Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 9 / 3

Assume the bounded set $\Omega$ is measurable, as needed to define its indicator in $L^q$. Put $w_j=f_j-f$ and choose a smooth cutoff function $\eta$ of compact support equal to one near $\Omega$. Weak convergence in the H1 space gives a uniform $H^1$ bound, by the Uniform boundedness principle, and therefore $u_j=\eta w_j$ is uniformly bounded in $H^1$ with fixed compact support.

Here is a direct Fourier proof of the local compactness in the Rellich-Kondrachov compactness theorem. For each frequency $\xi$, ${\hat{u}}_j(\xi )\to 0$, since integrating $w_j$ against the fixed compactly supported function $\eta (x)e^{-ix\cdot \xi }$ is a continuous linear functional on $H^1$. Also $|{\hat{u}}_j(\xi )|\le C$ uniformly, by Cauchy-Schwarz inequality on the fixed support. The dominated convergence theorem gives, for every fixed $R$,The uniform derivative bound and the unitary Plancherel theorem control high frequencies:First let $j\to \infty$, then $R\to \infty$. This proves $\Vert u_j{\Vert }_2\to 0$, and hence $\Vert {\chi }_{\Omega }{w}_j{\Vert }_2\to 0$.

Let $2^{\ast }=2n/(n-2)$. The Sobolev embedding theorem bounds $\Vert u_j{\Vert }_{2^{\ast }}$ uniformly. For $2<q<2^{\ast }$, choose $\theta \in (0,1)$ with $1/q=\theta /2+(1-\theta )/2^{\ast }$. Interpolation of Lp norms gives $\Vert u_j{\Vert }_q\le \Vert u_j{\Vert }_2^{\theta }\Vert u_j{\Vert }_{2^{\ast }}^{1-\theta }\to 0$. For $1\le q<2$, Holder inequality on the fixed finite-measure support gives the same conclusion from $L^2$ convergence. ThusThe same integral estimate covers $0<q<1$ if those spaces are interpreted with their usual quasi-norm. No conclusion at the critical exponent is asserted.

The appropriate Poincare-Wirtinger inequality, with no zero-boundary assumption, isAssume $\Omega$ is nonempty, bounded, connected and has the stated cone property. If this failed, after subtracting the mean and normalizing there would be $u_j\in W^{1,p}(\Omega )$ withThe sequence is bounded in $W^{1,p}$. By the permitted Rellich-Kondrachov compactness theorem, a subsequence converges strongly in $L^p$ to $u$. For every compactly supported test function, integration by parts and these two convergences show that each distributional derivative of $u$ is zero. A function with zero weak gradient is constant on a connected open set: mollification makes it locally constant, and connectedness identifies the local constants.

Convergence in $L^p$ on a finite-measure set also preserves the integral, so $u_{\Omega }=0$ and the constant is zero. But strong convergence preserves $\Vert u{\Vert }_p=1$, a contradiction. This proves the inequality; connectedness prevents different constants on separate components.