Mathematical analysis foundations and advanced techniques for functions of several variables

A Mathematicians and Other Scientists Andr´ e-Marie Amp` ere (1775–1836) Cesare Arzel` a (1847–1912) Eugenio Beltrami (1835–1899) Johann Bernoulli (1667–1748) Sergi Bernstein (1880–1968) Abram Besicovitch (1891–1970) Enrico Betti (1823–1892) Jacques Binet (1786–1856) Jean-Baptiste Biot (1774–1862) George Birkhoff (1884–1944) T Bonnesen Emile Borel (1871–1956) Luitzen E J Brouwer (1881–1966) Georg Cantor (1845–1918) Constantin Carath´eodory (1873–1950) Elie Cartan (1869–1951) Bonaventura Cavalieri (1598–1647) Pafnuty Chebycev (1821–1894) Rudolf Clausius (1822–1888) Charles Coulomb (1736–1806) Antoine Cournot (1801–1877) Georges de Rham (1903–1990) Paul Dirac (1902–1984) Peter Lejeune Dirichlet (1805–1859) Paul du Bois–Reymond (1831–1889) Dimitri Egorov (1869–1931) Leonhard Euler (1707–1783) Michael Faraday (1791–1867) Gyula Farkas (1847–1930) Werner Fenchel (1905–1986) Joseph Fourier (1768–1830) Ivar Fredholm (1866–1927) Guido Fubini (1879–1943) Carl Friedrich Gauss (1777–1855) J Willard Gibbs (1839–1903) Jacques Hadamard (1865–1963) William R Hamilton (1805–1865) Godfrey H Hardy (1877–1947) Felix Hausdorff (1869–1942) Hermann von Helmholtz (1821–1894) Heinrich Hertz (1857–1894) David Hilbert (18621943) William Hodge (19031975) Otto Hă older (18591937) Robert Hooke (1635–1703) Adolf Hurwitz (1859–1919) Christiaan Huygens (1629–1695) Carl Jacobi (18041851) Jean Pierre Kahane (1926 ) Erich Kă ahler (1906–2000) Leonid Kantorovich (1912–1986) Yitzhak Katznelson (1934– ) Harold Kuhn (1925– ) Joseph-Louis Lagrange (1736–1813) Henri Lebesgue (1875–1941) Adrien-Marie Legendre (1752–1833) Gottfried W von Leibniz (1646–1716) Beppo Levi (1875–1962) Tullio Levi–Civita (1873–1941) Rudolf Lipschitz (1832–1903) John E Littlewood (1885–1977) Hendrik Lorentz (1853–1928) Nikolai Lusin (1883–1950) Andrei Markov (1856–1922) James Clerk Maxwell (1831–1879) Adolph Mayer (1839–1903) Hermann Minkowski (1864–1909) Gaspard Monge (1746–1818) Oskar Morgenstern (1902–1976) Charles Morrey (1907–1984) Harald Marston Morse (1892–1977) John Nash (1928– ) Sir Isaac Newton (1643–1727) Otto Nikod´ ym (1887–1974) Emmy Noether (1882–1935) Marc-Antoine Parseval (1755–1836) Gabrio Piola (1794–1850) Sim´ eon Poisson (1781–1840) John Poynting (1852–1914) Johann Radon (1887–1956) Lord William Strutt Rayleigh (1842– 1919) Georg F Bernhard Riemann (18261866) Felix Savart (17911841) Erwin Schră odinger (18871961) Hermann Schwarz (1843–1921) Sergei Sobolev (1908–1989) Robert Solovay (1938– ) M Giaquinta and G Modica, Mathematical Analysis, Foundations and Advanced Techniques for Functions of Several Variables, DOI 10.1007/978-0-8176-8310-8, © Springer Science+Business Media, LLC 2012 395 396 A Mathematicians and Other Scientists Thomas Jan Stieltjes (1856–1894) Brook Taylor (1685–1731) Leonida Tonelli (1885–1946) Albert Tucker (1905–1995) Charles de la Vall´ee–Poussin (1866–1962) Giuseppe Vitali (1875–1932) John von Neumann (1903–1957) Karl Weierstrass (1815–1897) Hermann Weyl (1885–1955) Hassler Whitney (1907–1989) Ernst Zermelo (1871–1951) There exist many web sites dedicated to the history of mathematics, we mention, e.g., http://www-history.mcs.st-and.ac.uk/~history C Index 2-vector, 214 a.e., 308 – uniform convergence, 17 action – of a Lagrangian, 97, 157 action-angle variables, 197 adjoint – formal, 222 almost everywhere, 308 application – k-alternating, 216 – k-linear, 216 – bilinear, 213 – bilinear alternating, 213 – orientation preserving, 229 area formula – on submanifolds, 238 axiom of choice, 293–295 balance equation, Banach–Tarski paradox, 295 barycentric coordinates, 68 base point, 105 brachystochrone, 160 calibration, 187 catenoid, 160 co-phase space, 194 codifferential, 255 condition – Slater, 145 conditions – natural, 155 cone – convex – – dual, 107 – finite, 106 conformality relations, 183 conjugate exponent, 18 conservation – angular momentum, 186 – energy, 159, 181, 186, 194 – momentum, 186 conservation law, 185 constitutive equation, 4, 275 constraint – active, 110 – qualified, 110 constraints – holonomic, 172 – isoperimetric, 170 continuity equation, convex – duality, 87 convex body, 89 convex hull, 72 convex optimization – dual problem, 132, 140 – Kuhn–Tucker equilibrium conditions, 131 – Lagrangian, 131, 142 – primal problem, 130, 140 – saddle points, 143 – Slater condition, 145 – value function, 140 curl, 259, 261 curvature functional, 161, 162 – elastic lines, 164, 171 – variations – – normal, 163 – – tangential, 163 curve – minimal energy, 