| Abstract: |
| In the field of PDEs, it is frequently observed that solutions whose Morse index is small have a simpler shape in comparison to others. I report on a recent work with Francesca Gladiali from the University of Sassari (Italy) concerning a mixed boundary-value problem for a semilinear elliptic equation with a convex nonlinearity in a sector-like domain. Using cylindrical coordinates $(r, \theta, z)$, we investigate the shape of possibly sign-changing solutions whose derivative in $\theta$ vanishes at the boundary. We prove that any solution whose Morse index does not exceed 1 must either be axially symmetric (i.e., radial in the plane), or strictly monotone with respect to $\theta$. The proof is based on a rotating-plane argument. Classical references on Morse theory are those of Bott, Milnor, and Morse. More specific background on the talk's subject is found in papers by Gladiali, Pacella, Weth and in the book by Damascelli and Pacella. |
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