DIRECT METHODS IN THE CALCULUS OF VARIATIONS
CONTENTS INTRODUCTION THE DIRECT METHODS OF THE CALCULUS OF VARIATIONS CONVEX ANALYSIS AND THE SCALAR CASE CONVEX ANALYSIS LOWER SEMICONTINUITY AND EXISTENCE RESULTS THE ONE DIMENSIONAL CASE QUASICONVEX ANALYSIS AND THE VECTORIAL CASE QUASICONVEX FUNCTIONS QUASICONVEX ENVELOPES QUASICONVEX SETS LOWER SEMICONTINUITY AND EXISTENCE THEOREMS RELAXATION AND NON-CONVEX PROBLEMS RELAXATION THEOREMS SOME EXISTENCE THEOREMS FOR DI?ERENTIAL INCLUSIONS SOME EXISTENCE RESULTS FOR NON-QUASICONVEX INTEGRANDS MISCELLANEOUS H¨OLDER AND SOBOLEV SPACES SINGULAR VALUES SOME UNDERDETERMINED PARTIAL DI?ERENTIAL EQUATIONS EXTENSION OF LIPSCHITZ MAPS I CONVEX ANALYSIS AND THE SCALAR CASE CONVEX SETS AND CONVEX FUNCTIONS CONVEX SETS BASIC DE?NITIONS AND PROPERTIES SEPARATION THEOREMS CONVEX HULL AND CARATH´EODORY THEOREM EXTREME POINTS AND MINKOWSKI THEOREM CONVEX FUNCTIONS CONTINUITY OF CONVEX FUNCTIONS CONVEX ENVELOPE LOWER SEMICONTINUOUS ENVELOPE LEGENDRE TRANSFORM AND DUALITY SUBGRADIENTS AND DI?ERENTIABILITY OF CONVEX FUNCTIONS GAUGES AND THEIR POLARS CHOQUET FUNCTION WEAK LOWER SEMICONTINUITY PRELIMINARIES SOME APPROXIMATION LEMMAS NECESSARY CONDITION: THE CASE WITHOUT LOWER ORDER TERMS NECESSARY CONDITION: THE GENERAL CASE SU?CIENT CONDITION: A PARTICULAR CASE SU?CIENT CONDITION: THE GENERAL CASE WEAK CONTINUITY AND INVARIANT INTEGRALS WEAK CONTINUITY INVARIANT INTEGRALS EXISTENCE THEOREMS AND EULER-LAGRANGE EQUATIONS EXISTENCE THEOREMS EULER-LAGRANGE EQUATIONS SOME REGULARITY RESULTS AN EXISTENCE THEOREM THE EULER-LAGRANGE EQUATION THE CLASSICAL AND THE WEAK FORMS SECOND FORM OF THE EULER-LAGRANGE EQUATION SOME INEQUALITIES POINCAR´E-WIRTINGER INEQUALITY WIRTINGER INEQUALITY HAMILTONIAN FORMULATION REGULARITY LAVRENTIEV PHENOMENON II QUASICONVEX ANALYSIS AND THE VECTORIAL CASE POLYCONVEX, QUASICONVEX AND RANK ONE CONVEX FUNCTIONS DE?NITIONS AND MAIN PROPERTIES DE?NITIONS AND NOTATIONS MAIN PROPERTIES FURTHER PROPERTIES OF POLYCONVEX FUNCTIONS FURTHER PROPERTIES OF QUASICONVEX FUNCTIONS FURTHER PROPERTIES OF RANK ONE CONVEX FUNCTIONS EXAMPLES QUASIA?NE FUNCTIONS QUADRATIC CASE CONVEXITY OF SO (N) × SO (N) AND O (N) × O (N) INVARIANT FUNCTIONS POLYCONVEXITY AND RANK ONE CONVEXITY OF SO (N) × SO (N) AND O (N) × O (N) INVARIANT FUNCTIONS FUNCTIONS DEPENDING ON A QUASIA?NE FUNCTION THE AREA TYPE CASE THE EXAMPLE OF SVERAK THE EXAMPLE OF