A note on Serrin’s overdetermined problem.

*(English)*Zbl 1312.35010Summary: We consider the solution of the torsion problem
\[
-\Delta_u = N \text{ in } \Omega, \quad u = 0 \text{ on } \partial \Omega,
\]
where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\). Serrin’s celebrated symmetry theorem states that, if the normal derivative \(u_v\) is constant on \(\partial \Omega\), then \(\Omega\) must be a ball. In [G. Ciraolo, R. Magnanini and S. Sakaguchi, “Solutions of elliptic equations with a level surface parallel to the boundary: stability of the radial configuration”, Preprint, arXiv:1307.1257], it has been conjectured that Serrin’s theorem may be obtained by stability in the following way: first, for the solution \(u\) of the torsion problem prove the estimate
\[
r_e - r_i \leq C_t\Bigl(\max_{\Gamma_t} u-\min_{\Gamma_t} u\Bigr)
\]
for some constant \(C_t\) depending on \(t\), where \(r_e\) and \(r_i\) are the radii of an annulus containing \(\partial \Omega\) and \(\Gamma_t\) is a surface parallel to \(\partial \Omega\) at distance \(t\) and sufficiently close to \(\partial \Omega\) secondly, if in addition \(u_v\) is constant on \(\partial \Omega\), show that
\[
\max_{\Gamma_t} u-\min_{\Gamma_t} u = o(C_t) \quad \text{ as } t \to 0^{+}.
\]
The estimate constructed in [loc.cit.] is not sharp enough to achieve this goal. In this paper, we analyse a simple case study and show that the scheme is successful if the admissible domains \(\Omega\) are ellipses.

##### MSC:

35B06 | Symmetries, invariants, etc. in context of PDEs |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

35J61 | Semilinear elliptic equations |

35N05 | Overdetermined systems of PDEs with constant coefficients |