Steklov eigenvalues and quasiconformal maps of simply connected planar domains. (English) Zbl 1333.35274

Summary: We investigate isoperimetric upper bounds for sums of consecutive Steklov eigenvalues of planar domains. The normalization involves the perimeter and scale-invariant geometric factors which measure deviation of the domain from roundness. We prove sharp upper bounds for both starlike and simply connected domains for a large collection of spectral functionals including partial sums of the zeta function and heat trace. The proofs rely on a special class of quasiconformal mappings.


35Q74 PDEs in connection with mechanics of deformable solids
74K15 Membranes
30C62 Quasiconformal mappings in the complex plane
74H45 Vibrations in dynamical problems in solid mechanics
35P15 Estimates of eigenvalues in context of PDEs
35B45 A priori estimates in context of PDEs


SyFi; FEniCS
Full Text: DOI arXiv


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