| Abstract: |
| In this talk, we prove the hot spots conjecture for rotationally symmetric domains in $\mathbb{R}^{n}$
by the continuity method. More precisely, we show that the odd Neumann eigenfunction in $x_{n}$ associated with lowest nonzero
eigenvalue is a Morse function on the boundary, which has exactly two critical points and is monotone in the direction
from its minimum point to its maximum point. As a consequence, we prove that the Jerison and Nadirashvili`s conjecture 8.3 holds
true for rotationally symmetric domains and are also able to obtain a sharp lower bound for the Neumann eigenvalue. We will also discuss
some recent results on n-axes symmetry or hyperbolic drum type domains. |
|