181 – minimal length, 181 – rectifiable, 366 s-density, 381 decomposition of unity, 236 degree, 250, 252 derivative – co-normal, 155 – Radon–Nikodym, 358, 373 – strong in Lp , 33 399 400 Index – weak in Lp , 35 determinant, 217, 224 – Binet formula, 224 – Cauchy–Binet formula, 227 – Laplace formula, 224 differential form – Beltrami, 189 – Cartan, 196, 204 – Poincar´ e, 204 – symplectic, 204 Dirac’s delta, 337 direction – admissible, 110 Dirichlet – integral, 49, 155 – – generalized, 175 – principle, 33 – problem, divergence, 259, 261 dual basis, 220 dual of H01 (Ω), 46 duality, 220 eikonal, 190 energy method, 3, 5, equation – balance, – Carath´ eodory, 189 – Cauchy–Riemann, 54 – constitutive, – continuity, – equilibrium, – – in the sense of distributions, 45 – – in weak form, 45 – Euler–Lagrange, 98, 99 – – constrained, 172 – – strong form, 152 – – weak form, 152 – fundamental of simple fluids, 100 – geodetic, 177 – Hamilton, 194 – Hamilton’s canonical system, 99 – Hamilton–Jacobi, 195 – – complete integral, 200 – – reduced, 211 – heat, 3, 14 – Laplace, 1, – – in a disk, 11 – – in a rectangle, – Newton, 157, 185 parabolic, Poisson, Schră odinger, 210 – self-dual, 258 – wave, 6, 158 – – with viscosity, 15 equations – Maxwell, 276 equilibrium conditions – Euler–Lagrange, 99 – Kuhn–Tucker, 111, 117 essential – supremum, 16 Euler–Lagrange equation, 98 – constrained, 172 example – Hadamard, 13 – Lebesgue, 166 – Weierstrass, 167 exterior algebra, 213, 220 exterior differential, 233 extremal point – of a convex set, 76 family of sets – σ-algebra, 284 – σ-algebra generated, 284 – σ-algebra of Borel sets, 284 – algebra, 284 – Borel sets, 301 – semiring, 298 Fenchel transform, 138 field – dual slope, 195 – eikonal, 189 – Mayer, 189 – of extremals, 188 – of vectors – – Helmholtz decomposition, 273 – – Hodge–Morrey decomposition, 274 – optimal, 190 – slope, 188 fine covering, 371 first integral, 159 formula – area, 333, 384 – Binet, 224 – Cauchy–Binet, 227 – Cavalieri, 315 – change of variables, 335, 385 – coarea, 387 – disintegration, 376 – Fourier inversion, 30 – homotopy, 268 – integration by parts – – for absolutely continuous functions, 365 – Laplace, 224 – Parseval, 31 – Plancherel, 31 – Poisson, 12 – repeated integration, 325 – Tonelli – – repeated integration, 331 Fourier – inverse transform, 30 Index – inversion formula, 30 – transform, 28, 29, 31 free energy, 157 function – -regularized, 21 – p-summable, 18, 19 – absolutely continuous, 37, 364 – Banach indicatrix, 384 – biharmonic, 161 – Borel measurable, 303 – Cantor–Vitali, 292, 364 – convex, 76 – – bipolar, 139 – – closure, 135 – – effective domain, 133 – – polar, 138 – – proper, 133 – – regularization, 135 – convex l.s.c envelope, 147 – distance, 146 – – from a convex set, 73 – distribution, 316 – epigraph, 77, 133 – gauge, 146 – Hardy–Littlewood maximal, 348 – harmonic, – holomorphic, 54 – indicatrix, 133 – integrable, 312 – integral p-mean, 23 – integral mean, 23 – l.s.c., 134 – Lebesgue measurable, 309 – Lebesgue points, 352 – Lebesgue representative, 352 – Lipschitz-continuous, 365–367 – lower semicontinuous, 134 – measurable, 303 – of bounded variation, 363 – payoff, 127 – principal of Hamilton, 198 – quasiconvex, 78, 124 – rapidly decreasing, 29 – saddle point, 124 – simple, 307 – stereograohic projection, 392 – strictly convex, 76, 82 – summable, 312 – support, 78, 146 game – noncooperative, 128, 129 – optimal strategies, 122 – payoff, 122 – utility function, 122 – zero sum game, 122 Gauss map, 252 Grassmannian, 230 401 gravitational potential, 64 H −1 (Ω), 46 Haar’s basis, 64 Hadamard’s example, 13 Hamilton – minimal action principle, 97 – principal function, 198 Hamilton’s equations, 194 Hamiltonian, 98, 157 harmonic functions – formula of the mean, 12 – maximum principle, – Poisson’s formula, 12 harmonic oscillator, 156, 211 Hausdorff dimension, 380 heat equation, Helmholtz’s decomposition formula for fields, 273 Hodge operator, 230 homotopy map, 266 hyperplane – separating, 69 – support, 69 inequality – between means, 92 – Chebycev, 316 – discrete Jensen’s, 77, 80, 92 – entropy, 92 – Fenchel, 138 – Hadamard, 93 – Hardy–Littlewood inequality, 348 – Hardy–Littlewood weak estimate, 348 Hă older, 18, 92 interpolation, 24 – isoperimetric, 38 – Jensen, 24 – Kantorovich, 393 – Markov, 316 – Minkowski, 18, 92 – Poincar´ e, 40 – Poincar´ e–Wirtinger, 40 – weak-(1 − 1), 349 – Young, 92 infinitesimal generator, 178 inner measure, 336 inner variation, 180 integral – absolute continuity, 317 – along the fiber, 266 – as measure of the subgraph, 322 – functions