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Courant R. Dirichlet's Principle, Confomal Mapping and Minimal Surfaces
Courant R.  Dirichlet's Principle, Confomal Mapping and Minimal Surfaces


: Dirichlet's Principle, Confomal Mapping and Minimal Surfaces

: Courant R.

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ed2k: ed2k stats

: 1950

: 330

: 18.04.2008

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Analytic extension of minimal surfaces      118
Analytic functions over Riemann domains      76
Bilinear form, symmetric      13
Boundary conditions, natural      22 40
Boundary coordination      see "Coordination of boundary points"
Boundary element, crossing      87
Boundary element, crossing, simple      87
Boundary points, mapping of      02 115
Boundary strip      15
Boundary value problem      2 5 249
Boundary value problem of second type      249 259ff.
Boundary values on manifolds      202
Boundary values, free      40 see
Boundary values, minimal surface with partly free      206ff.
Boundary, partly free      21
Boundary, rectifiable      131
Boundary, rectifiable, unstable minimal surface with      236
Branch points      122ff.
Branch points, definition      122
Branch points, variation of      184
Catenoid      120
cell      65
Cell, Dirichlets Principle for      65
Chains      199
Chains, Schwarz'      208
Characteristic equation for minimal surface      97
Characteristic number of closed surface      82
Cohesion, condition of      145
Compactness      23
Comparison theory      283ff.
Concentric circles, lengths of images of      127
Conformal invariants      275ff.
Conformal mapping      21
Conformal mapping into circle of circular lune      21
Conformal mapping into circle of circular sector      21
Conformal mapping into circle of ellipse      21
Conformal mapping into circle of semicircle      21
Conformal mapping into circle of upper half-plane      21
Conformal mapping of boundary      92
Conformal mapping of domains not of genus zero      80
Conformal mapping of domains of genus zero on curved surfaces      67
Conformal mapping of doubly connected domains      38
Conformal mapping of minimal surface on parameter domain      97
Conformal mapping of multiply connected domains      167ff.
Conformal mapping of multiply connected domains on plane domains      178
Conformal mapping of multiply connected domains on Riemann surfaces      58
Conformal mapping of multiply connected domains on slit domains      308
Conformal mapping of simply connected domains      38
Conformal mapping of simply connected domains by Dirichlets Principle      40
Conformal mapping on circular domains      169 290
Conformal mapping on Riemann surfaces bounded by unit circles      183
Conformal mapping on slit domains      45
Conformal mapping on slit domains of domains of infinite connectivity      58
Conformal mapping on slit domains of domains of infinite genus      92
Conformal mapping on slit domains of multiply connected domains      308
Conformal mapping on slit domains of plane domains      55
Conformal mapping on slit domains of Riemann domains of higher genus      85
Conformal mapping, canonical      see "Normal domains"
Conformal mapping, connection with univalent functions      276
Conformal mapping, continuity theorems      191ff.
Conformal mapping, extremum problem in      289
Conformal mapping, invariance of Dirichlets integral under      20
Conformal mapping, radius      311ff.
Conformal mapping, recent developments      249
Conformal mapping, role of in solution of Plateaus problem      117
Conformal mapping, uniqueness theorems      see "Uniqueness theorems for conformed mapping"
Conformal mapping, variational problems for      192
Continuity theorem for absolute minimum of Dirichlet's integral      126
Continuity theorem for area of harmonic surfaces      134
Continuity theorem for area of minimal surfaces      131
Continuity theorem for potential flow with dipole      54
Continuity theorems for Dirichlets integral under deformation      241
Continuity theorems in conformal mapping      191ff.
Convergence in Frechet sense of curves      104
Convergence of quadratic functionals      14
Convergence, monotonic of domains      54
Convexity and conformal mapping      192
Coordination of boundary points      62
Coordination of boundary points, domains defined by      68
Cross cap      see "Projective plane"
Crossing point      87
Crossing point, multiple      91
Crossing point, simple      89
Deformation      see "Pinching"
Degeneration of sequence of surfaces      145
Descent, method of      123
Descent, safe      124
Descent, steepest      124
Dipole      46
Dipole for Biemann domain      77
