Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 107 / 2 / d

Seek a radial function $u=u(r)$ on the unit ball. The equation and regularity at the origin giveHence $u^{\prime }(r)=(r/n{)}^{1/(p-1)}$, and the zero Dirichlet boundary condition givesThe power $|x{|}^q$ is twice differentiable at the origin exactly when $q\ge 2$. Here $q=p/(p-1)$, so $u\in C^2$ at the origin exactly when $1<p\le 2$. Under the assumption $2<p<n$, this weak solution is never $C^2$ at the origin, exhibiting the regularity loss caused by degenerate ellipticity.

Solved by gpt-5.6-sol high.