Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 105 / 2 / f

Here is a direct use of the spherical representation that also avoids estimating separate coordinate derivatives at the poles. Put $A={\Delta }_{S^2}$, $T={\partial }_r^2+2r^{-1}{\partial }_r$, and write $d\omega$ for unit-sphere area. Then $f=Tu+r^{-2}Au$. Expanding its squared norm with the physical measure givesOn each boundary sphere $u=0$, so every angular derivative, including $Au$ and ${\nabla }_{S}u$, is zero there. Radial and angular integration by parts consequently giveIndeed the $u_{rr}$ term gives $\int |{\nabla }_S{u}_r{|}^2$ with boundary remainder $[\int u_{r}Au]=0$, while the $2u_r/r$ term is $-\int r^{-1}{\partial }_r|{\nabla }_{S}u{|}^2=-\int r^{-2}|{\nabla }_{S}u{|}^2$, its boundary remainder also being zero. Thuswhere the last step uses the preceding $H^1$ energy estimate and $r\ge 1/2$. All unmarked integrals in these two identities are with $drd\omega$.

To control every angular second derivative, the spherical Hessian identity isIt follows by integrating the derivative-commutation identity ${\nabla }^a{\nabla }_a{\nabla }_{b}w={\nabla }_{b}Aw+{\mathrm{Ric}}_b{}^c{\nabla }_{c}w$ and using $\mathrm{Ric}=h$ on the unit sphere. There is no sphere boundary. Hence the preceding bound controls ${\nabla }_S^{2}u$, ${\nabla }_S{u}_r$, and $u_{rr}=Tu-2u_r/r$ in their appropriate weighted norms.

In an orthonormal polar frame the Cartesian Hessian matrix has componentsand $D^{2}u(e_A,e_B)=r^{-2}({\nabla }_S^{2}u{)}_{AB}+r^{-1}{u}_r{\delta }_{AB}$. Since $r$ is bounded above and below, these bounds plus the first-order estimate proveThis is the direct spherical-shell H2 estimate. Intrinsic angular integration combines the suggested angular tests, with no spurious boundary at the polar coordinate singularities.