Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 107 / 2 / a

Assume an interior sphere condition at $y$, and assume the coefficients are bounded and the principal part defines a uniformly elliptic operator near its tangent Euclidean ball. Shrinking the Euclidean ball while keeping it tangent if necessary, arrange that $\overline{B_R(z)}\setminus \{y\}\subset \Omega$. Write $\nu =(y-z)/R$, the outward radial direction. The conclusion of the Hopf boundary point lemma in the regularity stated in the question isIf the one-sided normal derivative exists, this says ${\partial }_{\nu }u(y)>0$. For a $C^1$ boundary its outward normal agrees with the tangent Euclidean ball's radial normal. Continuity alone does not guarantee that the derivative exists, so its existence must be added when asserting a finite derivative.

On the annulus $R/2<r=|x-z|<R$, use the barrierThe operator has no zeroth-order term, and differentiation givesfor sufficiently large $k$: the quadratic-in-$k$ term is at least $k^2\lambda R^2$, while the remaining terms are bounded multiples of $k$.

On the inner sphere, compactness and strict inequality give a positive minimum of $u(y)-u(x)$. Choose $\epsilon >0$ small enough that $u-u(y)+\epsilon h\le 0$ there. The same inequality holds on the outer sphere since $h=0$. The weak maximum principle for elliptic operators yields $u-u(y)+\epsilon h\le 0$ throughout the annulus. Along the inward radius,This proves the lemma. The interior sphere condition supplies precisely the geometry needed for the barrier; no global boundary regularity is used.