Dipole for non-orient able surface      78
Dipole on boundary      46
Dipole potential      see "Dipole"
Dipole, direction of      46
Dipole, intensity of      46
Dirichlet metric      267
Dirichlet product      12 266
Dirichlets integral      5
Dirichlets integral for harmonic functions      266ff.
Dirichlets integral over circle      9
Dirichlets integral, absolute minimum as functional of boundary      126
Dirichlets integral, continuity under deformation      241
Dirichlets integral, first variation of      109 123 168
Dirichlets integral, first variation of, for surface      228
Dirichlets integral, first variation of, in space of harmonic vectors      110
Dirichlets integral, first variation of, under variation of parameter representation      108
Dirichlets integral, first variation of, under variation of surface      110
Dirichlets integral, invariance under conformal mapping      20
Dirichlets integral, minimized by minimal surface      100
Dirichlets integral, oscillation of functions appraised by      18
Dirichlets integral, oyer Riemann domains      76
Dirichlets integral, relative minimum of      115
Dirichlets integral, second variation for surface      236
Dirichlets integral, semicontinuity of      11
Dirichlets principle      5 318
Dirichlets Principle for cell      65
Dirichlets Principle for circle      11
Dirichlets Principle for circle with partly free boundary      21
Dirichlets Principle for free boundary values      40
Dirichlets Principle for general domains      23
Dirichlets Principle for special domains      21
Dirichlets Principle, connection with conformal mapping      46
Dirichlets Principle, connection with Plateaus problem      99ff.
Dirichlets Principle, exact formulation of      11
Dirichlets Principle, proof for circle      11ff.
Douglas      4 111 201
Douglas' problem      see "Problem of Douglas"
Eigenvalue problem      278ff. 286
Elevation of a wall      233 237
Equipotential lines      46 86
Euler characteristic      48 141
Euler equation of Plateaus problem (non-parametric)      95
Euler equations of Plateaus problem (parametric)      96
Extremal domain      297ff.
Extremal length, method of      313ff. 318
Frechet convergence      104
Free boundaries      40ff. see
Gamier      98
Gergonne      199
Greens formula      49 251
Greens function      1 261ff. 268 278
Greens function of a domain      250
Greens function, connection with mapping on circular domains      290
Greens function, electrostatic interpretation of      250 252
Greens function, extension to analytic function      251 258
Greens function, invariant under conformal mapping      275
Greens function, symmetry law for      250
Greens function, variation of      292ff.
Hadamards example      9
Hadamards variation formula      292ff.
Half-plane slit domain      46 61 85
Harmonic functions, construction by smoothing process      24
Harmonic functions, convergence of sequence of      16
Harmonic functions, Dirichlets integral for      266ff.
Harmonic functions, extension to analytic functions      97
Harmonic functions, over Riemann domains      76
Harmonic functions, semicontinuity of Dirichlets integral for      11
Harmonic measure      251 261 280 295
Harmonic surface, definition      97
Harmonic surface, length of images of concentric circles by      127
Harmonic surfaces, continuity of area of      184
Harnacks theorem      11
Hausdorff      76
Hilbert space      24
Inequalities      273ff.
Inequalities, triangle      14
Inequality, isoperimetric for minimal surfaces      129
Inequality, Poincares integral      31
Inequality, Schwarz      14
Integral equation      278 286
Invariants, conformal      275
Isometric parameters      97 99
Isoperimetric inequality for minimal surfaces      129
Kernel functions      268ff. 281ff.
Kernel functions, connection with orthonormal systems      292
Kernel functions, discontinuities of      277ff.
Kernel functions, method of      318
Klein bottle, mapping on slit domain of      82 90
Lavrentieffs method      310ff.
Least area, problem of      see "Plateau's problem"
Levy      122
Lewy      175
Linear function space      13
Mapping      see "Conformal mapping"
Mapping radius      311ff.
Mean curvature      96
Metric      314
Metric in space of analytic functions      266
Metric, Dirichlet      267
Metric, given by quadratic form      14
Mini-maxima      223
Minimal surface with free boundaries      40 199ff. 201ff.
Minimal surface with free boundaries, doubly connected      209
Minimal surface with free boundaries, multiply connected      211
Minimal surface with free boundaries, that are discontinuous      220
Minimal surface with partly free boundaries      206ff.
