Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 107 / 3 / a / Solution

The global Schauder estimate on a bounded $C^{2,\alpha }$ domain isHere $C$ depends on $n,\alpha$, the domain's boundary regularity and $\Vert c{\Vert }_{C^{0,\alpha }}$, but not on the particular solution or data. The $C^0$ term is needed before uniqueness has been established. The given $\psi$ is an extension of the boundary data to the closure.

For $c\le 0$, choose $d$ with $\Omega \subset \{|x_1|<d\}$ and set $M={\sup }_{\partial \Omega }|\psi |$, $F={\sup }_{\Omega }|f|$. The hinted functionis nonnegative, dominates $|\psi |$ on the boundary, and satisfiessince $x_1+d>0$. Therefore $(\Delta +c)(u-b)=f-(\Delta +c)b\ge 0$ and $(\Delta +c)(-u-b)=-f-(\Delta +c)b\ge 0$. Both functions are nonpositive on the boundary. The weak maximum principle for elliptic operators gives $|u|\le b$, hence the supremum norm barrier for an elliptic Dirichlet problem:Unlike the full Schauder estimate, this last constant depends only on a slab containing $\Omega$, and is independent of how negative $c$ is.