with discrete range, 337 – invariance under linear transformations, 331 – Lebesgue, 312 – linearity, 314 – Stieltjes–Lebesgue, 361 402 Index – – integration by parts, 363 – with respect to a discrete measure, 337 – with respect to a product measure, 330 – with respect to Dirac’s delta, 337 – with respect to the counting measure, 338 – with respect to the sum of measures, 337 integration by parts – for absolutely continuous functions, 365 isodiametric – inequality, 380, 389 Jensen inequality, 24 k-covectors, 221 – norm, 226 k-vectors, 217 – exterior product, 217 – norm, 226 – simple, 227 differential k-form, 233 – Brouwer degree, 251 – closed, 266 – codifferential, 255 – exact, 266 – exterior differential, 233 – harmonic, 258 – Helmholtz decomposition, 273 – Hodge–Morrey decomposition, 274 – inverse image, 234 – linking number, 253 – normal part, 257 – pull-back, 234, 240 – tangential part, 257 – volume of a hypersurface, 281 kinetic energy, 157 s-lower density, 381 Lagrange multiplier, 112, 170 Lagrangian, 97, 157 – null, 188 Laplace’s equation, 1, – weak form, 44 Laplace’s operator – on forms, 257 Laplacian – first eigenvalue, 171 lattice, 344 law – Amp` ere, 264 – Biot–Savart, 264 Legendre transform, 89, 90 Legendre’s polynomials, 64 lemma – du Bois–Reymond, 34, 168 – Farkas, 111 – Fatou, 314 – fundamental of the calculus of variations, 33 – Poincar´ e, 267 – Sard type, 388 linear programming, 116 – admissible solution, 116 – dual problem, 117 – duality theorem, 118 – feasible solution, 116 – objective function, 116 – optimality, 117 – primal problem, 117 linking number, 253 Lorentz’s metric, 278 map – harmonic, 174 – homotopy, 266 matrix – cofactor, 225, 249 – doubly stochastic, 94 – permutation, 94 – special symplectic, 202 – symplectic, 203 maximum principle – for elliptic equations, – for the heat equation, measure, 284 – σ-finite, 300 – absolutely continuous, 353 – Borel, 301 – Borel-regular, 301, 340 – conditional distribution, 377 – construction – – Method I, 298 – – Method II, 302 – counting, 300, 329 – derivative, 347 – – Radon–Nikodym, 358, 373 – Dirac, 342, 343 – disintegration, 376 – doubling property, 356 – Hausdorff, 378 – – s-densities, 381 – – spherical, 379 – inner-regular, 340, 342 – Lebesgue, 290, 301 – outer, 284 – outer-regular, 340 – product, 328 – Radon, 342 – restriction, 340 – singular, 353 – Stieltjes–Lebesgue, 361 – support, 343 method – energy, Index – Jacobi, 207 – separation of variables, 7, methods – direct, 164 – indirect, 164 metric – Lorentz, 278 minimal surfaces, 183 – parametric, 183 multiindex of length k, 215 multivectors – Hodge operator, 230 – product – – exterior, 220 – – scalar, 225 operator – biharmonic, 161 – codifferentiation, 255 – D’Alembert, – Hodge, 230 – Laplace, – – eigenvalues, 56 – – eigenvectors, 56 – – on forms, 257 – – variational characterization of eigenvalues, 57 – monotone, 80 – trace, 43 oriented – integral of a k-form, 239, 246 – plane, 230 outer measure – Lebesgue, 286 parabolic equation, parentheses – fundamental, 206 – Lagrange, 205 – Poisson, 206 Parseval’s formula, 31 permutation, 215 – signature, 215 – transposition, 215 permutation matrix, 94 Piola identities, 249 Plancherel formula, 31 Poincar´ e–Cartan integral, 196 point – Lebesgue, 352 – Nash, 129 Poisson’s equation, – weak form, 44 Poisson’s formula, 12 polyhedron, 72 potential – vector, 278 potential energy, 157 Poynting flux-energy vector, 280 principle – Hamilton’s minimal action, 97 – Dirichlet, 47–49 – Fermat, 156 – first of thermodynamics, 101 – Hamilton, 157 – Huygens, 192 – second of thermodynamics, 101 problem – diet, 115 – Dirichlet, 152 – – alternative, 55 – – eigenvvalues, 56 – – weak solution, 49 – investment management, 114 – isoperimetric, 170 – Neumann, 51, 155 – – weak form, 52 – optimal transportation, 115, 120 – with obstacle, 210 product – exterior, 214, 217 – – multivectors, 220 – triple, 262 – vector, 232 product measure, 328 property – doubling, 356 – mean, 25 – – for harmonic functions, 12 – universal of exterior product, 218 regularization – lower semicontinuous, 319 molliers, 21 upper semicontinuous, 319 Schră odinger’s equation, 210 self-dual equations, 258 set – μ-measurable – – following Carath´ eodory, 296 – σ-finite, 300 – Borel, 288, 301 – Cantor, 291 – Cantor ternary, 292 – contractible, 266 – convex, 67 – density, 352 – finite cone, 106 – – base cone, 106 – function, 283 – – σ-additive, 283 – – σ-subadditive, 283 – – additive, 283 – – countably additive, 283 – – monotone, 283 403 404 Index – Lebesgue measurable, 287 – Lebesgue nonmeasurable, 294 – measurable, 296 – – characterization, 288 – null set, 308 – perfect, 291 – polar, 87 – polyhedral, 72, 104 – polyhedron, 72, 104 – symmetric difference, 287 – zero set, 287 Slater condition, 145 space – L∞ , 16 – Lp , 19 – Sobolev, 33 Sturm–Liouville, 60 subdifferential, 79 submanifold – oriented, 240 surfaces – Gaussian curvature, 253 – minimal, 183 – – rotationally symmetric, 159 – of prescribed curvature, 155 symbols – Christoffel – – first kind, 177 – – second kind, 177 symplectic form, 204 symplectic group, 203 tensor – energy-momentum, 180 – Hamilton, 180 test – Carath´ eodory, 287 – – for measurability, 295 – – for measurability in metric spaces, 301 theorem – absolute continuity of the integral, 317 – alternative, 54 – Beppo Levi, 313 – Bernstein, 165 – Birkhoff, 94 – Brouwer, 250, 251 – Brunn–Minkowski, 96 – Carath´ eodory, 72 – Carath´ eodory’s construction, 299 – Carleson, 27 – circulation, 263 – construction of measures – – Method I, 299 – covering, 349 – – Besicovitch, 369, 371 – curl, 263 – de Rham, 270 – differentiation – – Lebesgue, 349, 358 – – Lebesgue–Besicovitch, 373 – – Lebesgue–Vitali, 348 – – under the integral sign, 318 – duality of linear programming, 118 – Egorov, 17 – existence of saddle points of von Neumann, 124 – Farkas–Minkowski, 108 – Federer–Whitney, 173 – Fredholm alternative, 108 – Fubini, 323, 325, 328, 330 – fundamental of calculus – – Lipschitz functions, 365 – Gauss–Bonnet, 253 – Gibbs – – on pure and mixed phases, 103 – Hardy–Littlewood, 349 – Helmholtz, 273 – Hodge–Morrey, 274 – integration of series, 317 – Jacobi, 201 – Kahane–Katznelson, 28 – Kakutani, 125 – Kirszbraun, 366 – Kolmogorov, 27 – Kuhn–Tucker, 111 – Lebesgue, 317 – – dominated convergence, 314 – Lebesgue decomposition, 354 – Lebesgue’s dominated convergence, 20 – Liouville, 195 – Lusin, 309, 341, 343 – Meyers–Serrin, 36 – minimax of von Neumann, 124 – monotone convergence – – for functions, 313 – – for measures, 285 – Motzkin, 75 – Nash, 129 – Noether, 185 – Perron–Frobenius, 113 – Poincar´ e recurrence, 195 – Poisson, 206 – Rademacher, 367 – Radon–Nikodym, 354 – regularity for 1-dimensional extremals, 168 – Rellich, 41 – repeated integration, 330 – Riesz, 345 – Sard type, 388 – Stokes, 247, 248 – Sturm–Liouville eigenvalue problem, 60 – Tonelli – – absolutely continuous curves, 366 – – repeated integration, 326 Index – total convergence, 317 – Vitali – – absolute continuity, 364 – – nonmeasurable sets, 294 – – on monotone functions, 362 – – Riemann integrability, 320 – Vitali–Lebesgue, 348 – Weierstrass representation formula, 190 thin plate, 161 total energy, 157 total variation, 363 transform – Fenchel, 138 – Fourier, 28 – Legendre, 89, 90 transformation – canonical, 203 – – exact, 204 – – Levi–Civita, 211 – – Poincar´ e, 211 – generalized canonical, 203 transition matrix, 113 s-upper density, 381 uniqueness – for the Dirichlet problem, – for the initial value problem, – for the parabolic problem, variable – cyclic, 197 – slack, 108, 117 variation – first, 152 – general, 179 – interior, 180 variational inequalities, 211 variational integral, 151 – admissible variations, 154 – extremal, 153 – stationary points, 180 – strongly stationary points, 180 Variational integrals – integrand, 151 variational integrals – regularity theorem, 168 vector calculus, 255 vector potential, 278 Vitali’s covering, 371 wave equation, – with viscosity, 15 weak estimate, 316 405 Spaces of Summable Functions and Partial Differential Equations This chapter aims at substantiating the abstract theory of Hilbert spaces developed in [GM3] After introducing the Laplace, heat and wave equations we present the classical method of separation of variables in the study of partial differential equations Then we introduce Lebesgue’s spaces of psummable functions and we continue with some elements of the theory of Sobolev spaces Finally, we present some basic facts concerning the notion of weak solution, the Dirichlet principle and the alternative theorem 1.1 Fourier Series and Partial Differential Equations 1.1.1 The Laplace, Heat and Wave Equations In our previous volumes [GM2, GM3, GM4] we discussed time by time partial differential equations, i.e., equations involving functions of several variables and some of their partial derivatives Among linear equations, i.e., equations for which the superposition principle holds, the following equations are particularly relevant, for instance, in classical physics: the Laplace equation, the heat equation and the wave equation They are respectively the prototypes of the so-called elliptic, parabolic