Regularity of stable solutions up to dimension 7 in domains of double revolution

We consider the class of semi-stable positive solutions to semilinear equations $-\Delta u=f(u)$ in a bounded domain $\Omega\subset\mathbb R^n$ of double revolution, that is, a domain invariant under rotations of the first $m$ variables and of the last $n-m$ variables. We assume $2\leq m\leq n-2$. When the domain is convex, we establish a priori $L^p$ and $H^1_0$ bounds for each dimension $n$, with $p=\infty$ when $n\leq7$. These estimates lead to the boundedness of the extremal solution of $-\Delta u=\lambda f(u)$ in every convex domain of double revolution when $n\leq7$. The boundedness of extremal solutions is known when $n\leq3$ for any domain $\Omega$, in dimension $n=4$ when the domain is convex, and in dimensions $5\leq n\leq9$ in the radial case. Except for the radial case, our result is the first partial answer valid for all nonlinearities $f$ in dimensions $5\leq n\leq 9$.