Suzuki T. Methods Of Geometry In The Theory Of Partial Diff Equations...2024

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Suzuki T. Methods Of Geometry In The Theory Of Partial Diff Equations...2024

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Textbook in PDF format 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 – δ 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. Preface Evolution of Geometric Objects Curves and Surfaces in R3 Surfaces Curves Curves on Surfaces Curvatures Differential Forms Tangent and Cotangent Spaces Frames and Covariant Derivatives Connections Fundamental Equation of Surfaces Conformal Geometry Static Recursive Hierarchy Point Vortices Mean Field Limit Liouville Integral Coverings of the Sphere Boltzmann Poisson Equation Method of Scaling Kinetic Recursive Hierarchy Smoluchowski Poisson Equation Trudinger Moser Inequality Quantized Blowup Mechanism Blowup in Finite Time Blowup in Infinite Time Formation of Collapses Improved ε-Regularity Scaling Limit Residual Vanishing Bounded Domains Exclusion of Boundary Blowup Points Global-in-Time Solution Initial Mass Quantization Collapse Dynamics Simplified System of Chemotaxis Diffusion Geometry 2D Normalized Ricci Flow Analytic Approach Geometric Argument Logarithmic Diffusion Benilan’s Inequality Concentration of Probability Measures Pre-Compactness of the Orbit Steady States Critical Manifolds Łojasiewicz Simon Inequality Convergence to the Steady State Non-Degeneracy Exponential Rate of Convergence Vanishing of the Center Manifold Differential Forms and Singularities Systems of Multiple Components Languages of Geometry Analytic Mechanics Symplectic and Poisson Manifolds Hamilton Jacobi Theory Symplectic and Poisson Structures of Biological Models Reaction Diffusion Systems Lotka Volterra Systems with Skew-Symmetry The Case without Linear Term The Case with Linear Term Integrable Systems and Differential Forms Integrable Systems of Order 1 Integrable Systems of Order 2 Integrable Systems of Order (N − 1) Interface Vanishing Geselowitz Equation Maxwell Equation Differential Forms in Higher Dimension 2-Forms on Euclidean Spaces 2-Forms on Minkowski Spaces 1-Forms d − δ Systems Theory of Transformations Non-Standard Elliptic Regularity Lipschitz Domains H1-Solution Sectional Curvatures Convex Domains Liouville’s Formulae Transformation of Variables Variational Formulae of Jacobian Volume Derivatives Filtration Flux of the Flow Stefan Condition as Heat Transfer Area Derivatives Hadamard’s Variational Formula An Abstract Theorem Green’s Function Lagrange Derivatives Euler Derivatives C1,1 Domains C2,θ Domains C2,1 Domains Neumann Problems First Variational Formula Second Variational Formula Second Fundamental Form on ∂Ω Perturbation of Eigenvalues Eigenvalue Problems Unilateral Derivatives and Rearrangements Characterization of Derivatives Reduction to the Abstract Theory Continuity of Eigenvalues and Eigenspaces First Derivatives Rearrangement of Eigenvalues Second Derivatives Bibliography Index