## Kodai Mathematical Journal

- Kodai Math. J.
- Volume 37, Number 3 (2014), 728-736.

### A note on Serrin's overdetermined problem

Giulio Ciraolo and Rolando Magnanini

#### Abstract

We consider the solution of the torsion problem

−Δ*u* = *N* in Ω, *u* = 0 on ∂Ω,

where Ω is a bounded domain in **R**^{N}.

Serrin's celebrated symmetry theorem states that, if the normal derivative *u*_{ν} is constant on ∂Ω, then Ω must be a ball. In [6], 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} ≤

for some constant *C*_{t} depending on *t*, where *r*_{e} and *r*_{i} are the radii of an annulus containing ∂Ω and Γ_{t} is a surface parallel to ∂Ω at distance *t* and sufficiently close to ∂Ω secondly, if in addition *u*_{ν} is constant on ∂Ω, show that

= *o*(*C*_{t}) as *t* → 0^{+}.

The estimate constructed in [6] 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 Ω are ellipses.

#### Article information

**Source**

Kodai Math. J., Volume 37, Number 3 (2014), 728-736.

**Dates**

First available in Project Euclid: 30 October 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.kmj/1414674618

**Digital Object Identifier**

doi:10.2996/kmj/1414674618

**Mathematical Reviews number (MathSciNet)**

MR3273893

**Zentralblatt MATH identifier**

1312.35010

#### Citation

Ciraolo, Giulio; Magnanini, Rolando. A note on Serrin's overdetermined problem. Kodai Math. J. 37 (2014), no. 3, 728--736. doi:10.2996/kmj/1414674618. https://projecteuclid.org/euclid.kmj/1414674618