and hyperbolic partial differential equations a Laplace’s and Poisson’s equation Laplace’s equation for a function u : Ω → R defined on an open set Ω ⊂ Rn , n ≥ 2, is n n ∂2u Δu := div ∇u = = uxi xi = i2 i=1 ∂x i=1 The operator Δ is called Laplace’s operator and the solutions of Δu = are called harmonic functions M Giaquinta and G Modica, Mathematical Analysis, Foundations and Advanced Techniques for Functions of Several Variables, DOI 10.1007/978-0-8176-8310-8_1, © Springer Science+Business Media, LLC 2012 6.3 Hausdorff Measures 379 (vi) Since the definition of Hδs (E) involves only the diameters of the sets of the covering of E, we may require that the covering is made by closed sets or convex and closed sets and even of open sets, since a closed set is the intersection of open sets with slightly larger diameters We may simplify even further and allow only coverings of E by balls ∞ s Hδ,sph (E) := inf ωs rjs E ⊂ ∪j B(xj , rj ) rj ≤ δ j=1 and s s (E) := lim Hδ,sph (E) Hsph δ→0 s (E) Hsph is again a Borel-regular measure, called the spherical Hausdorff measure However, if, for instance, E is an equilateral triangle in R2 with s (E) > Hδs (E), and, in general, one sees diameter less than δ, then Hδ,sph s that Hs and Hsph are different, although they agree on “sufficiently regular sets” This way we have at least two different ways of measuring s-dimensional sets in Rn , s < n Finally, it is easily seen that these two measures that not agree are in fact comparable: Since for every set E ⊂ Rn there exists a ball B that contains E and has diameter less than s diam(E), we see at once that Hs (E) ≤ Hsph (E) ≤ 2s Hs (E) ∀E ∈ B(Rn ) The following proposition collects some of the elementary properties of Hausdorff measures Hs 6.72 Proposition The Hausdorff measure Hs is Borel-regular; furthermore, Hs E is a Radon measure if Hs (E) < +∞ Moreover, the following hold: (i) Hs is invariant under translation and rotation, and is positively homogeneous of degree s, Hs (x + R(E)) = Hs (E), (ii) (iii) (iv) (v) Hs (λ E) = λs Hs (E) if R : Rn → Rn is linear and RT R = Id Hs = if s > n If ≤ s < t ≤ n, then Ht ≤ Hs More precisely, Ht (E) > implies Hs (E) = +∞ and Hs (E) < ∞ implies Ht (E) = s Hs (E) = if and only if H∞ (E) = n k If f : R → R is a Lipschitz map, then Lk (f (E)) ≤ (Lipf )k Hk (E) Proof By construction, see Proposition 6.2, Hs is a Borel-regular measure, while trivially Hs E is a Radon measure if Hs (E) < ∞ (i) This simply follows from the definition of Hs n (ii) Let s > n and let Q √ be a cube of side Since Q√can be covered by p cubes of side 1/p, we infer for δ ≥ n/p that Hsδ (Q) ≤ ωs 2−s ( n/p)s pn ≤ c(n, s)pn−s , hence for p → ∞, we get Hs (Q) = It follows that Hs (Rn ) = 380 Hausdorff and Radon Measures (iii) Since (diam E)t ≤ (diam E)s δt−s if diam E ≤ δ, we get Htδ (E) ≤ δt−s Hsδ (E) The claim follows at once letting δ → (iv) Clearly Hs∞ (E) ≤ Hs (E), hence Hs∞ (E) = if Hs (E) = Conversely, suppose that Hs∞ (E) = For any ∈]0, 1[ we can find a denumerable covering {Ej } of E with s diam(Ej ) < 2rj and ωs ∞ j=1 rj < Since the supremum of the rj ’s can be estimated by ( /ωs )1/s , we get Hsδ( ) (E) < σ and, letting with δ( ) := 2( /ωs )1/s → 0, we infer Hs (E) = (v) From diam(f (E)) ≤ Lip(f ) diam(E) we get Hk (f (E)) ≤ Lip(f ) Hk (E) The claim then follows since Hk = Lk in Rk , see Theorem 6.75 below 6.73 Remark In a less formal way, (iv) can be stated as follows: Hs (E) = if and only if for any > there exists a sequence of open sets {Ej } ∞ such that E ⊂ ∪j Ej and j=1 (diam Ej )s < We notice that (v) is convenient in order to estimate from below the Hausdorff measure Hs of a set To get an upper estimate, one usually estimates Hδs (E) by suitably choosing a covering of E The conclusion (iii) of Proposition 6.72 implies that for E ⊂ Rn , Hs (E) is finite and nonzero at most for a unique value of s, ≤ s ≤ n This motivates the following 6.74 Definition Let E ⊂ Rn The Hausdorff dimension of E is the number in [0, n] given by dimH (E) : = sup s | Hs (E) > = sup s | Hs (E) = +∞ = inf s | Hs (E) < ∞ = inf s | Hs (E) = Of course, the four different ways of defining dimH (E) agree because of (iii) of Proposition 6.72 that also implies that if < Hs (E) < +∞, then dimH (E) = s Notice, however, that not necessarily < Hs (E) < +∞ if dimH (E) = s 6.75 Theorem In Rn we have Hn = Hδn = Ln ∀δ > Proof We first prove that Ln (E) ≤ Hn δ (E) for any δ > Of course, it is not restrictive to assume Hn δ (E) < +∞ Consider a generic covering {Ej } of E with diam(Ej ) ≤ δ From the isodiametric inequality, see [GM4], we have ∞ Ln (E) ≤ j=1 