Past exam of the functional analytic methods for partial differential equations course of the University of Oxford / 2023 / 3 / a / iii

By the definition in (i), $u$ itself is an admissible test in the weak formulation. HenceThe absolute value makes a proof of positivity unnecessary. Holder inequality with exponents $6$ and $6/5$, followed by the zero-boundary Sobolev inequality, givesTaking square roots of both estimates proves exactly the requested chain:If $\Vert \nabla u{\Vert }_2=0$, the zero-boundary Poincaré inequality gives $u=0$ and the conclusion follows. Otherwise divide the first-to-last estimate by $\Vert \nabla u{\Vert }_2^{1/2}$ and square, obtaining $\Vert \nabla u{\Vert }_2\le C_S|E{|}^{5/6}$. Finally the same Poincaré inequality givesAll constants depend only on the fixed unit ball and the dimension, and are independent of $E$ and $u$. If $|E|=0$, testing the homogeneous equation with $u$ gives the same zero solution directly.