Linear and Nonlinear Partial Differential Equations

Erscheinungsjahr 2027
ISBN: 978-3-032-45372-3
Verlag: Springer

Buch, Englisch, Format (B × H): 155 mm x 235 mm

Reihe: Universitext

This book provides a thorough introduction to scalar linear and nonlinear elliptic, parabolic, and hyperbolic partial differential equations. It presents both classical and modern methods for proving the existence and regularity of solutions to boundary and initial-boundary value problems for first- and second-order equations.

The material is developed in a systematic and coherent way, progressing from elementary results for linear equations to advanced methods for nonlinear problems. A key feature is the careful derivation of the equations from physical principles. Part I covers first-order equations. Part II develops second-order theory, beginning with the wave equation and leading to elliptic and hyperbolic equations, including spectral methods, energy estimates, and the propagation of singularities. Part III focuses on nonlinear elliptic and parabolic equations, emphasizing monotonicity methods, variational approaches, regularity theory, and maximum principles. Modern techniques are based on Sobolev spaces and functional analysis, which are introduced with detailed proofs. The book also presents essential analytical tools needed for PDE theory.

Intended for graduate students with a strong background in calculus and a basic knowledge of functional analysis, the wide range of topics covered in the book also makes it of interest to researchers.

Zielgruppe

Graduate

Autoren/Hrsg.

Alber, Hans-Dieter

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen

Inhaltsverzeichnis

1 Basic Notions and Principles.- Part I Partial Differential Equations of First Order.- 2 Linear and Quasilinear Equations.- 3 Fully Nonlinear Equations of First Order.- Part II Linear Partial Differential Equations of Second Order.- 4 The Wave Equation and the Helmholtz Equation. Classification of Partial Differential Equations.- 5 Tools From Functional Analysis. The Dirichlet Boundary Value Problem for the Helmholtz Equation in One Dimension.- 6 Weak Derivatives. Sobolev Spaces. Weak Solutions of the Helmholtz
Equation.- 7 Boundary Value Problems in Two-Dimensional Circular Domains.- 8 Maximum Principle. Subsolutions. Perron’s Method- 9 Fundamental Solutions and Green’s Functions.- 10 The Integral Equation Method.- 11 Hilbert Space Methods for Elliptic Operators.- 12 Spectral Theory for Symmetric Operators and Eigenvalues.- 13 Function Spaces for Time-Dependent Problems.- 14 Linear Hyperbolic Equations: The Wave Equation on R.- 15 General Theory for Linear Hyperbolic Equations of Second Order.- 16 The Energy Inequality and Geometric Theory for Hyperbolic Equations.- Part III Nonlinear Partial Differential Equations of Second Order.- 17 Nonlinear Partial Differential Equations Modeling Diffusion and Elastic Deformation.- 18 Conditions on Nonlinearities in Differential Equations.- 19 Tools From the Duality Theory of Banach Spaces.- 20 The Nonlinear Elliptic Differential Operator.- 21 Solution of Nonlinear Elliptic Boundary Value Problems.- 22 Representations of Function Spaces for Parabolic Problems.- 23 The Nonlinear Parabolic Differential Operator.- 24 Solution of Nonlinear Parabolic Initial-Boundary Value Problems.- 25 -Regularity Theory for Linear and Nonlinear Elliptic Equations.- 26 Maximum Principles and 8-Estimates for Weak Solutions of Elliptic Equations.- A Bessel, Neumann, and Hankel Functions.- B Rules, Tools, and Approximation Results for Sobolev Spaces.- C Monotone Vector Fields and Convex Functions.- D Some Geometric Concepts for Domains in R.

Über Autor(innen)

Following a visiting professorship at Kansas State University, Hans-Dieter Alber was appointed professor at the University of Stuttgart and then, in 1988, at the Technical University of Darmstadt where he has remained ever since. His research focuses on linear elliptic and hyperbolic partial differential equations and systems, with particular emphasis on geometrical optics, high-frequency asymptotics, and the propagation of singularities. In the area of nonlinear equations, his work includes the study of hyperbolic conservation laws, incompressible fluid flow, and the investigation of mathematical models from materials science, a field in which he closely cooperated with researchers in mechanics and engineering. His book in the Springer Lecture Notes in Mathematics on initial-boundary value problems for materials with memory originated from this line of research. Another core area of his work is the study of phase-field models.