ALIBERT-DACOROGNA-MARCELLINI QUASICONVEX FUNCTIONS WITH SUBQUADRATIC GROWTH 5 3 10 THE CASE OF HOMOGENEOUS FUNCTIONS OF DEGREE ONE 5 3 11 SOME MORE EXAMPLES APPENDIX: SOME BASIC PROPERTIES OF DETERMINANTS POLYCONVEX, QUASICONVEX AND RANK ONE CONVEX ENVELOPES THE POLYCONVEX ENVELOPE DUALITY FOR POLYCONVEX FUNCTIONS ANOTHER REPRESENTATION FORMULA THE QUASICONVEX ENVELOPE THE RANK ONE CONVEX ENVELOPE SOME MORE PROPERTIES OF THE ENVELOPES ENVELOPES AND SUMS OF FUNCTIONS ENVELOPES AND INVARIANCES DUALITY FOR SO (N) × SO (N) AND O (N) × O (N) INVARIANT FUNCTIONS THE CASE OF SINGULAR VALUES THE KOHN-STRANG EXAMPLE THE SAINT VENANT-KIRCHHO?ENERGY FUNCTION THE CASE OF A NORM POLYCONVEX, QUASICONVEX AND RANK ONE CONVEX SETS SEPARATION THEOREMS FOR POLYCONVEX SETS APPENDIX: FUNCTIONS WITH ?NITELY MANY GRADIENTS THE DI?ERENT TYPES OF CONVEX HULLS THE DI?ERENT CONVEX HULLS THE DI?ERENT CONVEX ?NITE HULLS EXTREME POINTS AND MINKOWSKI TYPE THEOREM FOR POLYCONVEX, QUASICONVEX AND RANK ONE CONVEX SETS GAUGES FOR POLYCONVEX SETS CHOQUET FUNCTIONS FOR POLYCONVEX AND RANK ONE CONVEX SETS THE CASE OF POTENTIAL WELLS THE CASE OF A QUASIA?NE FUNCTION A PROBLEM OF OPTIMAL DESIGN LOWER SEMI CONTINUITY AND EXISTENCE THEOREMS IN THE VECTORIAL CASE NECESSARY CONDITION LOWER SEMICONTINUITY FOR QUASICONVEX FUNCTIONS WITHOUT LOWER ORDER TERMS LOWER SEMICONTINUITY FOR GENERAL QUASICONVEX FUNCTIONS FOR P = ? LOWER SEMICONTINUITY FOR GENERAL QUASICONVEX FUNCTIONS FOR 1 ?P < ? LOWER SEMICONTINUITY FOR POLYCONVEX FUNCTIONS WEAK CONTINUITY SU?CIENT CONDITION EXISTENCE THEOREM FOR QUASICONVEX FUNCTIONS EXISTENCE THEOREM FOR POLYCONVEX FUNCTIONS APPENDIX: SOME PROPERTIES OF JACOBIANS III RELAXATION AND NON-CONVEX PROBLEMS RELAXATION THEOREMS THE CASE WITHOUT LOWER ORDER TERMS THE GENERAL CASE 10 IMPLICIT PARTIAL DI?ERENTIAL EQUATIONS 10 1 INTRODUCTION 10 2 EXISTENCE THEOREMS 10 2 1 AN ABSTRACT THEOREM 10 2 2 A SU?CIENT CONDITION FOR THE RELAXATION PROPERTY 10 2 3 APPENDIX: BAIRE ONE FUNCTIONS 10 3 EXAMPLES 10 3 1 THE SCALAR CASE 10 3 2 THE CASE OF SINGULAR VALUES 10 3 3 THE CASE OF POTENTIAL WELLS 10 3 4 THE CASE OF A QUASIA?NE FUNCTION 10 3 5 A PROBLEM OF OPTIMAL DESIGN 11 EXISTENCE OF MINIMA FOR NON-QUASICONVEX INTEGRANDS 11 1 INTRODUCTION 11 2 SU?CIENT CONDITIONS 11 3 NECESSARY CONDITIONS 11 4 THE SCALAR CASE 11 4 1 THE CASE OF SINGLE INTEGRALS 11 4 2 THE CASE OF MULTIPLE INTEGRALS 11 5 THE VECTORIAL