Choose a <cutoff function> $\eta$ supported in $B_\rho(x)$, equal to one on $B_{\rho'}(x)$, and satisfying $|D\eta|\leq(\rho-\rho')^{-1}$. Use $\eta^2u$ as a <test function> in the <weak formulation>:
$$
0=\int D u\mathbin\cdot D(\eta^2u)
=\int\eta^2|Du|^2+2\int\eta u,Du\mathbin\cdot D\eta.
$$
The <Cauchy-Schwarz inequality> followed by <Young inequality> gives
$$
\int\eta^2|Du|^2\leq4\int u^2|D\eta|^2,
$$
and hence
$$
\int_{B_{\rho'}(x)}|Du|^2
\leq\frac{4}{(\rho-\rho')^2}\int_{B_\rho(x)}|u|^2.
$$
This is the <Caccioppoli inequality>. The numerical constant is convention-dependent and is normally absorbed into the displayed estimate; replacing the radii by fixed intermediate radii gives the stated form with one universal constant.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 107 / 1 / b / i / Solution

 Solution

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