Friedrich Sauvigny

This textbook focuses on the study of curves and surfaces, applying modern differential equation theory to geometric problems. By introducing isothermal parameters, it simplifies the fundamental equations of surface theory, leading to clear derivations of results like those of H Hopf and S Bernstein for surfaces of constant and vanishing mean curvature. Deviating from traditional approaches, the book first treats n-dimensional Riemannian spaces by a corresponding metric, then constructs Riemannian manifolds through transition conditions. The ultimate goal is to prove the Hadamard-Cartan theorem on the diffeomorphic character of the exponential mapping in Riemannian manifolds with nonpositive sectional curvature. By considering curves and surfaces in their optimal parametrization, the resulting ODEs and complex PDEs can be analytically solved, eliminating the need for intricate tensor calculus. The approach follows that of G Monge in his treatise L'Application de l'Analyse à la Géométrie, applying analytical techniques to geometric problems. Building on this foundation, the book uses modern theory of ODEs and PDEs to study the local and global results for curves and surfaces and their relevant curvatures.

Dieses Lehrbuch behandelt Lehrinhalte der Analysis für die ersten drei Semester des Bachelor-Studiums der Mathematik, Physik und Informatik. Es bietet eine moderne Darstellung der Differential- und Integralrechnung für Funktionen in einer und mehreren reellen sowie einer komplexen Variablen. Elementare Funktionen werden über komplexe Potenzreihen definiert und die Logarithmusfunktion auf ihrer Riemannschen Fläche betrachtet. Nachdem die eindimensionale Integration mittels reeller und komplexer Stammfunktionen durchgeführt ist, wird über uneigentliche n-dimensionale Riemannsche Integrale die Integration auf Mannigfaltigkeiten mit Differentialformen vorgestellt. Mit dem Lebesgueschen Integral und dessen Maßtheorie wird der Banachraum der p-fach integrablen Funktionen eingeführt. Für gewöhnliche Differentialgleichungen werden Existenz-, Eindeutigkeits- und Stabilitätsfragen beantwortet. In einem Kapitel zur Variationsrechnung wird über Geodätische der n-dimensionale Riemannsche Raum präsentiert. Ferner wird das Stieltjes-Integral mit BV-Belegungsfunktionen behandelt und die Differentiation absolut stetiger Funktionen durchgeführt. Schließlich wird der stetige Dualraum zum Lebesgueraum der p-fach integrablen Funktionen über den Rieszschen Darstellungssatz bestimmt.

In diesem Lehrbuch wird der Spektralsatz für selbstadjungierte Operatoren aus dem Resultat der Linearen Algebra über die Diagonalisierung Hermitescher Matrizen hergeleitet. Dabei werden Lebesgue-Stieltjes-Integrale verwendet und der Auswahl- sowie der Konvergenzsatz von Helly über monotone Funktionen bereitgestellt. Wir konstruieren die Spektralschar durch eine technisch aufwändige Approximation, wobei die Stieltjes-Umkehrformel im Zentrum des Beweises steht. Ein Ergebnis hiervon ist, dass selbstadjungierte Operatoren nicht nur ein diskretes, sondern auch ein kontinuierliches Spektrum besitzen. Die auftretenden Streueigenwerte können hierbei nicht durch Variationsmethoden gewonnen werden. Dann wenden wir uns der zentralen Frage zu, welche elliptischen Differentialoperatoren eine selbstadjungierte Fortsetzung besitzen und somit im Geltungsbereich des Spektralsatzes liegen. Hier unterscheiden wir zwischen stabilen elliptischen Differentialoperatoren auf beschränkten Gebieten und denen auf dem ganzen Raum, wie etwa dem Schrödingeroperator. Auch Laplace-Beltrami-Operatoren und der Schwarzsche Operator für Minimalflächen werden im obigen Sinne als selbstadjungiert erkannt. Am Ende dieses Buches geben wir eine Einführung in die Störungstheorie selbstadjungierter Operatoren. Hier weisen wir die analytische Abhängigkeit der Spektralschar vom Störungsparameter nach. Dieses Werk zur Spektraltheorie ist insbesondere für das fortgeschrittene Mathematik- und Physikstudium geeignet, Kenntnisse in der Funktionalanalysis und der Theorie elliptischer Differentialgleichungen werden vorausgesetzt.

Minimal Surfaces is the first volume of a three volume treatise on minimal surfaces (Grundlehren Nr. 339-341). Each volume can be read and studied independently of the others. The central theme is boundary value problems for minimal surfaces. The treatise is a substantially revised and extended version of the monograph Minimal Surfaces I, II (Grundlehren Nr. 295 & 296). The first volume begins with an exposition of basic ideas of the theory of surfaces in three-dimensional Euclidean space, followed by an introduction of minimal surfaces as stationary points of area, or equivalently, as surfaces of zero mean curvature. The final definition of a minimal surface is that of a nonconstant harmonic mapping X: \Omega\to\R^3 which is conformally parametrized on \Omega\subset\R^2 and may have branch points. Thereafter the classical theory of minimal surfaces is surveyed, comprising many examples, a treatment of Björling´s initial value problem, reflection principles, a formula of the second variation of area, the theorems of Bernstein, Heinz, Osserman, and Fujimoto. The second part of this volume begins with a survey of Plateau´s problem and of some of its modifications. One of the main features is a new, completely elementary proof of the fact that area A and Dirichlet integral D have the same infimum in the class C(G) of admissible surfaces spanning a prescribed contour G. This leads to a new, simplified solution of the simultaneous problem of minimizing A and D in C(G), as well as to new proofs of the mapping theorems of Riemann and Korn-Lichtenstein, and to a new solution of the simultaneous Douglas problem for A and D where G consists of several closed components. Then basic facts of stable minimal surfaces are derived; this is done in the context of stable H-surfaces (i.e. of stable surfaces of prescribed mean curvature H), especially of cmc-surfaces (H = const), and leads to curvature estimates for stable, immersed cmc-surfaces and to Nitsche´s uniqueness theorem andTomi´s finiteness result. In addition, a theory of unstable solutions of Plateau´s problems is developed which is based on Courant´s mountain pass lemma. Furthermore, Dirichlet´s problem for nonparametric H-surfaces is solved, using the solution of Plateau´s problem for H-surfaces and the pertinent estimates.

