PROBLEMS IN GENERAL PHYSICS
CONTENTS INTRODUCTION HISTORICAL OVERVIEW SOME GLOBAL RESULTS, RECENT DEVELOPMENTS PURPOSE OUTLINE PDE THEORY THE FOURIER TRANSFORM AND SOBOLEV SPACES SYMMETRIC HYPERBOLIC SYSTEMS LINEAR AND NON-LINEAR WAVE EQUATIONS GEOMETRY, GLOBAL HYPERBOLICITY AND UNIQUENESS GEOMETRY AND GLOBAL HYPERBOLICITY UNIQUENESS GENERAL RELATIVITY CONSTRAINT EQUATIONS AND LOCAL EXISTENCE CAUCHY STABILITY EXISTENCE OF A MAXIMAL GLOBALLY HYPERBOLIC DEVELOPMENT PATHOLOGIES, STRONG COSMIC CENSORSHIP PART I BACKGROUND FROM THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS FUNCTIONAL ANALYSIS MEASURABILITY DUALITIES THE FOURIER TRANSFORM SCHWARTZ FUNCTIONS, THE FOURIER TRANSFORM THE FOURIER INVERSION FORMULA SOBOLEV SPACES MOLLI?ERS WEAK DIFFERENTIABILITY, W K;P SPACES TEMPERATE DISTRIBUTIONS, H S SPACES SOBOLEV EMBEDDING BASIC INEQUALITIES GAGLIARDO–NIRENBERG INEQUALITIES GRÖNWALL’S LEMMA THE BASIC ENERGY INEQUALITY EXISTENCE LINEAR WAVE EQUATIONS LINEAR ALGEBRA EXISTENCE OF SOLUTIONS TO LINEAR WAVE EQUATIONS LOCAL EXISTENCE, NON-LINEAR WAVE EQUATIONS TERMINOLOGY PRELIMINARIES LOCAL EXISTENCE CONTINUATION CRITERION, SMOOTH SOLUTIONS STABILITY PART II BACKGROUND IN GEOMETRY, GLOBAL HYPERBOLICITY AND UNIQUENESS 10 BASIC LORENTZ GEOMETRY MANIFOLDS LORENTZ GEOMETRY 10 2 1 LORENTZ METRICS 10 2 2 COVARIANT DIFFERENTIATION 10 2 3 COORDINATE EXPRESSIONS FOR CURVATURE 10 2 4 BASIC CAUSALITY CONCEPTS 10 2 5 GEODESICS 10 2 6 GLOBAL HYPERBOLICITY 10 2 7 CAUCHY SURFACES 10 2 8 TECHNICAL OBSERVATIONS 11 CHARACTERIZATIONS OF GLOBAL HYPERBOLICITY EXISTENCE OF A CAUCHY HYPERSURFACE BASIC CONSTRUCTIONS SMOOTH TIME FUNCTIONS SMOOTH TEMPORAL FUNCTIONS ADAPTED TO CAUCHY HYPERSURFACES 11 5 AUXILIARY OBSERVATIONS CONTENTS 12 UNIQUENESS OF SOLUTIONS TO LINEAR WAVE EQUATIONS PRELIMINARY TECHNICAL OBSERVATIONS UNIQUENESS OF SOLUTIONS TO TENSOR WAVE EQUATIONS PART III GENERAL RELATIVITY 13 THE CONSTRAINT EQUATIONS INTRODUCTION, EQUATIONS THE CONSTRAINT EQUATIONS CONSTRAINT EQUATIONS, NON-LINEAR SCALAR ?ELD CASE 14 LOCAL EXISTENCE GAUGE CHOICE INITIAL DATA EXISTENCE OF A GLOBALLY HYPERBOLIC DEVELOPMENT TWO DEVELOPMENTS ARE EXTENSIONS OF A COMMON DEVELOPMENT 15 CAUCHY STABILITY SOBOLEV SPACES ON MANIFOLDS BACKGROUND SOLUTIONS CAUCHY STABILITY IN GENERAL RELATIVITY 16 EXISTENCE OF A MAXIMAL GLOBALLY HYPERBOLIC DEVELOPMENT BACKGROUND FROM SET THEORY PART IV PATHOLOGIES, STRONG COSMIC CENSORSHIP 17 PRELIMINARIES STRONG COSMIC CENSORSHIP 17 2 1 THE ASYMPTOTICALLY ?AT CASE 17 2 2 THE COSMOLOGICAL CASE 17 2 3 GENERICITY CONSTRUCTION OF EXTENSIONS IN THE UNIMODULAR CASE SKETCH OF THE PROOF OF EXISTENCE OF INEQUIVALENT EXTENSIONS 18 CONSTANT MEAN CURVATURE CALCULUS OF VARIATIONS CONSTANT MEAN CURVATURE HYPERSURFACES CONDITIONS ENSURING MAXIMALITY CONDITIONS ENSURING INEXTENDIBILITY 19 INITIAL DATA UNIMODULAR LIE GROUPS CURVATURE 20 EINSTEIN’S VACUUM EQUATIONS MODEL METRICS CONSTRUCTING A SPACETIME ELEMENTARY PROPERTIES OF DEVELOPMENTS CAUSAL GEODESIC COMPLETENESS AND INCOMPLETENESS 21 CLOSED UNIVERSE RECOLLAPSE RECOLLAPSE FOR AN OPEN SET OF INITIAL DATA 22 ASYMPTOTIC BEHAVIOUR THE WAINWRIGHT–HSU VARIABLES RELATION BETWEEN THE TIME COORDINATES TERMINOLOGY, ASYMPTOTIC BEHAVIOUR CRITERIA ENSURING CURVATURE BLOW UP LIMIT CHARACTERIZATION OF THE TAUB SOLUTIONS 22 6 ASYMPTOTIC BEHAVIOUR, BIANCHI TYPE I AND II TYPE VI0 SOLUTIONS TYPE VII0 SOLUTIONS BIANCHI TYPE VIII AND IX 22 10 CURVATURE BLOW UP 23 LRS BIANCHI CLASS A SOLUTIONS BIANCHI TYPE I BIANCHI VII0 BIANCHI TYPE VI0 BIANCHI TYPE II, VIII AND IX 24 EXISTENCE OF EXTENSIONS CONSTRUCTION OF AN EMBEDDING BASIC PROPERTIES OF THE EXTENSIONS SCC, UNIMODULAR VACUUM CASE 25 EXISTENCE OF INEQUIVALENT EXTENSIONS