Nausica Aldeghi: Inequalities between mixed Dirichlet–Neumann eigenvalues of the Laplacian
Time: Wed 2021-10-20 13.15 - 14.15
Lecturer: Nausica Aldeghi (SU)
Laplace eigenvalues can be interpreted as frequencies of vibrating membranes: intuitively, mixed Dirichlet–Neumann eigenvalues mean that the membrane is only partially attached. By a variational argument, enlarging the attached portion increases the corresponding frequencies. However, in general, these frequencies do not depend monotonously on the length of the attached portion.
In this seminar, we investigate the monotonicity of these frequencies: given the Laplacian on a bounded convex plane domain with piecewise smooth boundary, we prove an inequality between the lowest eigenvalues obtained by imposing a Dirichlet boundary condition on different smooth parts of the boundary and a Neumann boundary condition on the corresponding remainders of the boundary. We show how this inequality depends heavily on the geometry of the domain, and in particular on the shape of the single smooth parts of the boundary and the value of the angles between them. We also see how this result applies to polygons, and in particular to triangles, giving a partial answer to a recent conjecture raised by Siudeja.