From Steklov to Neumann via homogenisation

Area of Science:

• Mathematical analysis
• Partial differential equations
• Spectral theory

Background:

• Steklov and Neumann eigenvalues are crucial in analyzing boundary value problems.
• Homogenization techniques are used to study properties of perforated domains.
• Understanding eigenvalue behavior is key in various physics and engineering applications.

Purpose of the Study:

• To establish a novel link between Steklov and Neumann eigenvalues.
• To explore eigenvalue problems with dynamical boundary conditions via homogenization.
• To derive new isoperimetric inequalities and construct novel planar domains.

Main Methods:

• Homogenization limit of Steklov eigenvalue problems on periodically perforated domains.
• Analysis of an intermediate eigenvalue problem with spectral parameters in the domain and boundary.
• Modification of the energy method with quantitative estimates for harmonic functions.

Main Results:

• A new interpolation between Steklov and Neumann eigenvalues was established.
• Isoperimetric type bounds for Neumann eigenvalues were recovered from Steklov eigenvalue bounds.
• Construction of planar domains with enhanced first perimeter-normalized Steklov eigenvalues.

Conclusions:

• The homogenization approach provides a powerful tool for linking different spectral problems.
• The study advances the understanding of eigenvalue problems in perforated domains.
• New theoretical insights and practical constructions in spectral geometry were achieved.