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

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

The range of $v\in C^0(\overline{B})$ is compact. Continuity and positivity of $\lambda ,\Lambda$ therefore give constantsThe coefficients ${\alpha }_{ij}(x)=a_{ij}(x,v(x))$ meet the hypotheses of part (a), which gives a unique weak solution. Its global De Giorgi–Nash–Moser estimate gives $u\in C^{0,\mu }(\overline{B})$ for some $\mu =\mu (v)\in (0,1)$.

The Leray-Schauder fixed point theorem says that if $X$ is a Banach space and $T:X\to X$ is continuous and compact, and the setis bounded, then $T$ has a fixed point.

For $v\in C^0(\overline{B})$, let $T(v)$ be the solution from part (i). The weak maximum principle givesOn this fixed bounded range of values, $\lambda$ and $\Lambda$ have common positive lower and finite upper bounds. The global De Giorgi-Nash-Moser theorem therefore bounds $T(v)$ uniformly in one Hölder space. The compact embedding $C^{0,\mu }(\overline{B})\hookrightarrow C^0(\overline{B})$ makes $T$ compact. Uniform convergence $v_k\to v$, the continuity of $a_{ij}$ on the relevant compact set, energy bounds, and uniqueness of the limiting linear problem show that $T(v_k)\to T(v)$ uniformly, so $T$ is continuous.

The same maximum-principle bound controls every solution of the Leray–Schauder homotopy after using boundary data $t\phi$. Thus the homotopy set is bounded, and the Leray-Schauder fixed point theorem supplies $u=T(u)$. By definition,and $u\in W^{1,2}(B)\cap C^0(\overline{B})$.