Ln (Ej ) ≤ ωn 2−n ∞ (diam Ej )n , j=1 hence, by taking the infimum among all coverings, we get Ln (E) ≤ Hn δ (E) ∀δ > Proving the opposite inequality is more complicated First we notice that it suffices to prove that Hn (E) ≤ Ln (E) for bounded sets E In this case, as in (ii) of Proposition 6.72, we see that Hn is a Radon measure that has the doubling property by the n-homogeneity and the invariance under translations Given δ > and A ⊃ E with Ln (A) ≤ Ln (E) + , Vitali’s covering theorem, Theorem 6.64, yields a covering of E 6.3 Hausdorff Measures 381 made of closed and disjoint closed balls B(xi , ri ) with xi ∈ E, ri ≤ δ, B(xi , ri ) ⊂ A and Hn (E \ ∪i B(xi , ri )) = From the subadditivity of Hn δ we infer ∞ ∞ Hn δ (E) ≤ Hn δ (B(xi , ri )) ≤ ωn i=1 ∞ i=1 and, letting first δ → and then Ln (B(xi , ri )) ≤ Ln (A) ≤ Ln (E) + , rin = i=1 → 0, we conclude the proof We notice that in Rn , we have, instead, Hs = Hδs if s < n and δ > 6.3.1 Densities a Densities and Hausdorff measures The Radon–Nikodym derivative of a Borel measure with respect to a Hausdorff measure is meaningless since Hs (B(x, r)) = +∞ ∀s < n ∀x and ∀r > A suitable replacement is the so-called s-dimensional density Let λ be a Borel measure in Rn and < s ≤ n The upper s-dimensional density and the lower s-dimensional density of λ at x are defined by θs∗ (λ, x) := lim sup r→0 λ(B(x, r)) , rs θ∗s (λ, x) := lim inf r→0 λ(B(x, r)) , rs respectively If the two values agree, the common value θs (λ, x) := lim r→0 λ(B(x, r)) rs is called the s-density of λ at x In the previous definitions it is irrelevant whether the balls are open or closed Again, arguing as in Proposition 6.41, we have the following 6.76 Proposition Let λ : Rn → R be a Borel measure in Rn The functions x → θ∗s (λ, x) and x → θ∗s (λ, x) are Borel functions As for Radon measures, the following result is very useful in many instances that we shall not discuss here 6.77 Theorem Let λ be a Borel-regular measure, E ⊂ Rn and t ≥ (i) If θ∗s (λ, x) > t for all x ∈ E, then t Hs (E) ≤ λ(A) for any open set A ⊃ E In particular, if λ is a Radon measure, then t Hs (E) ≤ λ(E) (ii) If θs∗ (λ, x) ≤ t ∀x ∈ E, then λ(E) ≤ 2s t Hs (E) 382 Hausdorff and Radon Measures Proof (i) We may and assume λ(A) finite and t > Fix δ > 0; the family B := B(x, r) B(x, r) closed, x ∈ E, < r < δ/2, B(x, r) ⊂ A, λ(B(x, r) > t ωs r s finely covers E Let B be the subfamily selected according to Proposition 6.65 Since λ(A ∩ B(x, 5r)) > for any B = B(x, r) ∈ B , B is denumerable since each ball B ∈ B has positive radius, and λ(B) > Let B = {Bn }, Bn := B(xn , rn ), be the family of these balls It follows that ∞ n Hsδ (E) ≤ ωs ris + 5s ωs i=1 ris (6.38) i=n+1 On the other hand, ∞ ris ≤ ωs i=1 t ∞ λ(B(xi , ri )) ≤ i=1 1 λ(∪i B(xi , ri )) ≤ λ(A) < +∞, t t hence, letting n → ∞ in (6.38), ∞ Hs (E) ≤ ωs ris ≤ i=1 λ(A) t The last part of claim (i) follows since every Radon measure in Rn is outer-regular, see Proposition 6.2 (ii) We decompose E as E = ∪k Ek with Ek := x ∈ E λ(B(x, r)) ≤ tωs r s ∀r ∈]0, 1/k[ Clearly, Ek ⊂ Ek+1 , hence λ(Ek ) → λ(E) (even if the Ek ’s are nonmeasurable) It suffices then to prove λ(Ek ) ≤ 2s tHs (Ek ) ∀k = 1, 2, For this, fix δ < 1/(2k) and consider a covering of E with sets {Cj } with diam(Cj ) < δ and such that Cj ∩ Ek = ∅ Let B(xj , rj ) be a ball with center in Ek ∩ Cj and radius rj := diam(Cj ) that contains Cj Then rj < 1/k and ∞ λ(E) ≤ λ(∪j Cj ) ≤ ∞ λ(Cj ) ≤ j=1 λ(B(xj , rj )) j=1 ∞ ≤ t ωs ∞ rjs = t ωs j=1 (diam Cj )s j=1 By taking the infimum on the coverings we finally get λ(E) ≤ 2s t Hsδ (E) ≤ 2s t Hs (E) 6.78 Corollary Let λ be a Borel-regular measure in Rn and let < s ≤ n Then the following hold: (i) If λ is finite on Rn , then θ∗s (λ, x) < +∞ for Hs -a.e x ∈ Rn (ii) If λ is a Radon measure and λ(E) = 0, then θs (λ, x) = for Hs -a.e x ∈ E (iii) If E is λ-measurable and λ(E) < ∞, then θs (λ E, x) = for Hs a.e x ∈ Rn \ E 6.4 Area and Coarea Formulas 383 (iv) If E is λ-measurable and λ(E) < ∞, then 2−s ≤ θ∗s (E, x) ≤ for Hs -a.e x ∈ E Proof (i) If At := {x | θ ∗s (λ, x) > t}, we have Hs (At ) ≤ for t → +∞ λ(Rn ) t and the claim follows (ii) Since λ is Radon, then for each t > 0, for Et := x ∈ B | θ ∗s (λ, x) > t we have Hs (Et ) ≤ tλ(Bt ) = 0, hence {x | θ ∗s (λ, x) > 0} has zero Hs -measure (iii) Define At := {x ∈ E c θ ∗s (λ E, x) > t} for t > It suffices to prove that Hs (At ) = for all t > Conclusion (i) of Theorem 6.77 yields t Hs (At ) ≤ λ E(A) = λ(E ∩ A) for all open sets A that contain E c Since E is λ-measurable and with finite measure then λ E is a finite Borel measure, hence outer-regular, see