GEOMETRY AND PHYSICS

By G. GIOND

CONTENTS INTRODUCTION SINGULARITY LOCUS AS PARAMETER THE MAIN STEPS OF REDUCTION A FEW DE?NITIONS AN ALGORITHM IN EIGHT STEPS SIMPLE EXAMPLES OF REDUCED FUCHSIAN EQUATIONS REDUCTION AND APPLICATIONS PART I FUCHSIAN REDUCTION FORMAL SERIES THE OPERATOR D AND ITS ?RST PROPERTIES THE SPACE A?AND ITS VARIANTS FORMAL SERIES WITH VARIABLE EXPONENTS RELATION OF A?TO THE INVARIANT THEORY OF BINARY FORMS PROBLEMS GENERAL 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 XIV CONTENTS PART II THEORY OF FUCHSIAN PARTIAL DI?ERENTIAL 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 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 SOLUTION OF FUCHSIAN ELLIPTIC BOUNDARY-VALUE PROBLEMS BASIC LP RESULTS FOR EQUATIONS WITH DEGENERATE CHARACTERISTIC FORM SCHAUDER REGULARITY FOR FUCHSIAN PROBLEMS SOLUTION OF A MODEL FUCHSIAN OPERATOR PART III APPLICATIONS APPLICATIONS IN ASTRONOMY NOTIONS ON STELLAR MODELING POLYTROPIC MODEL POINT-SOURCE MODEL APPLICATIONS IN GENERAL RELATIVITY THE BIG-BANG SINGULARITY AND AVD BEHAVIOR GOWDY SPACE-TIMES SPACE-TIMES WITH TWIST APPLICATIONS IN DI?ERENTIAL GEOMETRY FE?ERMAN–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 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 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 PART IV BACKGROUND RESULTS DISTANCE FUNCTION AND H¨OLDER SPACES THE DISTANCE FUNCTION H¨OLDER SPACES ON C2+? DOMAINS INTERIOR ESTIMATES FOR THE LAPLACIAN PERTURBATION OF COE?CIENTS NASH–MOSER INVERSE FUNCTION THEOREM NASH–MOSER THEOREM WITHOUT SMOOTHING NASH–MOSER THEOREM WITH SMOOTHING SOLUTIONS REFERENCES INTRODUCTION THIS INTRODUCTION DE?NES FUCHSIAN REDUCTION, OR REDUCTION FOR SHORT, ILLUS- TRATES IT WITH A NUMBER OF SIMPLE EXAMPLES AND OUTLINES ITS MAIN SUCCESSES THE TECHNICAL ASPECTS OF THE THEORY ARE DEVELOPED IN THE SUBSEQUENT CHAP- TERS THE UPSHOT OF REDUCTION IS A PARAMETERIZED REPRESENTATION OF SOLUTIONS OF NONLINEAR DI?ERENTIAL EQUATIONS, IN WHICH SINGULARITY LOCUS MAY BE ONE OF THE PARAMETERS WE ?RST SHOW, ON A VERY SIMPLE EXAMPLE, THE ADVANTAGES OF SUCH A REPRESENTATION WE THEN DESCRIBE THE MAIN STEPS OF THE REDUCTION PRO- CESS IN GENERAL TERMS, AND SHOW IN CONCRETE SITUATIONS HOW THIS REDUCTION IS ACHIEVED WE CLOSE THIS INTRODUCTION WITH A SURVEY OF THE IMPACT OF REDUCTION ON APPLICATIONS 1 1 SINGULARITY LOCUS AS PARAMETER CONSIDER THE ODE DU DT = U2 THE SOLUTION TAKING THE VALUE A FOR T = T0 IS U(A, T0, T) = A 1 ?A(T ?T0) THE SET OF SOLUTIONS MAY BE PARAMETERIZED BY TWO PARAMETERS (A, T0) THE PROCEDURE IS QUITE SIMILAR TO SOLVING THE ALGEBRAIC EQUATION S2 + T2 = 1 IN THE FORM S = S(T) = ± ? 1 ?T2, IN WHICH T PLAYS THE ROLE OF A PARAMETER, OR LOCAL COORDINATE BUT UNLIKE THE REPRESENTATION S = S(T), THE REPRESENTATION U = U(A, T0, T) IS REDUNDANT: ONLY ONE PARAMETER SU?CES TO DESCRIBE THE GENERAL SOLUTION INDEED, LET B = A/(1 + AT0); WE OBTAIN U(A, T0, T) = B 1 ?BT 1 INTRODUCTION THE PARAMETER B GIVES THE POSITION OF THE (ONLY) SINGULAR POINT OF THE SOLUTION, NAMELY 1/B QUITE GENERALLY, ?NDING THE GENERAL SOLUTION OF A DI?ERENTIAL EQUATION AMOUNTS TO ?NDING A SET OF PARAMETERS THAT LABEL ALL SOLUTIONS CLOSE TO A GIVEN ONE THE PROCESS IS COMPARABLE TO ?NDING LOCAL COORDINATES ON A MANIFOLD TAKING SINGULARITY LOCUS AS ONE OF THE PARAMETERS, ONE OBTAINS A PARAMETERI- ZATION WITHOUT REDUNDANCY, UNLIKE THE PARAMETERIZATION BY T0 AND THE CAUCHY DATA AT TIME T0 1 2 THE MAIN STEPS OF REDUCTION A COMPLETE APPLICATION OF THE REDUCTION TECHNIQUE TO A SPECI?C PROBLEM F[U] =