The textbook provides an in-depth exploration of elliptic, parabolic, and hyperbolic equations across multiple variables. In its second volume, it focuses on functional analytic methods and their applications in differential geometry, offering updated and revised content that enhances understanding of complex mathematical concepts.
Foundations and Integral Representations
Focusing on geometric and complex variable methods, this revised second edition delves into integral representations of partial differential equations. It provides an in-depth exploration of topics like Brouwer's mapping degree, making it a valuable resource for those studying advanced mathematical concepts. This volume serves as a comprehensive guide for understanding the intricacies of the subject.
This volume is the first in a three-part treatise on minimal surfaces, focusing on boundary value problems. It serves as a revised and expanded version of earlier monographs. The book opens with fundamental concepts of surface theory in three-dimensional Euclidean space, introducing minimal surfaces as stationary points of area or surfaces with zero mean curvature. A minimal surface is defined as a nonconstant harmonic mapping that is conformally parametrized and may have branch points. The classical theory of minimal surfaces is explored, featuring numerous examples, Björling’s initial value problem, reflection principles, and important theorems by Bernstein, Heinz, Osserman, and Fujimoto. The second part addresses Plateau’s problem and its modifications, presenting a new elementary proof that the area and Dirichlet integral share the same infimum for admissible surfaces spanning a prescribed contour. This leads to a simplified solution for minimizing both area and Dirichlet integral, along with new proofs of Riemann and Korn-Lichtenstein's mapping theorems, and a solution to the simultaneous Douglas problem for contours with multiple components. The volume also covers stable minimal surfaces, deriving curvature estimates and presenting uniqueness and finiteness results. Additionally, it develops a theory of unstable solutions to Plateau’s problems based on Courant’s mountain pass lemma and solves Dirichlet’s problem for non
This encyclopedic work covers the whole area of Partial Differential Equations - of the elliptic, parabolic, and hyperbolic type - in two and several variables. Emphasis is placed on the connection of PDEs and complex variable methods. This second volume addresses Solvability of operator equations in Banach spaces; Linear operators in Hilbert spaces and spectral theory; Schauder's theory of linear elliptic differential equations; Weak solutions of differential equations; Nonlinear partial differential equations and characteristics; Nonlinear elliptic systems with differential-geometric applications. While partial differential equations are solved via integral representations in the preceding volume, this volume uses functional analytic solution methods.
This comprehensive two-volume textbook covers the whole area of Partial Differential Equations - of the elliptic, parabolic, and hyperbolic type - in two and several variables. Special emphasis is placed on the connection of PDEs and complex variable methods. In this first volume the following topics are treated: Integration and differentiation on manifolds, Functional analytic foundations, Brouwer's degree of mapping, Generalized analytic functions, Potential theory and spherical harmonics, Linear partial differential equations. We solve partial differential equations via integral representations in this volume, reserving functional analytic solution methods for Volume Two.