Inequalities Between Neumann and Dirichlet Laplacian Eigenvalues on Planar Domains

Explore a seminar on spectral geometry that delves into inequalities between Neumann and Dirichlet Laplacian eigenvalues on planar domains. Learn about the generalization of Payne's classical inequality from 1955, which states that below the k-th eigenvalue of the Dirichlet Laplacian, there exist at least k+2 eigenvalues of the Neumann Laplacian for convex domains. Discover how this theorem has been extended to all simply connected planar Lipschitz domains, supporting a long-standing conjecture. Gain insights into the novel variational principle used in the proof and its implications for spectral geometry. Examine the connections to Lie-Hamilton systems on the plane and their applications in differential equations.