MATHEMATICAL LECTURE NOTES
CONTENTS PART I PROBLEMS WITH MULTIPLE SCALES FROM DISCRETE SYSTEMS TO CONTINUOUS VARIATIONAL PROBLEMS: AN INTRODUCTION DISCRETE PROBLEMS WITH LIMIT ENERGIES DE?NED ON SOBOLEV SPACES DISCRETE FUNCTIONALS EQUIVALENT ENERGIES ON SOBOLEV FUNCTIONS CONVEX ENERGIES NON-CONVEX ENERGIES WITH SUPERLINEAR GROWTH A GENERAL CONVERGENCE THEOREM CONVERGENCE OF MINIMUM PROBLEMS MORE EXAMPLES ENERGIES DEPENDING ON SECOND-DI?ERENCE QUOTIENTS LIMIT ENERGIES ON DISCONTINUOUS FUNCTIONS: TWO EXAMPLES THE BLAKE-ZISSERMAN MODEL SECOND-DI?ERENCE INTERACTIONS LENNARD-JONES POTENTIALS RENORMALIZATION-GROUP APPROACH GENERAL CONVERGENCE RESULTS FUNCTIONS OF BOUNDED VARIATION NEAREST-NEIGHBOUR INTERACTIONS LONG-RANGE INTERACTIONS REFERENCES RELAXATION FOR BULK AND INTERFACIAL ENERGIES INTRODUCTION LOWER SEMICONTINUITY - THE DIRECT METHOD OF THE CALCULUS OF VARIATIONS RELAXATION PRELIMINARIES FUNCTIONS OF BOUNDED VARIATION AND SPECIAL FUNCTIONS OF BOUNDED VARIATION NOTATION MEASURE SPACES STRUCTURE OF BV FUNCTIONS APPROXIMATE DISCONTINUITY POINTS AND APPROXIMATE JUMP POINTS SPECIAL FUNCTIONS OF BOUNDED VARIATION BV (SBV ) FUNCTIONS IN ONE DIMENSION LOWER SEMICONTINUITY AND RELAXATION IN ONE DIMENSION A CHARACTERIZATION OF LSC FOR FUNCTIONALS DE?NED ON SBV (I) RELAXATION ON BV (I) A SU?CIENT CONDITION FOR LSC FOR FUNCTIONALS DE?NED ON SBV (I) RELAXATION IN BV (I) LOWER SEMICONTINUITY AND RELAXATION IN HIGHER DIMENSION LOWER SEMICONTINUITY AND RELAXATION IN HIGHER DIMENSION AND FOR VECTOR-VALUED FUNCTIONS UPPER BOUND FOR THE RELAXED FUNCTIONAL DE GIORGI-LETTA CRITERION TO PROVE MEASURE PROPERTIES OF THE RELAXED FUNCTIONAL INTEGRAL REPRESENTATION PROOF OF THE MAIN THEOREM RELAXATION IN HIGHER DIMENSION (ISOTROPY ASSUMPTIONS) APPENDIX APPENDIX A: LOWER SEMICONTINUOUS ENVELOPE APPENDIX B: PROOF OF (73) APPENDIX C: PROOF OF STEP 3 IN PROPOSITION 67 CONVERGENCE OF DIRICHLET FORMS ON FRACTALS CONSTRUCTION OF AN ENERGY ON THE GASKET DIRICHLET FORMS ON ?NITELY RAMI?ED FRACTALS MAIN PROPERTIES OF RENORMALIZATION AND HARMONIC EXTENSION HOMOGENIZATION ON THE GASKET HOMOGENIZATION ON GENERAL FRACTALS HOMOGENIZATION IN PERFORATED DOMAINS APPEARENCE OF A “TERM ETRANGE” DIRICHLET PROBLEMS IN DOMAINS WITH LOW CONCENTRATION OF HOLES BASIC NOTATION AND SETTING OF THE PROBLEM THE HOMOGENIZATION THEOREM HOMOGENIZATION PROBLEMS IN PERFORATED DOMAIN WITH OSCILLATING FOURIER BOUNDARY CONDITIONS FORMAL ASYMPTOTIC ANALYSIS OF THE PROBLEM MAIN ESTIMATES AND RESULTS AUXILIARY RESULTS JUSTI?CATION OF ASYMPTOTICS HOMOGENIZATION OF RANDOM NON STATIONARY PARABOLIC OPERATORS THE SETUP FACTORIZATION OF THE EQUATION A PRIORI ESTIMATES FOR THE FACTORIZED EQUATION AUXILIARY PROBLEMS HOMOGENIZATION OF THE FACTORIZED EQUATION HOMOGENIZATION OF THE ORIGINAL EQUATION EQUATIONS WITH DI?USION DRIVING PROCESS PART II PROBLEMS WITH CONCENTRATION -CONVERGENCE FOR CONCENTRATION PROBLEMS TWO CLASSICAL EXAMPLES SOBOLEV INEQUALITY CAPACITY AND ISOPERIMETRIC INEQUALITY FOR THE CAPACITY THE GENERAL PROBLEM VARIATIONAL FORMULATION GENERALIZED SOBOLEV INEQUALITY CONCENTRATION ? +-CONVERGENCE THE CONCENTRATION RESULT IN TERMS OF ? +-CONVERGENCE IDENTI?CATION OF THE CONCENTRATION POINT THE GREEN’S FUNCTION AND THE ROBIN FUNCTION ASYMPTOTIC EXPANSION IN ? +-CONVERGENCE THE RESULT HARMONIC TRANSPLANTATION IRREGULAR DOMAINS GAMMA-CONVERGENCE OF GRADIENT ?OWS AND APPLICATIONS TO GINZBURG-LANDAU VORTEX DYNAMICS PRESENTATION OF THE GINZBURG-LANDAU MODEL GAMMA-CONVERGENCE ?-CONVERGENCE OF GINZBURG-LANDAU THE ABSTRACT RESULT FOR ?-CONVERGENCE OF GRADIENT-?OWS THE ABSTRACT SITUATION INTERPRETATION IDEA OF THE PROOF APPLICATION TO GINZBURG-LANDAU REMARKS PROOF OF THE ADDITIONAL CONDITIONS FOR GINZBURG-LANDAU A PRODUCT-ESTIMATE FOR GINZBURG-LANDAU APPLICATION TO THE DYNAMICS PROOF OF THE CONSTRUCTION 2?) EXTENSIONS OF THE METHOD COLLISIONS SECOND ORDER QUESTIONS - STABILITY ISSUES PDE ANALYSIS OF CONCENTRATING ENERGIES FOR THE GINZBURG- LANDAU EQUATION THE MONOTONICITY FORMULA SCALING PROPERTIES CONSEQUENCES FOR µ? REGULARITY THEOREMS FOR (GL)? BOCHNER, SMALL ENERGY REGULARITY AND CLEARING-OUT POTENTIAL AND MODULUS ESTIMATES ?µ IS A MINIMAL SURFACE CLASSICAL AND WEAK NOTIONS OF MEAN CURVATURE DERIVING THE CURVATURE EQUATION