Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 107 / 5 / a / Solution

Adopt the sign convention that a weak supersolution for $Lu=D_i(a^{ij}{D}_{j}u)$ satisfies $Lu\le 0$, or $\int a^{ij}{D}_{j}u{D}_i\phi \ge 0$ for every nonnegative compactly supported test function. For a nonnegative supersolution, the Weak Harnack inequality gives some $q>0$ and $C_H\ge 1$, depending only on $n,\lambda ,\Lambda$, such thatHere $|E{|}^{-1}{\int }_E$ denotes the average over $E$; equivalently the right side gives a lower bound for the essential infimum. One may take a sufficiently small positive exponent, so no assumption $q\ge 1$ is needed. The inequality is unchanged under translations and dilations of the Euclidean balls.

Since a Sobolev space element is defined up to changes on a null set, the infimum is initially essential. After choosing a continuous representative it is the ordinary infimum. Without choosing such a representative, an arbitrary pointwise infimum cannot be bounded by an integral.