Regularity of periodic solutions for first order partial differential operators

Author(s)

Nguyen Thi Hoang Yen

• Document type: Master's Dissertation
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2023-03-06

Examining board members

Adalberto Panobianco Bergamasco; Gabriela Del Valle Planas; José Ruidival Soares dos Santos Filho; Sergio Luis Zani

• Advisor: Adalberto Panobianco Bergamasco

Abstract

We consider the equation Lu = f, where L is a linear partial differential operator acting on periodic functions (or distributions). A problem of interest is the following: given a smooth periodic function f (satisfying some natural conditions), find a smooth periodic function u satisfying Lu = f. On the other hand, let u be a periodic distribution such that Lu = f is smooth. If, for every choice of f , we have u smooth, we say that the operator L is globally hypoelliptic. We will analyze the global hypoellipticity of some operators. Finally, we will study the effect, on the global hypoellipticity, of lower order perturbations. More precisely, if L is globally hypoelliptic, then L - c, where c is a complex number, is likewise globally hypoelliptic? Most of the results presented here deal with first-order operators in two variables. In some of the results the operators may either be of arbitrary order or act on more variables. (AU)

• FAPESP's process: 20/14135-9 - Existence of periodic solutions for first-order partial differential equations
• Grantee: Nguyen Thi Hoang Yen
• Support Opportunities: Scholarships in Brazil - Master