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Solvability of Vekua-type periodic operators and applications to classical equations

Alexandre Kirilov, Wagner Augusto Almeida de Moraes, Pedro Meyer Tokoro

Abstract

In this note, we investigate Vekua-type periodic operators of the form $Pu=Lu-Au-B\bar u$, where $L$ is a constant coefficient partial differential operator. We provide a complete characterization of the necessary and sufficient conditions for the solvability and global hypoellipticity of $P$. As an application, we provide a comprehensive characterization of Vekua-type operators associated with classical wave, heat, and Laplace equations.

Solvability of Vekua-type periodic operators and applications to classical equations

Abstract

In this note, we investigate Vekua-type periodic operators of the form , where is a constant coefficient partial differential operator. We provide a complete characterization of the necessary and sufficient conditions for the solvability and global hypoellipticity of . As an application, we provide a comprehensive characterization of Vekua-type operators associated with classical wave, heat, and Laplace equations.
Paper Structure (5 sections, 6 theorems, 52 equations)

This paper contains 5 sections, 6 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Necessary and sufficient conditions for solvability
  3. Case 1:
  4. Case 2:
  5. Applications

Key Result

Theorem 2

The operator $P$ is solvable if and only if the following Diophantine condition holds: there exists $\gamma > 0$ such that

Theorems & Definitions (13)

  • Definition 1
  • Theorem 2
  • proof
  • Corollary 3
  • proof
  • Theorem 4
  • proof
  • Example : Laplace operator
  • Theorem 5: Heat Operator
  • proof
  • ...and 3 more