Abstract
In this paper, we prove the hot spots conjecture for long rotationally symmetric domains in Rn by the continuity method. More precisely, we show that the odd Neumann eigenfunction in xn 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.
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