Boundary value problems beyond singular integrals
Lead Research Organisation:
University of Birmingham
Department Name: School of Mathematics
Abstract
This project will develop a new framework for solving the Dirichlet and Neumann boundary value problems for magnetic Schrödinger operators on Lipschitz domains. It will build on recent developments in harmonic analysis and functional calculus that extend the Calderón-Zygmund theory of singular integrals.
Organisations
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509590/1 | 01/10/2016 | 30/09/2021 | |||
| 1809483 | Studentship | EP/N509590/1 | 01/10/2016 | 30/03/2020 | Andrew Turner |
| Description | So far the work of this award has proven that the boundary value problems for the generalised Schrodinger operator with singular potentials are solvable and in doing so this has established a framework to extend these results for the full magnetic Schrodinger operator. An other result from this work is the control of the non-tangential maximal function which allows for a stronger notion of convergence at the boundary. There are also partial results regarding the equivalence of the solutions generated with this method and those from the classical case known as energy solution when the boundary data is in both boundary spaces. |
| Exploitation Route | Some ways to move forward with this work is to allow the perturbations to depend on the normal direction t or to allow the perturbation to be degenerate by allowing some growth at infinity. Other generalisations are to add a parabolic term or to consider boundary data with different regularity. |
| Sectors | Other |
| URL | https://arxiv.org/abs/2001.11901 |