Theorem 6.1 Therefore, t Hs (At ) ≤ inf λ E(A) | A ⊃ E c , A open =λ E(E c ) = (iv) Since E has finite measure, Hs E is a Radon measure, hence for Ct := {x ∈ E | θ ∗s (E, x) > t} we have t Hs (Ct ) ≤ Hs E(Ct ) = Hs (Ct ) It follows Hs (Ct ) = if t > since Hs (Ct ) < ∞ Similarly, if Et := {x ∈ E | θ ∗s (E, x) ≤ t}, then (ii) of Theorem 6.77 yields Hs (Et ) = Hs E(Et ) ≤ 2s tHs (Et ), i.e., Hs (Et ) = if t < 2−s since Hs (Et ) < +∞ 6.79 ¶ Let E ⊂ Rn The upper and lower s-densities of E at a point x are defined by θ ∗s (E, x) := θ ∗s (Hs E, x), θ∗s (E, x) := θ∗s (Hs E, x), respectively Suppose that E is Hs measurable and Hs (E) < +∞ Show that θ s (E, x) = for Hs -a.e x ∈ Rn \ E 6.4 Area and Coarea Formulas We conclude this chapter by proving two important formulas, the so-called area and coarea formulas The formulas hold true for Lipschitz maps, but in the sequel we restrict ourselves to the case of C maps 384 Hausdorff and Radon Measures Figure 6.5 Frontispieces of two monographs dealing with geometric measure theory 6.4.1 The area formula 6.80 Theorem (Area formula) Let Ω be an open set in Rn , f : Ω → RN , N ≥ n ≥ 1, a map of class C and A ⊂ Ω an Ln -measurable set Then the function y ∈ RN → H0 (A ∩ f −1 (y)) that for every y counts the points that are mapped into y is Hn -measurable and J(Df )(x) dx = H0 (A ∩ f −1 (y)) dHn (y), (6.39) RN A where J(Df )(x) := det(Df (x)T Df (x)) (6.40) In particular, Hn (f (A)) = J(Df )(x) dx (6.41) A if f is injective in A The function y → H0 (A ∩ f −1 (y)) =: N (f, A, y) is called the multiplicity function or Banach indicatrix Notice that f (A) = y ∈ RN N (f, A, y) = Since J(Df ) is continuous and nonnegative, the theorem states that both integrals in (6.39) exist, finite or infinite, and agree By approximating a nonnegative function u by simple functions and using Beppo Levi’s theorem we readily infer the following 6.4 Area and Coarea Formulas 385 6.81 Theorem (Formula of change of variables) Let f : Ω ⊂ Rn → RN , n ≤ N , Ω open, be a function of class C and let u : Rn → R be either a Ln -measurable and nonnegative function, or a function such that |u| J(Df ) is summable Then y→ u(x) x∈f −1 (y) is Hn -measurable and u(x) dHn (y) u(x) J(Df )(x) dx = Ω RN (6.42) x∈f −1 (y) In particular, if v : RN → R is Hn -measurable and nonnegative, then v(y) N (f, A, y) dHn (y) v(f (x)) J(Df )(x) dx = (6.43) RN A Proof If u = χA is the characteristic function of a measurable set A, then (6.42) and (6.39) agree since χA (x) = H0 (A ∩ f −1 (y)) x∈f −1 (y) By linearity, (6.42) holds for simple functions If u is measurable and nonnegative, it is the limit of a nondecreasing sequence of simple functions; passing to the limit, we infer the result by Beppo Levi’s theorem If |u|J(Df ) is summable, we decompose u as u = u+ − u− and conclude by subtracting (6.42) for u+ e u− Proof of Theorem 6.80 We shall prove the theorem when A ⊂⊂ Ω The general case follows easily by invading Ω with a sequence of compact sets Ωk ⊂⊂ Ω, with Ω = ∪k Ωk , Ωk ⊂ Ωk+1 , writing the formula for Ak := A ∩ Ωk and passing to the limit Let A ⊂⊂ Ω It is not restrictive to assume that f is Lipschitz in Ω, i.e., there is a constant L such that |f (x) − f (y)| ≤ L |x − y| ∀x, y ∈ A Step f (A) is Hn -measurable Let {Kh } be a sequence of compact sets such that Kh ⊂ A, Kh ⊂ Kh+1 and Ln (A) = Ln (∪h Kh ) Since f is continuous, we have f (∪h Kh ) = ∪h f (Kh ), hence f (∪h Kh ) is a Borel set as denumerable union of compact sets Now, f (A) = f (∪h Kh ) ∪ f (A \ ∪h Kh ) and Hn (f (A \ ∪ Kh )) ≤ Ln Hn (A \ ∪h Kh ) = Ln Ln (A \ ∪h Kh ) = 0; h therefore, f (A) is the union of a Borel set and of a set of zero Hn measure, hence f (A) is Hn -measurable Step y → H0 (A ∩ f −1 (y)) is Hn -measurable and RN H0 (A ∩ f −1 (y)) dHn (y) ≤ Ln Ln (A) (6.44) This is proved by constructing a sequence {gk } of functions from RN into R that are Hn -measurable and that converge a.e to the function y → H0 (A ∩ f −1 (y)) Decompose RN into union of cubes {Qki } with disjoint interiors points, sides that are parallel to the coordinate axes, congruent and with sides of length 2−k Set ∞ gk (y) := χf (A∩Qk ) (y), i=1 i y ∈ RN Since f (A ∩ Qki ) is Hn -measurable for all i, see Step 1, gk is Hn -measurable for all k Moreover, gk (y) ≤ gk+1 (y) ∀y We now remark the following: 386 Hausdorff and Radon Measures If f −1 (y) ∩ A = ∅, then gk (y) = = H0 (f −1 (y) ∩ A) for all k If H0 (A ∩ f −1 (y)) < ∞, then for sufficiently large k, each point of f −1 (y) ∩ A belongs to exactly one of the cubes Qki , hence gk (y) = H0 (A ∩ f −1 (y)) If H0 (A ∩ f −1 (y)) = +∞, trivially