Principal Eigenvalues and Behavior of Weighted p-Laplacian with Robin Conditions
Analysis
This paper addresses a gap in the spectral theory of the p-Laplacian, specifically the less-explored Robin boundary conditions on exterior domains. It provides a comprehensive analysis of the principal eigenvalue, its properties, and the behavior of the associated eigenfunction, including its dependence on the Robin parameter and its far-field and near-boundary characteristics. The work's significance lies in providing a unified understanding of how boundary effects influence the solution across the entire domain.
Key Takeaways
- •Proves existence, uniqueness, simplicity, and isolation of the principal eigenvalue.
- •Analyzes the dependence of the principal eigenvalue on the Robin parameter, recovering Neumann and Dirichlet limits.
- •Describes the far-field behavior of the eigenfunction with a universal algebraic decay rate.
- •Provides a unified understanding of boundary effects through gradient estimates and a characteristic length scale.
“The main contribution is the derivation of unified gradient estimates that connect the near-boundary and far-field regions through a characteristic length scale determined by the Robin parameter, yielding a global description of how boundary effects penetrate into the exterior domain.”