Minimal surface, analytic extension of      118
Minimal surface, branchpoints of      122
Minimal surface, characteristic equation for      97
Minimal surface, conformal mapping on parameter domain of      97
Minimal surface, continuity of area of      131
Minimal surface, definition      98
Minimal surface, general construction of      see "Problem of Douglas"
Minimal surface, isoperimetric inequality for      129
Minimal surface, non-orientable      164
Minimal surface, not of genus zero      160ff.
Minimal surface, properties of free boundaries of      218
Minimal surface, reflection across line segment      119
Minimal surface, reflection across plane      218
Minimal surface, represented by stationary vector      112ff.
Minimal surface, solves Plateaus problem      100 116
Minimal surface, spanning closed manifold      213ff.
Minimal surface, surface of mean curvature zero      96
Minimal surface, trace on boundary manifold of      218
Minimal surface, unstable      199ff. 223ff.
Minimal surface, unstable, represented as stationary point of function of finite number of variables      226ff.
Minimal surface, unstable, with polygonal boundary      226ff. 235
Minimal surface, unstable, with rectifiable boundary      236
Minimal surfaces, infinitely many in same boundary      119ff.
Minimizing sequence      102
Minimizing sequence, definition      8
Minimizing sequence, generalized      215
Minimizing sequence, normalization of      103
Moduli of domains of finite connectivity      61
Moduli of doubly connected domains      315
Moduli, geometric representation of      61
Moebius strip      142
Morse      4 134 200
Multiply connected domains      56
Nehari      277
Neumann series      282
Neumanns function      259ff.
Neumanns function, definition of      261
Neumanns function, representation in terms of Greens function      262
Norm in space of analytic functions      266
Normal domains      45 144
Normal domains, canonical mappings on      253
Normal domains, definition of      45
Normalization of slit domains      92
Orientability      141
Orientability of Riemann domains      77
Orientable covering surface of nonorientable surface      78
Orthonormal systems      291ff.
Orthonormal systems, connection with kernel functions      292
Oscillation of functions      18
Oscillation of functions of vectors      101
Parallel-slit domains      see "Slit domains"
Parameter domains for Douglas problem      164
Parameters, isometric      97 99
Period      38
Pinching      153ff.
Pinching about point      241
Pinching about segment      242
Plane domains, conformal mapping of      55
Plateaus problem      95ff.
Plateaus problem, connection with Dirichlets Principle      99ff.
Plateaus problem, Euler equation (non-parametric)      95
Plateaus problem, Euler equations (parametric)      96
Plateaus problem, linearization of      97
Plateaus problem, method of safe descent for      124
Plateaus problem, solution of      98ff.
Plateaus problem, solution of, by minimal surface      100
Plateaus problem, solution of, by soap film      96
Plateaus problem, solution of, depends continuously on boundary      126ff.
Plateaus problem, solution of, for rectifiable boundary      131
Plateaus problem, solution of, role of conformal mapping in      117
Plateaus problem, statement of      95
Plateaus problem, uniqueness question for      119
Poincare      276
Poincares integral inequality      31
Poissons integral      9
Polyhedral domains      66
potential flow      46
Potential flow with dipole, continuity theorem for      54
Primend      64 72
Problem of Douglas      100 141ff.
Problem of Douglas, Douglas sufficient condition for      149ff.
Problem of Douglas, parameter domains for      164
Problem of Douglas, statement of      141
Problem of Douglas, sufficient condition for      142
Projective plane, mapping on slit domain of      82 90
Quadratic differential      290
Quadratic functionals      13
Quadratic functionals, convergence of      14
Quadratic functionals, properties of      13
Rado      4 116
Reflection principle for minimal surface, across line segment      119
Reflection principle for minimal surface, across plane      218
Retraction      234
Riemann      96 199
Riemann domains      64ff.
Riemann domains of genus zero      80
Riemann domains, abstract definition      76
Riemann domains, definition by non-overlapping cells      78
Riemann domains, important cases of      66
Riemann domains, mapping theorems for      85
Riemann surface      30 66
Riemanns mapping theorem      100 308
Riemanns mapping theorem, proved by solution of Plateaus problem      117
Schlicht-artig      80
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