq R/2$. If $\|f\|_2\leq r$, the closed ball $B_R(0)$ is mapped into itself.
For $w,z\in B_R(0)$, factor the <cubic gradient nonlinearity> as
$$
|Dw|^2w-|Dz|^2z
=|Dw|^2(w-z)+(Dw+Dz)\mathbin\cdot D(w-z),z.
$$
The same <Sobolev embedding theorem> and the <Holder inequality> imply
$$
\||Dw|^2w-|Dz|^2z\|_2
\leq CR^2\|w-z\|_{H^2}.
$$
The elliptic estimate therefore yields
$$
\|\Phi(w)-\Phi(z)\|_{H^2}
\leq C'R^2\|w-z\|_{H^2}.
$$
Shrinking $R$ further makes $C'R^2<1$, so $\Phi$ is a <contraction mapping> of the closed ball. This ball is complete because $H^2(U)\cap H_0^1(U)$ is a <Banach space>. The <contraction mapping theorem> gives a fixed point $u=\Phi(u)$, and its defining equation is
$$
-\Delta u-|Du|^2u=f,qquad u|_{\partial U}=0.
$$
Thus the nonlinear <elliptic boundary value problem> has a solution for sufficiently small $\|f\|_2$.
Back to article page
