Methods of Geometry in the Theory of Partial Differential Equations

Mathematical models are used to describe the essence of the real world, and their analysis induces new predictions filled with unexpected phenomena.

In spite of a huge number of insights derived from a variety of scientific fields in these five hundred years of the theory of differential equations, and its extensive developments in these one hundred years, several principles that ensure these successes are discovered very recently.

This monograph focuses on one of them: cancellation of singularities derived from interactions of multiple species, which is described by the language of geometry, in particular, that of global analysis.

Five objects of inquiry, scattered across different disciplines, are selected in this monograph: evolution of geometric quantities, models of multi-species in biology, interface vanishing of d systems, the fundamental equation of electro-magnetic theory, and free boundaries arising in engineering.

The relaxation of internal tensions in these systems, however, is described commonly by differential forms, and the reader will be convinced of further applications of this principle to other areas.

Contents:

• Evolution of Geometric Objects:

  • Curves and Surfaces in R3
  • Static Recursive Hierarchy
  • Kinetic Recursive Hierarchy
  • Diffusion Geometry

• Differential Forms and Singularities:

  • Systems of Multiple Components
  • Interface Vanishing

• Theory of Transformations:

  • Non-Standard Elliptic Regularity
  • Liouville's Formulae
  • Hadamard's Variational Formula
  • Perturbation of Eigenvalues

Readership: Mathematics - Researchers and graduate students interested in Global Analysis / Analysis on Manifolds, Partial Differential Equations. Applications - Researchers/professionals working on the following area turbulence in fluid mechanics, models in statistical mechanics and in astrophysics, diffusion geometry in the context of the Poincare conjecture, cooperative systems in ecology, fundamental equation of electro-magnetic theory, non-standard elliptic regularity in partial differential equations, free boundary problems in material science and engineering, and mathematical foundation of the theory of domain perturbations.
Key Features:

• This book provides with new insights to mathematical analysis of nonlinear phenomena
• Theoretical basis for technical tools developed so far for their study are founded in the languages of geometry, that is, curvatures and differential forms
• This method spreads in a wide area of applied and theoretical sciences: theory of turbulence in fluid mechanics, models in statistical mechanics in astrophysics, diffusion geometry in the context of the Poincare conjecture, cooperative systems in ecology, fundamental equation of electro-magnetic theory, non-standard elliptic regularity in partial differential equations, free boundary problems in material science and engineering, and mathematical foundation of the theory of domain perturbations
• Once their geometric background is taken, however, one sees their common structure and get insights to reveal hidden phenomena, that is, cancellation of singularities. This point of view evokes new scientific research based on mathematical theories

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