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

With only $u\in L^2(\Omega )$ available, use a distributional weak solution: for every test function $\eta \in C_c^{\infty }(\Omega )$ requireAll terms are meaningful because the smooth coefficients are bounded on the compact support of the test function and $u$ is locally integrable. This is exactly $\Delta u={\sum }_i{D}_i(b^{i}u)+cu+f$ as an identity of distributions. No first weak derivative of $u$ is assumed in this definition, and no boundary condition is being imposed.

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.

Choose nested interior open sets $U\subset \subset V\subset \subset W\subset \subset \Omega$. For small $\sigma$, part (b) gives a smooth equation on $V$ of the formThe divergence-forcing interior H1 estimate isOne can obtain this estimate directly by testing the smooth equation against ${\eta }^2{u}_{\sigma }$ and applying Young inequality, with a cutoff function equal to one on $U$ and supported in $V$. The approximate identity and the $L^2$ convolution bound give, uniformly in $\sigma$,with analogous bounds for $(cu{)}_{\sigma }$ and $f_{\sigma }$. The coefficients need only be bounded on $W$. Therefore $u_{\sigma }$ is bounded in $H^1(U)$. It converges to $u$ in $L^2(U)$, and weak sequential compactness in a Hilbert space in this Sobolev space gives $u\in H^1(U)$. Since $U$ was arbitrary, $u\in H_{\mathrm{loc}}^1(\Omega )$.

Now expand the distributional derivative:Its right side belongs to $L_{\mathrm{loc}}^2$, so the interior $H^2$ elliptic regularity estimate gives $u\in H_{\mathrm{loc}}^2$. More generally, if $u\in H_{\mathrm{loc}}^m$ for an integer $m\ge 1$, multiplication by the smooth coefficients puts the right side in $H_{\mathrm{loc}}^{m-1}$. Interior elliptic regularity then gives $u\in H_{\mathrm{loc}}^{m+1}$. This elliptic regularity bootstrap proves $u\in H_{\mathrm{loc}}^k$ for every integer $k$.

For any nonnegative integer $\ell$, choose $k>\ell +n/2$. The Sobolev embedding theorem on compact interior subsets gives $u\in C_{\mathrm{loc}}^{\ell }$. Taking all $\ell$ proves $u\in C^{\infty }(\Omega )$, for its smooth representative. The first gain from $L^2$ to $H^1$ is the step supplied by the mollified equation; assuming $H^1$ in the initial definition would miss that step.