Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 9 / 6 / b / Solution

Fix $x$ with $\mathrm{dist}(x,\partial \Omega )>\sigma$ and use ${\eta }_x(y)={\varphi }_{\sigma }(x-y)$ as the test function in part (a). Its support lies compactly inside $\Omega$. Differentiating the convolution under the integral is allowed, and $D_{y_i}{\eta }_x=-D_{x_i}{\varphi }_{\sigma }(x-y)$ while ${\Delta }_y{\eta }_x={\Delta }_x{\varphi }_{\sigma }(x-y)$. ConsequentlyThusat every stated interior point. This is the convolution derivative identity. The convolutions are local, so no integrability of the smooth coefficients all the way to the boundary is needed. In particular, $(b^{i}u{)}_{\sigma }$ is the convolution of the product; it must not be replaced by $b^i{u}_{\sigma }$ for a variable coefficient.