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Nonlinear Analysis

Characterization of f-extremal disks

Jose Espinar: IMPA, Rio de Janeiro

Location:  Hill 705
Date & time: Tuesday, 28 February 2017 at 1:40PM - 1:41PM

In this talk we show uniqueness for overdetermined elliptic problems defined on topological disks with regular boundary, i.e., positive solutions \[u\]to \[ Delta u + f(u)=0\]in \[ Omega subset (M^2,g)\]so that \[u = 0\]and \[ frac{ partial u}{ partial vec eta} = cte \]along \[ partial Omega\], \[ vec eta\]the unit outward normal along \[ partial Omega\]under the assumption of the existence of a candidate family. To do so, we adapt the G'alvez-Mira generalized Hopf-type Theorem to the realm of overdetermined elliptic problem. In particular, this gives a positive answer to the Schiffer conjecture for the first Dirichlet eigenvalue and classifies simply-connected harmonic domains, also called { it Serrin Problem}) in \[ mathbb S ^2\].

This is a joint work with L. Mazet.

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