Research lectures, seminars and events
The events listed in this area are research seminars, workshops and lectures hosted by Durham University departments and research institutes. If you are not a member of the University, but wish to enquire about attending one of the events please contact the organiser or host department.
Analysis and/of PDE: Courant-sharp Robin eigenvalues for the square
Let $\Omega$ be a planar, bounded, connected, open set with Lipschitz boundary. Let $u$ be an eigenfunction of the Laplacian on $\Omega$ with either a Dirichlet, Neumann or Robin boundary condition. We are interested in the number of nodal domains of $u$.
If an eigenfunction $u$ associated with the $k$--th eigenvalue has exactly $k$ nodal domains, then we call it a Courant-sharp eigenfunction. In this case, we call the corresponding eigenvalue Courant-sharp.
The Courant-sharp Dirichlet, respectively Neumann, eigenvalues of the square are known due to Pleijel, B\'erard--Helffer, respectively Helffer--Persson-Sundqvist.
We discuss whether the Robin eigenvalues of the square are Courant-sharp.
This is based on joint work with B. Helffer (Universit\'e de Nantes).
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