Kichenassamy S. Fuchsian Reduction: Applications to Geometry, Cosmology and Mathematical Physics

Boston: Birkhäuser, 2007. — xv+293 p. — (Progress in Nonlinear Differential Equations and Their Applications, Vol. 71). — ISBN: 978-0-8176-4637-0.Fuchsian reduction is a method for representing solutions of nonlinear PDEs near singularities. The technique has multiple applications including soliton theory, Einstein's equations and cosmology, stellar models, laser collapse, conformal geometry and combustion. Developed in the 1990s for semilinear wave equations, Fuchsian reduction research has grown in response to those problems in pure and applied mathematics where numerical computations fail.
This work unfolds systematically in four parts, interweaving theory and applications. The case studies examined in Part III illustrate the impact of reduction techniques, and may serve as prototypes for future new applications. In the same spirit, most chapters include a problem section. Background results and solutions to selected problems close the volume.
This book can be used as a text in graduate courses in pure or applied analysis, or as a resource for researchers working with singularities in geometry and mathematical physics.
(перевод)
Фуксова редукция является методом представления решений нелинейных УрЧП вблизи особенностей. Этот метод имеет множество применений, включая теорию солитонов, уравнения и космологию Эйнштейна, звездные модели, лазерный коллапс, конформную геометрию и горение. Разработанное в 1990-х годах для полулинейных волновых уравнений, исследование редукции Фукса выросло в ответ на те проблемы в чистой и прикладной математике, где численные вычисления терпят неудачу.
Эта работа систематически разворачивается в четырех частях, переплетая теорию и приложения. Тематические исследования, рассмотренные в части III, иллюстрируют воздействие методов сокращения выбросов и могут служить прототипами для будущих новых применений. В том же духе большинство глав содержат задачный раздел. Закрывают том фоновые результаты и решения избранных задач.
Эта книга может быть использована как пособие в аспирантуре по чистому или прикладному анализу, так и в качестве источника для исследователей, работающих с особенностями в геометрии и математической физике.Preface
Introduction
Singularity locus as parameter
The main steps of reduction
A few definitions
An algorithm in eight steps
Simple examples of reduced Fuchsian equations
Reduction and applicationsFuchsian Reduction
Formal Series
The Operator D and its first properties
The space Al and its variants
Formal series with variable exponents
Relation of Al to the invariant theory of binary forms
ProblemsGeneral Reduction Methods
Reduction of a single equation
Introduction of several time variables and second reduction
Semilinear systems
Structure of the formal series with several time variables
Resonances, instability, and group invariance
Stability and parameter dependence
ProblemsTheory of Fuchsian Partial Differential Equations
Convergent Series Solutions of Fuchsian Initial-Value Problems
Theory of linear Fuchsian ODEs
Initial-value problem for Fuchsian PDEs with analytic data
Generalized Fuchsian systems
Notes
Problems
Fuchsian Initial-Value Problems in Sobolev Spaces
Singular systems of ODEs in weighted spaces
A generalized Fuchsian ODE
Fuchsian PDEs: abstract results
Optimal regularity for Fuchsian PDEs
Reduction to a symmetric system
Problems
Solution of Fuchsian Elliptic Boundary-Value Problems
Basic L^p results for equations with degenerate characteristic form
Schauder regularity for Fuchsian problems
Solution of a model Fuchsian operator
ProblemsApplications
Applications in Astronomy
Notions on stellar modeling
Polytropic model
Point-source model
Problems
Applications in General Relativity
The big-bang singularity and AVD behavior
Gowdy space-times
Space-times with twist
Problems
Applications in Differential Geometry
Fefferman–Graham metrics
First Fuchsian reduction and construction of formal solutions
Second Fuchsian reduction and convergence of formal solutions
Propagation of constraint equations
Special cases
Conformal changes of metric
Loewner–Nirenberg metrics
Problems
Applications to Nonlinear Waves
From blowup time to blowup pattern
Semilinear wave equations
Nonlinear optics and lasers
Weak detonations
Soliton theory
The Liouville equation
Nirenberg’s example
Problems
Boundary Blowup for Nonlinear Elliptic Equations
A renormalized energy for boundary blowup
Hardy–Trudinger inequalities
Variational characterization of solutions with boundary blowup
Construction of the partition of unity
ProblemsBackground Results
Distance Function and Hölder Spaces
The distance function
Hölder spaces on C2+α domains
Interior estimates for the Laplacian
Perturbation of coeffcients
Nash–Moser Inverse Function Theorem
Nash–Moser theorem without smoothing
Nash–Moser theorem with smoothingSolutions

Index