CASE 11 5 1 THE CASE OF SINGULAR VALUES 11 5 2 THE CASE OF QUASIA?NE FUNCTIONS 11 5 3 THE SAINT VENANT-KIRCHHO?ENERGY 11 5 4 A PROBLEM OF OPTIMAL DESIGN 11 5 5 THE AREA TYPE CASE 11 5 6 THE CASE OF POTENTIAL WELLS IV MISCELLANEOUS 12 FUNCTION SPACES 12 1 INTRODUCTION 12 2 MAIN NOTATION 12 3 SOME PROPERTIES OF H¨OLDER SPACES 12 4 SOME PROPERTIES OF SOBOLEV SPACES 12 4 1 DE?NITIONS AND NOTATIONS 12 4 2 IMBEDDINGS AND COMPACT IMBEDDINGS 12 4 3 APPROXIMATION BY SMOOTH AND PIECEWISE A?NE FUNCTIONS 13 SINGULAR VALUES 13 1 INTRODUCTION 13 2 DE?NITION AND BASIC PROPERTIES 13 3 SIGNED SINGULAR VALUES AND VON NEUMANN TYPE INEQUALITIES 14 SOME UNDERDETERMINED PARTIAL DI?ERENTIAL EQUATIONS 14 1 INTRODUCTION 14 2 THE EQUATIONS DIV U = F AND CURL U = F 14 2 1 A PRELIMINARY LEMMA 14 2 2 THE CASE DIV U = F 14 2 3 THE CASE CURL U = F 14 3 THE EQUATION DET ?U = F 14 3 1 THE MAIN THEOREM AND SOME COROLLARIES 14 3 2 A DEFORMATION ARGUMENT 14 3 3 A PROOF UNDER A SMALLNESS ASSUMPTION 14 3 4 TWO PROOFS OF THE MAIN THEOREM 15 EXTENSION OF LIPSCHITZ FUNCTIONS ON BANACH SPACES 15 1 INTRODUCTION 15 2 PRELIMINARIES AND NOTATION 15 3 NORMS INDUCED BY AN INNER PRODUCT 15 4 EXTENSION FROM A GENERAL SUBSET OF E TO E 15 5 EXTENSION FROM A CONVEX SUBSET OF E TO E CHAPTER THE DIRECT METHODS OF THE CALCULUS OF VARIATIONS THE MAIN PROBLEM THAT WE WILL BE INVESTIGATING THROUGHOUT THE PRESENT MONO- GRAPH IS THE FOLLOWING CONSIDER THE FUNCTIONAL I (U) := ? F (X, U (X) , ?U (X)) DX WHERE - ? ?RN, N ?1, IS A BOUNDED OPEN SET AND A POINT IN ? IS DENOTED BY X = (X1, , XN) ; - U : ? ?RN, N ?1, U = U1, , UN , AND HENCE ?U = ?UJ ?XI 1?J?N 1?I?N ?RN×N ; - F : ? × RN × RN×N ?R, F = F (X, U, ?) , IS A GIVEN FUNCTION WE SAY THAT THE PROBLEM UNDER CONSIDERATION IS SCALAR IF EITHER N = 1 OR N = 1; OTHERWISE WE SPEAK OF THE VECTORIAL CASE ASSOCIATED TO THE FUNCTIONAL I IS THE MINIMIZATION PROBLEM (P) M := INF {I (U) : U ?X} , MEANING THAT WE WISH TO ?ND U ?X SUCH THAT M = I (U) ?I (U) FOR EVERY U ?X HERE X IS THE SPACE OF ADMISSIBLE FUNCTIONS (IN MOST PARTS, IT IS THE SOBOLEV SPACE U0 + W 1,P ?; RN , WHERE U0 IS A GIVEN FUNCTION) ?2GU (2) WHEN TURNING OUR ATTENTION TO THE CASE N > N = 1 (IN OUR TERMINOLOGY, IT IS STILL PART OF THE SCALAR CASE), THE DIRICHLET INTEGRAL SURELY PLAYS A CENTRAL ROLE; WE HAVE THERE F (X, U, ?) = F (?) := 2 |?|2 A NATURAL GENERALIZATION IS WHEN 1 < P < ?AND F (X, U, ?) = F (?) := P |?