Spectral geometry of the magnetic $p$-Laplacian
We investigate the interplay between the geometry of a planar domain and the spectrum of the nonlinear $p$-Laplacian, subject to a homogeneous magnetic field and Dirichlet boundary conditions. First, we establish a lower bound for the bottom of the spectrum on the whole plane, extending the classical Landau-level estimate to the nonlinear regime. Second, we prove Pólya-type bounds for the first eigenvalue on bounded convex planar domains in terms of the area, perimeter, minimal width and magnetic field strength, and show that they are asymptotically sharp along sequences of suitable thinning domains. Finally, we post some remarks with open problems.
math.AP↗