Past exam of the fixed point methods for nonlinear partial differential equations course of the University of Oxford / 2023 / 2 / c / Solution

We apply the closed-bounded-domain version of the Schauder fixed-point theorem in the Banach space $L^2(\Omega )$. Write $m=|\Omega |$ and use $G,T,C_P$ from part (b)(ii). ChooseThe set $C$ is nonempty, bounded, closed and a convex set. For closedness of its positivity condition, if $w_j\ge 0$ and $w_j\to w$ in $L^2$, then pointwise $w^{-}\le |w-w_j|$, whence $\Vert w^{-}{\Vert }_2\to 0$.

The weak maximum principle already gives $Tw\ge 0$. Our estimates give, for $w\in C$,Thus $T(C)\subset C$.

To prove continuity, use the pointwise square root estimate $|\sqrt{s}-\sqrt{t}{|}^2\le |s-t|$ for $s,t\ge 0$. The Cauchy-Schwarz inequality then givesHence $T$ is a continuous map on $C$, even though the square root need not have a bounded derivative near zero.

Finally,The Rellich-Kondrachov theorem says that $H_0^1(\Omega )$ embeds compactly into $L^2(\Omega )$ on a bounded smooth domain. For $n>2$, $2<2n/(n-2)$, and for $n=2$ every finite target exponent is available, so the required compact embedding holds in every dimension in the question. Thus $T(C)$ is relatively compact in $L^2(\Omega )$.

All hypotheses of the Schauder fixed-point theorem are met. It gives $u=Tu\in C$, and part (b)(ii) converts this fixed point into the required weak solution:Only existence is needed. The construction does not assume that $\Omega$ is small or that $T$ is a contraction mapping.