Notes on the Calculus of Variations and
                                  Optimization
                                 Preliminary Lecture Notes
                                  Adolfo J. Rumbos
                               c
                               
Draft date November 14, 2017
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                            Contents
                            1 Preface                                                                               5
                            2 Variational Problems                                                                  7
                                2.1   Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . .        7
                                2.2   Linearized Minimal Surface Equation . . . . . . . . . . . . . . . .          12
                                2.3   Vibrating String     . . . . . . . . . . . . . . . . . . . . . . . . . . .   14
                            3 Indirect Methods                                                                     19
                                3.1   Geodesics in the plane . . . . . . . . . . . . . . . . . . . . . . . .       19
                                3.2   Fundamental Lemmas . . . . . . . . . . . . . . . . . . . . . . . .           25
                                3.3   The Euler–Lagrange Equations . . . . . . . . . . . . . . . . . . .           31
                            4 Convex Minimization                                                                  47
                                4.1   Gˆateaux Differentiability . . . . . . . . . . . . . . . . . . . . . . .      47
                                4.2   AMinimization Problem . . . . . . . . . . . . . . . . . . . . . . .          52
                                4.3   Convex Functionals . . . . . . . . . . . . . . . . . . . . . . . . . .       53
                                4.4   Convex Minimization Theorem . . . . . . . . . . . . . . . . . . .            62
                            5 Optimization with Constraints                                                        65
                                5.1   Queen Dido’s Problem . . . . . . . . . . . . . . . . . . . . . . . .         65
                                5.2   Euler–Lagrange Multiplier Theorem . . . . . . . . . . . . . . . .            67
                                5.3   An Isoperimetric Problem . . . . . . . . . . . . . . . . . . . . . .         78
                            A Some Inequalities                                                                    87
                                A.1 The Cauchy–Schwarz Inequality . . . . . . . . . . . . . . . . . . .            87
                            B Theorems About Integration                                                           89
                                B.1 Differentiating Under the Integral Sign . . . . . . . . . . . . . . .           89
                                B.2 The Divergence Theorem . . . . . . . . . . . . . . . . . . . . . .             90
                            C Continuity of Functionals                                                            93
                                C.1 Definition of Continuity . . . . . . . . . . . . . . . . . . . . . . .          93
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