UNIFIED MATHEMATICS

By EMMA HERNANDEZ

CONTENTS ELEMENTS OF HILBERT SPACE THEORY HILBERT SPACE SOME CONSTRUCTION PRINCIPLES OF HILBERT SPACES DIRECT SUMS OF HILBERT SPACES DUAL SPACES TENSOR PRODUCTS OF HILBERT SPACES SOBOLEV LATTICES SOBOLEV CHAINS SOBOLEV LATTICES FROM TENSOR PRODUCTS OF SOBOLEV CHAINS LINEAR PARTIAL DIFFERENTIAL EQUATIONS WITH CONSTANT COEF?CIENTS PARTIAL DIFFERENTIAL EQUATIONS IN H1 @ C E/ FIRST STEPS TOWARDS A SOLUTION THEORY THE TARSKI–SEIDENBERG THEOREM AND SOME CONSEQUENCES REGULARITY LOSS 0; : : : ; 0/ CLASSI?CATION OF PARTIAL DIFFERENTIAL EQUATIONS THE CLASSICAL CLASSI?CATION OF PARTIAL DIFFERENTIAL EQUATIONS ELLIPTIC, PARABOLIC, HYPERBOLIC? EVOLUTIONARY EXPRESSIONS IN CANONICAL FORM FUNCTIONS OF @ AND CONVOLUTIONS SYSTEMS AND SCALAR EQUATIONS 3 1 10 CAUSALITY 3 1 11 INITIAL VALUE PROBLEMS 3 1 12 SOME APPLICATIONS TO LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF MATHEMATICAL PHYSICS TRANSPORT EQUATION ACOUSTICS THERMODYNAMICS ELECTRODYNAMICS ELASTODYNAMICS FLUID DYNAMICS QUANTUM MECHANICS PARTIAL DIFFERENTIAL EQUATIONS IN H1 JDJ/ EXTENSION OF THE SOLUTION THEORY TO H1 JDJ/ SOME APPLICATIONS TO LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF MATHEMATICAL PHYSICS HELMHOLTZ EQUATION IN R3 HELMHOLTZ EQUATION IN R2 CAUCHY–RIEMANN OPERATOR WAVE EQUATION IN R2 (METHOD OF DESCENT) PLANE WAVES LINEARIZED NAVIER–STOKES EQUATIONS ELECTRO- AND MAGNETOSTATICS FORCE-FREE MAGNETIC FIELDS BELTRAMI FIELD EXPANSIONS CONVOLUTIONS IN H1 JDJ/; 2 RNC1 INTEGRAL REPRESENTATIONS OF CONVOLUTIONS WITH FUNDAMENTAL SOLUTIONS AN INTEGRAL REPRESENTATION OF THE SOLUTION OF THE TRANSPORT EQUATION POTENTIALS, SINGLE AND DOUBLE LAYERS IN R3 ELECTRO- AND MAGNETOSTATICS (BIOT–SAVART’S LAW) CAUCHY’S INTEGRAL FORMULA INTEGRAL REPRESENTATIONS OF SOLUTIONS OF THE HELMHOLTZ EQUATION IN R3 RETARDED POTENTIALS INTEGRAL REPRESENTATIONS OF SOLUTIONS OF THE TIME- HARMONIC MAXWELL EQUATIONS LINEAR EVOLUTION EQUATIONS LINEAR OPERATOR EQUATIONS IN SOBOLEV LATTICES POLYNOMIALS OF COMMUTING OPERATORS POLYNOMIALS OF COMMUTING, SELFADJOINT OPERATORS EVOLUTION EQUATIONS WITH POLYNOMIALS OF OPERATORS AS COEF?CIENTS CLASSI?CATION OF OPERATOR POLYNOMIALS WITH TIME DIFFERENTI- ATION CAUSALITY OF EVOLUTIONARY PROBLEMS ABSTRACT INITIAL VALUE PROBLEMS FIRST-ORDER-IN-TIME EVOLUTION EQUATIONS IN SOBOLEV LATTICES SOME EVOLUTION EQUATIONS OF MATHEMATICAL PHYSICS SCHRÖDINGER TYPE EQUATIONS THE SELFADJOINT LAPLACE OPERATOR SOME PERTURBATIONS BOUNDED PERTURBATIONS RELATIVELY BOUNDED PERTURBATIONS (THE COULOMB POTENTIAL) HEAT EQUATION THE SELFADJOINT OPERATOR CASE PRESCRIBED DIRICHLET AND NEUMANN BOUNDARY DATA TRANSMISSION INITIAL BOUNDARY VALUE PROBLEM STEFAN BOUNDARY CONDITION LOWER ORDER PERTURBATIONS DIRICHLET AND NEUMANN BOUNDARY CONDITION WAVE EQUATION REVERSIBLE HEAT TRANSPORT THE ELECTRIC BOUNDARY CONDITION SOME DECOMPOSITION RESULTS THE EXTENDED MAXWELL SYSTEM THE VECTORIAL WAVE EQUATION FOR THE ELECTROMAGNETIC FIELD THE RIGID BOUNDARY CONDITION FREE BOUNDARY CONDITION SHEAR AND PRESSURE WAVES PLATE DYNAMICS THERMO-ELASTICITY A “ROYAL ROAD” TO INITIAL BOUNDARY VALUE PROBLEMS A CLASS OF EVOLUTIONARY MATERIAL LAWS EVOLUTIONARY DYNAMICS AND MATERIAL LAWS THE SHAPE OF EVOLUTIONARY PROBLEMS WITH MATERIAL LAWS SOME SPECIAL CASES MATERIAL LAWS VIA DIFFERENTIAL EQUATIONS COUPLED SYSTEMS INITIAL VALUE PROBLEMS MEMORY PROBLEMS SOME APPLICATIONS REVERSIBLE HEAT TRANSFER MODELS OF THERMOELASTICITY THERMO-PIEZO-ELECTRO-MAGNETISM