| Some Remarks on the History and Objectives of the Calculus of Variations |
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1 | (14) |
| 1 Direct Methods of the Calculus of Variations |
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15 | (20) |
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1.1 The Fundamental Theorem of the Calculus of Variations |
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15 | (5) |
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1.2 Applying the Fundamental Theorem in Banach Spaces |
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20 | (5) |
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1.2.1 Sequentially Lower Semicontinuous Functionals |
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22 | (3) |
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1.3 Minimising Special Classes of Functions |
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25 | (5) |
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1.3.1 Quadratic Functionals |
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28 | (2) |
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1.4 Some Remarks on Linear Optimisation |
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30 | (1) |
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1.5 Ritz's Approximation Method |
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31 | (4) |
| 2 Differential Calculus in Banach Spaces |
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35 | (19) |
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35 | (1) |
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2.2 The Frechet Derivative |
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36 | (10) |
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43 | (1) |
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2.2.2 Some Properties of Frechet Derivatives |
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44 | (2) |
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2.3 The Gateaux Derivative |
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46 | (3) |
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49 | (2) |
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2.5 The Assumptions of the Fundamental Theorem of Variational Calculus |
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51 | (1) |
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2.6 Convexity of f and Monotonicity of f' |
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52 | (2) |
| 3 Extrema of Differentiable Functions |
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54 | (9) |
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3.1 Extrema and Critical Values |
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54 | (1) |
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3.2 Necessary Conditions for an Extremum |
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55 | (5) |
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3.3 Sufficient Conditions for an Extremum |
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60 | (3) |
| 4 Constrained Minimisation Problems (Method of Lagrange Multipliers) |
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63 | (14) |
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4.1 Geometrical Interpretation of Constrained Minimisation Problems |
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63 | (3) |
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4.2 Ljusternik's Theorems |
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66 | (6) |
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4.3 Necessary and Sufficient Conditions for Extrema Subject to Constraints |
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72 | (3) |
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75 | (2) |
| 5 Classical Variational Problems |
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77 | (65) |
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77 | (3) |
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5.2 Hamilton's Principle in Classical Mechanics |
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80 | (27) |
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5.2.1 Systems with One Degree of Freedom |
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81 | (14) |
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5.2.2 Systems with Several Degrees of Freedom |
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95 | (10) |
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5.2.3 An Example from Classical Mechanics |
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105 | (2) |
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5.3 Symmetries and Conservation Laws in Classical Mechanics |
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107 | (6) |
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5.3.1 Hamiltonian Formulation of Classical Mechanics |
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107 | (2) |
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5.3.2 Coordinate Transformations and Integrals of Motion |
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109 | (4) |
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5.4 The Brachystochrone Problem |
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113 | (3) |
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5.5 Systems with infinitely Many Degrees of Freedom: Field Theory |
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116 | (8) |
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5.5.1 Hamilton's Principle in Local Field Theory |
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117 | (5) |
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5.5.2 Examples of Local Classical Field Theories |
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122 | (2) |
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5.6 Noether's Theorem in Classical Field Theory |
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124 | (6) |
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5.7 The Principle of Symmetric Criticality |
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130 | (12) |
| 6 The Variational Approach to Linear Boundary and Eigenvalue Problems |
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142 | (29) |
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6.1 The Spectral Theorem for Compact Self-Adjoint Operators. Courant's Classical Minimax Principle. Projection Theorem . |
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142 | (6) |
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6.2 Differential Operators and Forms |
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148 | (4) |
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6.3 The Theorem of Lax-Milgram and Some Generalisations |
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152 | (4) |
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6.4 The Spectrum of Elliptic Differential Operators in a Bounded Domain. Some Problems from Classical Potential Theory |
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156 | (3) |
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6.5 Variational Solution of Parabolic Differential Equations. The Heat Conduction Equation. The Stokes Equations |
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159 | (12) |
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6.5.1 A General Framework for the Variational Solution of Parabolic Problems |
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161 | (5) |
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6.5.2 The Heat Conduction Equation |
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166 | (1) |
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6.5.3 The Stokes Equations in Hydrodynamics |
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167 | (4) |
| 7 Nonlinear Elliptic Boundary Value Problems and Monotonic Operators |
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171 | (21) |
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7.1 Forms and Operators - Boundary Value Problems |
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171 | (2) |
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7.2 Surjectivity of Coercive Monotonic Operators. Theorems of Browder and Minty |
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173 | (5) |
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7.3 Nonlinear Elliptic Boundary Value Problems. A Variational Solution |
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178 | (14) |
| 8 Nonlinear Elliptic Eigenvalue Problems |
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192 | (49) |
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192 | (3) |
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8.2 Determination of the Ground State in Nonlinear Elliptic Eigenvalue Problems |
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195 | (10) |
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8.2.1 Abstract Versions of Some Existence Theorems |
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195 | (8) |
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8.2.2 Determining the Ground State Solution for Nonlinear Elliptic Eigenvalue Problems |
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203 | (2) |
