Page 240 - INTRODUCTION TO THE CALCULUS OF VARIATIONS
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Index
Absolutely continuous functions, 39 Ellipticity
Ascoli-Arzela Theorem, 12, 36, 141 uniform, 124
Enneper surface, 133, 137, 146
Banach fixed point theorem, 148, 149 Equicontinuity, 12
Bernstein problem, 147 Equiintegrability, 20
Bolza example, 87, 109 Euler-Lagrange equation, 2, 8, 9, 22,
Bolzano-Weierstrass Theorem, 20 45, 46, 48—50, 52—56, 59—
Brachistochrone, 1, 4, 55 62, 66—68, 72—76, 80, 92,
Brunn-Minkowski theorem, 10, 160, 93, 97, 98, 100, 106, 111—
163, 164 113, 116, 124, 125, 128, 129,
135, 141, 160
Canonical form, 62 Exact field, 75—77
Carathéodory theorem, 42, 107
Catenoid, 5, 133, 134 Fermat principle, 3, 56
Cauchy-Schwarz inequality, 17, 122 Fourier series, 20, 155
Conformal mapping, 129, 140, 141, Fubini theorem, 33
143, 145 Fundamental lemma of the calculus
Conformal parameters, 136 of variations, 23, 49, 81, 95,
Convergence 113, 136, 144
in the sense of distributions, 105
strong, 18 Gaussian curvature, 132
weak, 18
weak*, 18 Hölder continuous functions, 14, 15
Convex envelope, 42, 107, 108 Hölder inequality, 17, 22, 23, 30, 31,
Convex function, 46 33, 38, 90, 101, 103
Courant-Lebesgue lemma, 141 Hahn-Banach theorem, 30
Cycloid, 4, 55 Hamilton-Jacobi equation, 46, 69—
72, 77, 78
Difference quotient, 120, 121, 125 Hamiltonian, 46, 62, 66—69, 71
Dirac mass, 26, 107 Hamiltonian system, 66—68, 72, 115
Dirichlet integral, 2, 5, 9, 10, 79—81, Harmonic function, 137, 141
85, 95, 111, 117, 141, 143 Harnack theorem, 141
DuBois-Reymond equation, 59 Helicoid, 133
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