Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hodge Decomposition and Potentials in Variable Exponent Lebesgue and Sobolev Spaces

Anna Balci, Swarnendu Sil, Mikhail Surnachev

Abstract

The objective of this work is to establish a systematic study of boundary value problems within the framework of differential forms and variable exponent spaces. Specifically, we investigate the Hodge Laplacian and related first order systems like the div-curl systems, Hodge-Dirac systems, and Bogovskii-type problems in the context of variable exponent spaces. Our approach yields both existence theorems and elliptic estimates. These estimates provide key results such as the Hodge decomposition theorem, Gaffney inequality, and gauge fixing. These findings are crucial for advancing the nonlinear theory related to problems involving differential forms.

Hodge Decomposition and Potentials in Variable Exponent Lebesgue and Sobolev Spaces

Abstract

The objective of this work is to establish a systematic study of boundary value problems within the framework of differential forms and variable exponent spaces. Specifically, we investigate the Hodge Laplacian and related first order systems like the div-curl systems, Hodge-Dirac systems, and Bogovskii-type problems in the context of variable exponent spaces. Our approach yields both existence theorems and elliptic estimates. These estimates provide key results such as the Hodge decomposition theorem, Gaffney inequality, and gauge fixing. These findings are crucial for advancing the nonlinear theory related to problems involving differential forms.
Paper Structure (53 sections, 113 theorems, 526 equations)

This paper contains 53 sections, 113 theorems, 526 equations.

Table of Contents

  1. Introduction
  2. Boundary value problems for differential forms
  3. Variable exponent spaces and the log-Hölder condition
  4. Results
  5. Gauge fixing and nonlinear problems
  6. Organization
  7. Lebesgue and Sobolev Spaces of Differential Forms on Manifolds
  8. Bundle of differential forms
  9. Variable exponent Lebesgue and Sobolev spaces in Rn
  10. Variable exponent Lebesgue spaces on manifolds
  11. Admissible coordinate systems
  12. Variable exponent Sobolev spaces on manifolds
  13. Approximation results for variable exponent Sobolev spaces
  14. Integration-by-parts formula
  15. Partial Sobolev spaces
  16. ...and 38 more sections

Key Result

Theorem 1

Consider the boundary value problem

Theorems & Definitions (189)

  • Theorem : Main estimate for the Hodge Laplacian
  • Theorem : Hodge decomposition
  • Theorem : Solvability of first order systems
  • Theorem : Hodge-Dirac system
  • Theorem : Poincaré lemma with full Dirichlet data
  • Corollary
  • Definition 2.1
  • Lemma 2.2: Morrey1966, Lemma 7.5.1
  • Lemma 2.3
  • proof
  • ...and 179 more