Local solvability of rotationally invariant differential forms

• Grant number: 20/14106-9
• Support Opportunities: Scholarships in Brazil - Master
• Start date: April 01, 2021
• End date: February 28, 2023
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Paulo Leandro Dattori da Silva
• Grantee: Fernanda Martins Simão
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated research grant: 18/14316-3 - Geometric theory of PDE and multidimensional complex analysis, AP.TEM

Abstract

We will deal with local solvability of classes of linear partial differential operators. More precisely, we will deal with solvability of classes of first order differential equations in context of differential forms. Let \Omega=A(z)dz+B(z)d\bar{z} be a smooth differential 1-form defined in a neighborhood of the origin in \mathbb{R}^2. We say that \Omega is rotationally invariant if \Omega\wedge R^*_\alpha\Omega=0 for all rotation of angle \alpha, R_\alpha, of \mathbb{R}^2. Let \Omega be a rotationally invariant smooth differential 1-form, singular at (0,0) and elliptic for all (x,y)\in\mathbb{R}^2\setminus{(0,0)}. We are interested in studying equations in the formdu\wedge\Omega=\eta\wedge\Omega,where \eta is a smooth differential 1-form defined in a neighborhood of the origin of \mathbb{R}^2. The relation between the order of the vanishing of the 2-forms\Omega\wedge\overline{\Omega} and \Omega\wedge(\bar{z}dz+zd\bar{z}) has influence in the solvability. (AU)

Academic Publications

(References retrieved automatically from State of São Paulo Research Institutions)

SIMÃO, Fernanda Martins. Hypoellipticity of rotationally invariant differential forms with a singularity. 2023. Master's Dissertation - Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB) São Carlos.