- Consideration of the first eigenvalue of the p-Laplacian on a compact submanifold in the unit sphere (arXiv)
Author: Mateus Nunes Soares, Fabio Reis dos Santos
Summary: The integer inequality of the singularity p-Laplacian is 3/2
2. About the Dirichlet to Neumann mapping of the p-Laplacian on the metric space (arXiv)
Author: Ryan Gibala, Nageswari Shanmugalingam
Summary: In this note, we construct a Dirichlet-to-Neumann mapping from the Besov space of functions to the dual of this class. A Besov space is a function on the boundary of a bounded, locally compact, uniform domain with a doubling measure that supports the p-Poincaré inequality, and which has a codimensional relationship with the measure on the domain. Radon measurements are also provided. . Build this map according to the following recipe. First, we show that the solution of the Dirichlet problem for the p-Laplacian over a domain with given boundary data in Besov space gives rise to an operator that exists in the duality of Besov space. Conversely, we show that in a homogeneous Newton-Sobolev space there is a solution to the p-Laplacian Neumann problem with Neumann boundary data given by a continuous linear function that belongs to the duality of the Besov space. We also obtain the bounds of that operator norm in terms of the norm of the trace and extension operators that relate the Newton-Sobolev function on the domain to the Besov function on the boundary. △
