Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 107 / 5 / a / Solution

Write $u=\phi +w$ with $w\in W_0^{1,2}(B)$. On this space defineBoundedness of the coefficients makes $\mathcal{B}$ bounded, and ellipticity giveswhich is coercive by the Poincare inequality. The functional $z\mapsto -\mathcal{B}(\phi ,z)$ is bounded, so the Lax-Milgram theorem gives a unique $w$ and hence a unique weak solution $u$.

The supplied global De Giorgi-Nash-Moser theorem and the weak maximum principle give $u\in C^{0,\beta }(\overline{B})$ for some $\beta =\beta (n,\gamma ,\Gamma )$. Finally, test the weak equation with $u{\psi }^2$. Ellipticity, the coefficient bound, and Young inequality yieldRearranging gives