This two-volume textbook provides comprehensive coverage of partial differential equations, spanning elliptic, parabolic, and hyperbolic types in two and several variables. In this second volume, special emphasis is placed on functional analytic methods and applications to differential geometry. The following topics are treated: 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 characteristicsnonlinear elliptic systemsboundary value problems from differential geometryThis new second edition of this volume has been thoroughly revised and a new chapter on boundary value problems from differential geometry has been added. In the first volume, partial differential equations by integral representations are treated in a classical way. This textbook will be of particular use to graduate and postgraduate students interested in this field and will be of interest to advanced undergraduate students. It may also be used for independent study.

This two-volume textbook provides comprehensive coverage of partial differential equations, spanning elliptic, parabolic, and hyperbolic types in two and several variables. In this first volume, special emphasis is placed on geometric and complex variable methods involving integral representations. The following topics are treated: • integration and differentiation on manifolds• foundations of functional analysis• Brouwer's mapping degree• generalized analytic functions • potential theory and spherical harmonics • linear partial differential equationsThis new second edition of this volume has been thoroughly revised and a new section on the boundary behavior of Cauchy’s integral has been added. The second volume will present functional analytic methods and applications to problems in differential geometry. This textbook will be of particular use to graduate and postgraduate students interested in this field and will be of interest to advanced undergraduate students. It may also be used for independent study.

Minimal Surfaces is the first volume of a three volume treatise on minimal surfaces (Grundlehren Nr. 339-341). Each volume can be read and studied independently of the others. The central theme is boundary value problems for minimal surfaces. The treatise is a substantially revised and extended version of the monograph Minimal Surfaces I, II (Grundlehren Nr. 295 & 296). The first volume begins with an exposition of basic ideas of the theory of surfaces in three-dimensional Euclidean space, followed by an introduction of minimal surfaces as stationary points of area, or equivalently, as surfaces of zero mean curvature. The final definition of a minimal surface is that of a nonconstant harmonic mapping X: \Omega\to\R^3 which is conformally parametrized on \Omega\subset\R^2 and may have branch points. Thereafter the classical theory of minimal surfaces is surveyed, comprising many examples, a treatment of Björling´s initial value problem, reflection principles, a formula of the second variation of area, the theorems of Bernstein, Heinz, Osserman, and Fujimoto. The second part of this volume begins with a survey of Plateau´s problem and of some of its modifications. One of the main features is a new, completely elementary proof of the fact that area A and Dirichlet integral D have the same infimum in the class C(G) of admissible surfaces spanning a prescribed contour G. This leads to a new, simplified solution of the simultaneous problem of minimizing A and D in C(G), as well as to new proofs of the mapping theorems of Riemann and Korn-Lichtenstein, and to a new solution of the simultaneous Douglas problem for A and D where G consists of several closed components. Then basic facts of stable minimal surfaces are derived; this is done in the context of stable H-surfaces (i.e. of stable surfaces of prescribed mean curvature H), especially of cmc-surfaces (H = const), and leads to curvature estimates for stable, immersed cmc-surfaces and to Nitsche´s uniqueness theorem andTomi´s finiteness result. In addition, a theory of unstable solutions of Plateau´s problems is developed which is based on Courant´s mountain pass lemma. Furthermore, Dirichlet´s problem for nonparametric H-surfaces is solved, using the solution of Plateau´s problem for H-surfaces and the pertinent estimates.

Dieses zweibändige Lehrbuch stellt das Gesamtgebiet der partiellen Differentialgleichungen - vom elliptischen,parabolischen und hyperbolischen Typ - in zwei und mehreren Veränderlichen vor. Im vorliegenden zweiten Band werden folgende Themen behandelt: Lösbarkeit von Operatorgleichungen im Banachraum, lineare Operatoren im Hilbertraum und Spektraltheorie, Schaudersche Theorie linearer elliptischer Differentialgleichungen, schwache Lösungen elliptischer Differentialgleichungen, nichtlineare partielle Differentialgleichungen und Charakteristikentheorie, nichtlineare elliptische Systeme mit differentialgeometrischen Anwendungen. Während im vorausgehenden Band die partiellen Differentialgleichungen mit Integraldarstellungen im Mittelpunkt standen, werden nun funktionalanalytische Lösungsmethoden vorgestellt. Dieses Lehrbuch kann daher für einen mehrsemestrigen Kurs verwendet werden. Fortgeschrittene Leser können jedes Kapitel auch unabhängig voneinander studieren.

Unter Berücksichtigung der Vorlesungen von E. Heinz