Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 107 / 3

Assume $c\le 0$, bounded coefficients and uniform ellipticity near the boundary point, an interior tangent ball there, and enough regularity for the boundary normal derivative. The Hopf boundary point lemma says that a nonnegative maximum at $x_0$ with strictly smaller values inside the tangent ball satisfiesfor the outward unit normal. If $c=0$, the maximum can have either sign. Without existence of a classical derivative, one states a strict one-sided derivative bound.

The PDF also asks to deduce the strong maximum principle for elliptic operators; that request is missing from the TeX transcription. Suppose $v$ attains a nonnegative global maximum $M$ inside a connected domain but is not constant. A ball in $\{v<M\}$ can be chosen relatively compact in the domain and tangent to $\{v=M\}$ at an interior point $z$: use the distance from a nearby point to the closed maximum set. The Hopf boundary point lemma on that ball gives a nonzero outward derivative at $z$, contradicting $Dv(z)=0$ at an interior maximum. Therefore $v$ is constant. When $c=0$, the same proof works for a maximum of either sign.

For zero Dirichlet boundary condition, the global Schauder estimate isThe constant depends on $n,\alpha$, domain regularity, the ellipticity lower bound, and coefficient $C^{0,\alpha }$ norms. No sign restriction on $c$ is needed. Nonzero boundary data contribute the $C^{2,\alpha }$ norm of an extension.

Nonnegativity lets us dispense with a sign restriction on $c$. Set $c_0=\min (c,0)$ and $L_0=a^{ij}{D}_{ij}+b^i{D}_i+c_0$. ThenThe strong minimum principle for elliptic operators forces $u$ to vanish identically if it has an interior zero. Otherwise it is strictly positive:Connectedness is part of the usual domain hypothesis; on a disconnected open set this conclusion applies separately to each component.

Take ${\Omega }^{\prime }$ nonempty. If the bound failed, normalization would give nonnegative $v_j$ with supremum one, zero boundary values, ${\inf }_{{\Omega }^{\prime }}{v}_j\to 0$, and $\Vert L{v}_j{\Vert }_{C^{0,\alpha }}\to 0$. The global Schauder estimate bounds them in the Hölder space $C^{2,\alpha }$. By compact embedding of Hölder spaces, a subsequence converges in $C^{2,\beta }$, $0<\beta <\alpha$, to $v\ge 0$ with $Lv=0$, zero boundary data and supremum one.

Points approaching the infima have a subsequence converging in $\overline{{\Omega }^{\prime }}\subset \Omega$. Uniform convergence gives an interior zero of $v$, and part (c), the strong minimum principle for elliptic operators after replacing $c$ by $\min (c,0)$, gives $v=0$. This contradicts its supremum one, proving