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

Use the real zero-trace Sobolev space $H_0^1(\Omega )$ with norm $\Vert \nabla u{\Vert }_2$. The Poincaré inequality on the bounded smooth domain gives $\Vert u{\Vert }_2\le C_P\Vert \nabla u{\Vert }_2$, making this an equivalent Sobolev norm. The weak formulation is to find $u\in H_0^1(\Omega )$ with $u\ge 0$ almost everywhere andThe zero Dirichlet boundary condition is encoded in the Sobolev trace. Formally, the identity follows by multiplying the partial differential equation by a test function and using integration by parts; it then extends to all $v\in H_0^1(\Omega )$ by density.

To check all terms, boundedness of $\Omega$ gives finite Lebesgue measure $m=|\Omega |$. The Cauchy-Schwarz inequality gives $u\in L^1(\Omega )$ andThus the right side is a bounded linear functional of $v$, since its absolute value is at most $\Vert 1+\sqrt{u}{\Vert }_2{C}_P\Vert \nabla v{\Vert }_2$. The left side is finite by the Cauchy-Schwarz inequality. These estimates work for every stated $n\ge 2$.

Let $P=\{w\in L^2(\Omega ):w\ge 0\text{ almost everywhere}\}$. For $g\in L^2(\Omega )$, define $Gg=z\in H_0^1(\Omega )$ byThe Lax-Milgram theorem guarantees a unique $z$: the bilinear form on the left is bounded and coercive in the norm $\Vert \nabla \cdot {\Vert }_2$, and the right side is a bounded linear functional by the Poincaré inequality. Testing with $z$ givesDefine $T:P\to L^2(\Omega )$ by $Tw=G(1+\sqrt{w})$. This actually maps into $P$. Indeed the weak maximum principle says that a weak solution $z\in H_0^1(\Omega )$ of $-\Delta z=g\ge 0$ is nonnegative almost everywhere. Here $1+\sqrt{w}\ge 0$ and belongs to $L^2(\Omega )$. Directly, taking the negative part $z^{-}=\max \{-z,0\}$ as a test function gives $-\Vert \nabla z^{-}{\Vert }_2^2={\int }_{\Omega }g{z}^{-}dx\ge 0$, so $z^{-}=0$ by the Poincaré inequality.

The required fixed point formulation isEvery such fixed point belongs automatically to $H_0^1(\Omega )$ and satisfies part (b)(i). Conversely, uniqueness in the Lax-Milgram theorem makes every nonnegative weak solution a fixed point of $T$.