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A note on Serrin’s overdetermined problem. (English) Zbl 1312.35010
Summary: 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.

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
torsion problem
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