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Local solvability of rotationally invariant differential forms

Grant number: 20/14106-9
Support type:Scholarships in Brazil - Master
Effective date (Start): April 01, 2021
Effective date (End): March 31, 2023
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Analysis
Principal researcher:Paulo Leandro Dattori da Silva
Grantee:Fernanda Martins Simão
Home 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)

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