|P THE MINIMAL SURFACE IN NON-PARAMETRIC FORM ENTERS ALSO IN THIS FRAMEWORK; WE HAVE IN THIS CASE F (X, U, ?) = F (?) := 1 + |?|2 IN GEOMETRICAL TERMS, THE INTEGRAL REPRESENTS THE AREA OF THE SURFACE GIVEN BY (X, U (X)) ?RN+1 WHEN X ?? ?RN (3) IN THE VECTORIAL CASE N, N ?2, THE ?RST EXAMPLE IS THE CASE OF MINIMAL SURFACES IN PARAMETRIC FORM, A GEOMETRICAL FRAMEWORK MORE GENERAL THAN THE PRECEDING ONE IN THIS CASE, WE HAVE N = N + 1 AND THEREFORE THE MATRIX ? ?R(N+1)×N WE DENOTE BY ADJN ? ?RN+1 THE VECTOR FORMED BY ALL THE N × N MINORS OF THE MATRIX ? FINALLY, WE LET F (X, U, ?) = F (?) := |ADJN ?| , WHERE | | STANDS FOR THE EUCLIDEAN NORM IN GEOMETRICAL TERMS, THE INTEGRAL REPRESENTS THE AREA OF THE SURFACE GIVEN BY U (X) ?RN+1 WHEN X ?? ?RN; MOREOVER, ADJN ?U REPRESENTS THE NORMAL TO THE SURFACE OTHER IMPORTANT EXAMPLES IN THE VECTORIAL CASE ARE MOTIVATED BY NON-LINEAR ELASTICITY A PARTICULARLY SIMPLE ONE IS WHEN N = N AND F (X, U, ?) = F (?) := G (DET ?) , DIRECT METHODS IN THE CALCULUS OF VARIATIONS ?; RN , IF WE WANT TO BE ABLE TO WRITE THE DI?ERENTIAL EQUATION NATURALLY ASSOCIATED TO THE MINIMIZATION PROBLEM AND KNOWN AS THE EULER-LAGRANGE EQUATION THIS TURNS OUT TO BE A STRATEGY TOO HARD TO IMPLEMENT IN MOST PROBLEMS, PARTICULARLY THOSE DEALING WITH PARTIAL DERIVATIVES (I E N > 1) THE ESSENCE OF THE DIRECT METHODS OF THE CALCULUS OF VARIATIONS IS TO SPLIT THE PROBLEM INTO TWO PARTS FIRST TO ENLARGE THE SPACE OF ADMISSIBLE FUNCTIONS, FOR EXAMPLE BY CONSIDERING SPACES SUCH AS THE SOBOLEV SPACES W 1,P SO AS TO GET A GENERAL EXISTENCE THEOREM AND THEN TO PROVE SOME REGULARITY RESULTS THAT SHOULD SATISFY ANY MINIMIZER OF (P) IN THE PRESENT BOOK, WE ARE ESSENTIALLY CONCERNED ONLY WITH THE ?RST PROBLEM IN MOST CASES, THE SPACE OF ADMISSIBLE FUNCTIONS IS X = U0 + W 1,P CONVEX ANALYSIS AND THE SCALAR CASE WE START WITH THE SCALAR CASE WHERE N = 1 OR N = 1 THE ?RST ONE CORRESPONDS TO THE CASE OF ONE SINGLE INDEPENDENT VARIABLE AND IS MUCH EASIER TO DEAL WITH, IN PARTICULAR FROM THE POINT OF VIEW OF REGULARITY IT IS DISCUSSED IN THE GENERAL DIRECT METHODS IN THE CALCULUS OF VARIATIONS I=1 ?I = WE THEN CONCLUDE THIS BRIEF ACCOUNT ON CONVEX SETS BY RECALLING THE NOTION OF EXTREME POINTS OF A CONVEX SET AND MINKOWSKI THEOREM, ENSURING THAT IF E IS COMPACT AND EEXT DENOTES THE SET OF EXTREME POINTS OF CO E, THEN CO E = CO EEXT DIRECT METHODS IN THE CALCULUS OF VARIATIONS