| Abstract: |
| In this talk, we will discuss the hot spots conjecture for n-axes symmetric domain
in $R^{n}$ by continuity method. More precisely, we show that the
odd Neumann eigenfunctions with lowest nonzero eigenvalues are the
Morse functions on the boundary that have exactly two
non-degenerate critical points and the eigenfunctions are
monotone in the direction from minimum point to maximum point. As
a consequence, we show that hot spots conjecture holds for such
kind of domains provided that the second Neumann eigenvalue is simple.
And we also settle the Jerison and Naridashvili`s conjecture
for the domains with n-axes of symmetry or hyperbolic drum.
Finally, associated with odd Neumann eigenfunctions we obtain a sharp estimate for nonzero least Neumann eigenvalue. |
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