Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 107 / 4

The Weak Harnack inequality states that there are $p>0$ and $C=C(n,\lambda ,\Lambda )$ such that every nonnegative weak supersolution satisfiesThe radii may be replaced by any fixed nested pair of balls, with the constant adjusted accordingly.

On a ball $B_{2r}(x_0)⋐B$, bothare nonnegative solutions and hence supersolutions. Combining the Weak Harnack inequality with the local boundedness estimate for subsolutions gives an oscillation decay estimatewhere $\theta$ depends only on $n,\lambda ,\Lambda$. Iteration yields Hölder continuity with some exponent $\mu \in (0,1)$. The local $L^2$-to-$L^{\infty }$ estimate controls the initial oscillation and gives

First replace $u$ by $u+\epsilon$ and later let $\epsilon \downarrow 0$. In the supersolution inequality use the nonnegative test functionwhere $\eta$ is compactly supported. Ellipticity and Young inequality give the Logarithmic Caccioppoli inequalityChoose $\eta =1$ on $B_{\rho }$, supported in $B$, with $|D\eta |\le C(1-\rho {)}^{-1}$. Letting $\epsilon \downarrow 0$ and using Fatou lemma gives

NormalizeThen $0\le v_k\le 1$, and $v_k$ remains a weak supersolution. The ordinary Caccioppoli inequality with a cutoff supported in $B_R$ gives a uniform $W^{1,2}(B_{\rho })$ bound whenever $\rho <R<1$. The Rellich-Kondrachov compactness theorem and a diagonal subsequence therefore give, for every $\rho <1$,The strong convergence preserves nonnegativity. Passing to the limit in the linear supersolution inequality against every nonnegative compactly supported test function shows that $v$ is a weak supersolution in $B$.