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8.3 Ljusternik-Schnirelman Theory for Compact Manifolds |
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205 | (12) |
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8.3.1 The Topological Basis of the Generalised Minimax Principle |
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205 | (2) |
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8.3.2 The Deformation Theorem |
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207 | (3) |
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8.3.3 The Ljusternik-Schnirelman Category and the Genus of a Set |
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210 | (5) |
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8.3.4 Minimax Characterisation of Critical Values of Ljusternik-Schnirelman |
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215 | (2) |
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8.4 The Existence of Infinitely Many Solutions of Nonlinear Elliptic Eigenvalue Problems |
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217 | (24) |
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8.4.1 Sphere-Like Constraints |
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217 | (3) |
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8.4.2 Galerkin Approximation for Nonlinear Eigenvalue Problems in Separable Banach Spaces |
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220 | (5) |
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8.4.3 The Existence of Infinitely Many Critical Points as Solutions of Abstract Eigenvalue Problems in Separable Banach Spaces |
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225 | (3) |
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8.4.4 The Existence of Infinitely Many Solutions of Nonlinear Eigenvalue Problems |
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228 | (13) |
| 9 Semilinear Elliptic Differential Equations. Some Recent Results on Global Solutions |
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241 | (99) |
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241 | (6) |
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9.2 Technical Preliminaries |
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247 | (19) |
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9.2.1 Some Function Spaces and Their Properties |
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247 | (5) |
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9.2.2 Some Continuity Results for Niemytski Operators |
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252 | (4) |
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9.2.3 Some Results on Concentration of Function Sequences |
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256 | (6) |
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9.2.4 A One-dimensional Variational Problem |
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262 | (4) |
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9.3 Some Properties of Weak Solutions of Semilinear Elliptic Equations |
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266 | (17) |
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9.3.1 Regularity of Weak Solutions |
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266 | (12) |
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9.3.2 Pohozaev's Identities |
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278 | (5) |
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9.4 Best Constant in Sobolev Inequality |
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283 | (4) |
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9.5 The Local Case with Critical Sobolev Exponent |
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287 | (7) |
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9.6 The Constrained Minimisation Method Under Scale Covariance |
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294 | (8) |
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9.7 Existence of a Minimiser I: Some General Results |
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302 | (10) |
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302 | (2) |
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9.7.2. Necessary and Sufficient Conditions |
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304 | (1) |
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9.7.3 The Concentration Condition |
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305 | (3) |
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308 | (2) |
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9.7.5 Growth Restrictions on the Potential |
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310 | (2) |
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9.8 Existence of a Minimiser II: Some Examples |
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312 | (10) |
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9.8.1 Some Non-translation-invariant Cases |
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313 | (3) |
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9.8.2 Spherically Symmetric Cases |
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316 | (3) |
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9.8.3 The Translation-invariant Case Without Spherical Symmetry |
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319 | (3) |
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9.9 Nonlinear Field Equations in Two Dimensions |
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322 | (10) |
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9.9.1 Some Properties of Niemytski Operators on Eq |
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323 | (3) |
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9.9.2 Solution of Some Two-Dimensional Vector Field Equations |
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326 | (6) |
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9.10 Conclusion and Comments |
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332 | (5) |
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332 | (2) |
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334 | (1) |
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335 | (2) |
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9.11 Complementary Remarks |
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337 | (3) |
| 10 Thomas-Fermi Theory |
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340 | (23) |
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340 | (2) |
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10.2 Some Results from the Theory of Lp Spaces (1 less than or equal to p less than or equal to infinity) |
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342 | (2) |
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10.3 Minimisation of the Thomas-Fermi Energy Functional |
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344 | (7) |
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10.4 Thomas-Fermi Equations and the Minimisation Problem for the TF Functional |
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351 | (6) |
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10.5 Solution of TF Equations for Potentials of the Form V(x) = Σkj=1 |
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357 | (4) |
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10.6 Remarks on Recent Developments in Thomas-Fermi and Related Theories |
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361 | (2) |
| Appendix A Banach Spaces |
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363 | (8) |
| Appendix B Continuity and Semicontinuity |
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371 | (2) |
| Appendix C Compactness in Banach Spaces |
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373 | (7) |
| Appendix D The Sobolev Spaces Wm,p(Ω) |
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380 | (11) |
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D.1 Definition and Properties |
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380 | (5) |
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D.2 Poincare's Inequality |
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385 | (1) |
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D.3 Continuous Embeddings of Sobolev Spaces |
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386 | (2) |
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D.4 Compact Embeddings of Sobolev Spaces |
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388 | (3) |
| Appendix E |
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391 | (4) |
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391 | (1) |
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E.2 Some Properties of Weakly Differentiable Functions |
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392 | (1) |
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E.3 Proof of Theorem 9.2.3 |
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393 | (2) |
| References |
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395 | (10) |
| Index of Names |
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405 | (2) |
| Subject Index |
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407 | |
