Seminars & Colloquia Calendar

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.