Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 105 / 4

Extend $u$ by zero to ${\mathbb{R}}^n$; its compact support inside the ball makes the extension smooth. The Fourier transform of a derivative givesThe assumption says that $L_0$ is a uniformly elliptic operator, soMoreover,The Plancherel theorem therefore yieldsAll integrands vanish outside the original support where appropriate, so

Write $Lu=L_{0}u+(a^{ij}-A^{ij})D_{ij}u$, with repeated indices summed. The triangle inequality and the Cauchy-Schwarz inequality over the $n^2$ coefficient pairs giveChooseThen the strict coefficient bound in the question implies

The continuous coefficients are uniformly continuous on a compact neighborhood of $\overline{W}$. For every $x_0\in \overline{W}$, choose a ball $B_r(x_0)⋐U$ small enough thatfor all $i,j$. Part 4(b), with the frozen symmetric matrix $A=(a^{ij}(x_0))$, then applies on this ball.

Choose a finite collection of these balls and a smooth partition of unity $({\eta }_k)$ that sums to one near $\overline{W}$, with each ${\eta }_k$ supported in its corresponding ball. Applying part 4(b) to ${\eta }_{k}u$ givesSince $(a^{ij})$ is symmetric, the Leibniz rule gives the commutator formulaThe coefficients and the finitely many derivatives of the cutoff functions are bounded, soFinally $u={\sum }_k{\eta }_{k}u$ near $\overline{W}$. Summing the finite set of local estimates provesThis is the coefficient-freezing interior second-derivative estimate.

Fix $W⋐V⋐U$. The standard local regularization of the maximal graph norm domain supplies smooth compactly supported approximants $u_k$ on $V$ such thatApply part 4(c), with $V$ as the outer domain, to $u_k-u_m$. It givesThus $(u_k)$ is Cauchy in $H^2(W)$. Its $H^1$ limit is $u$, so $u\in H^2(W)$. Since $W⋐U$ was arbitrary, the definition of a Local Sobolev space givesThis is the Interior H2 regularity for continuous nondivergence coefficients.