Elliptic boundary value problems in non-smooth domains via a harmonic analysis and geometric measure theory approach

• Grant number: 16/22593-1
• Support Opportunities: Research Grants - Visiting Researcher Grant - International
• Duration: February 04, 2017 - February 12, 2017
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Gustavo Hoepfner
• Grantee: Gustavo Hoepfner
• Visiting researcher: Marius Mitrea
• Visiting researcher institution: University of Missouri, Columbia (UM), United States
• Host Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil
• Associated research grant: 12/03168-7 - Geometric theory of PDE and several complex variables, AP.TEM

Abstract

This research proposal is envisioned as a collaborative effort between faculty members Gustavo Hoepfner (Universidade Federal de São Carlos), Paulo Liboni (Universidade Estadual de Londrina), Irina Mitrea (Temple University) and Marius Mitrea (University of Missouri-Columbia). The three main research directions of this proposal, rooted in the areas of Harmonic Analysis (HA), Geometric Measure Theory (GMT) and Partial Differential Equations (PDE), are: A. Develop a new generation of Calderón-Zygmund theory for multi-layer potential operators in uniformly rectifiable (UR) domains. This includes mapping properties, non-tangential maximal function estimates, and Carleson measure estimates for multi-layers of double and single type in UR domains; B. Identify algebraic and geometric measure theoretic settings for which boundary multi-layer potential operators of double and single type are invertible on appropriate function spaces on the boundary suitably adapted to the higher order setting; C. Establish spectral properties of layer potentials associated with second and higher order elliptic problems in UR domains. (AU)