limk→∞ gk (y) = +∞ In conclusion, gk (y) ↑ H0 (A ∩ f −1 (y)) ∀y ∈ RN Finally, taking into account Beppo Levi’s theorem, RN H0 (A ∩ f −1 (y)) dHn (y) = lim k→∞ RN Hn (f (A ∩ Qki )) = lim k→∞ gk (y) dHn (y) i ≤ lim sup k→∞ Ln Ln (A ∩ Qki ) i = Ln Ln (A), i.e., (6.44) Step Let B := x ∈ Rn J(Df )(x) > and let t > We now prove the following There exist a decomposition of B into disjoint Borel sets Bj and injective linear maps Tj : Rn → RN such that the following hold: (i) f|Bj is injective −1 (ii) Lip(f|Bj ◦ Tj−1 ) ≤ t and Lip(Tj ◦ fB ) ≤ t j (iii) We have | det Tj | ≤ J(Df )(x) ≤ tn | det Tj | ∀ x ∈ Ej (6.45) tn Since Df (x0 ), x0 ∈ B, has maximal rank, the implicit function theorem yields r0 > such that f|B(x0 ,r0 ) is injective; moreover, if Tx0 : Rn → RN is the linear )(0) = Id, hence tangent map x → Df (x0 )(x), then Tx0 is invertible and D(f ◦ Tx−1 −1 Lip(f|B(x0 ,r0 ) ◦Tx−1 ) ≤ t, Lip(Tx0 ◦f|B(x0 ,r0 ) ) ≤ t, and (6.45) holds for all x ∈ B(x0 , r0 ) possibly for a smaller r0 In this way, we find a denumerable covering {B(xi , ri )} of B for which (i), (ii) and (iii) hold with Bj := B(xj , rj ) and Tj := Txj Then, by choosing inductively B1 := B(x1 , r1 ), T1 := Tx1 , B2 := B(x2 , r2 ) \ B1 , T2 := Tx2 , B3 := B(x3 , r(x3 )) \ (B1 ∪ B2 ) and so on, we find the requested decomposition Step Let A ⊂ B := {x ∈ Rn | J(Df )(x) > 0}, and let {Bj }, Tj be as in Step If we set Aj := A ∩ Bj , we have Hn (f (Aj )) = Hn (f|Bj ◦ Tj−1 ◦ Tj (Aj )) ≤ tn Hn (Tj (Aj )), −1 Hn (Tj (Aj )) = Hn (Tj ◦ f|B ◦ f|Bj (Aj )) ≤ tn Hn (f (Aj )), j J(Tj )Ln (Aj ) ≤ tn J(Df )(x) dx ≤ tn J(Tj )Ln (Aj ) Aj and, because of the area formula for linear maps, Theorem 5.100, we have Hn (Tj (Aj )) = J(Tj ) Ln (Aj ) and, therefore, 6.4 Area and Coarea Formulas 387 J(Tj )Ln (Aj ) ≤ Hn (f (Aj )) ≤ tn J(Tj )Ln (Aj ), tn J(Tj )Ln (Aj ) ≤ J(Df )(x) dx ≤ tn J(Tj )Ln (Aj ) tn Aj For t → 1, we conclude J(Df ) dx = Hn (f (Aj )) Aj and, summing in j, we infer the area formula, since H0 (A ∩ f −1 (y)) = H0 (Aj ∩ f −1 (y)) = j χf (Aj ) (y) j and RN H0 (A ∩ f −1 (y)) = Hn (f (Aj )) j Step A ⊂ {x | J(Df )(x) = 0} In this case it suffices to prove that Hn (f (A)) = Given > 0, let g : Rn → RN × Rn , g (x) := (f (x), x) Since f (x) is the first factor of g and the projection map (x, y) → x has Lipschitz constant 1, Hn (f (A)) ≤ Hn (g (A)) )2 On the other hand, J(Dg = det(Df T Df + J(Dg ) → J(Dg0 ) = J(Df ) Step then yields Hn (g (A)) = Id), hence J(Dg ) and, for J(Dg )(x) dx → A → 0, J(Df ) dx = A Step To conclude the proof, it suffices to decompose A as A = A ∩ x J(Df )(x) = A ∩ x J(Df )(x) > and apply Step and Step 6.4.2 The coarea formula 6.82 Theorem (Coarea formula) Let f : Ω → RN , Ω ⊂ Rn open, N ≤ n, be a function of class C (Ω), and let A ⊂ Ω be an Ln -measurable function Then for LN -a.e y ∈ RN , the set A ∩ f −1 (y) is Hn−N measurable, the function y → Hn−N (A ∩ f −1 (y)) is LN -measurable and J(Df )(x) dLn (x) = A RN Hn−N (A ∩ f −1 (y)) dLN (y) (6.46) Here J(Df )(x) := det(Df (x)Df (x)T ) (6.47) Again we notice that Theorem 6.82 states that if one of the two integrals in (6.46) exists, then the other exists too, and they are equal irrespective of their finiteness By an argument similar to that of Theorem 6.81, the following result can be proved starting from Theorem 6.82 388 Hausdorff and Radon Measures 6.83 Theorem Let f : Ω ⊂ Rn → RN , Ω open, N ≤ n, be a function of class C , and let u : Ω → R be a nonnegative and Ln -measurable function or, alternatively, let |u| J(Df ) be Ln -summable We have u(x) dHn−N (x) dLN (y) (6.48) u(x) J(Df )(x) dx = Rn RN f −1 (y) Theorems 6.82 and 6.83 may be seen as a “curvilinear” extension of Fubini’s theorem By applying (6.46) with A = B := {x ∈ Ω | J(Df )(x) = 0}, we infer Hn−N (B ∩ f −1 (y)) = for LN -a.e y ∈ RN On the other hand, if x ∈ B c , the implicit function theorem yields an open neighborhood Ux of x in Rn such that Ux ∩f −1 (y) is an Hn−N -submanifold of class C of Rn We can therefore state the following 6.84 Corollary (Sard-type theorem) Let f : Ω ⊂ Rn → RN , n ≥ N , be a map of class C (Ω) and let B := {x ∈ Ω | J(Df )(x) = 0} = {x | Rank Df (x) < N } For LN -a.e y ∈ RN we can decompose f −1 (y) as f −1 (y) = (f −1 (y) \ B) ∪ (f −1 (y) ∩ B) where Hn−N (B ∩ f −1 (y)) = and f −1 (y) \ B is a C (n-N)-submanifold of Rn Proof of Theorem 6.82 We divide the proof in eight steps Step We prove the theorem for linear maps of maximal rank Assume f : Rn → RN is a linear map of rank N From the polar decomposition formula, f = (f f T )1/2 U ∗ , where U ∗ U = IdRN and (f f T ) is an isomorphism Therefore, if Rn = RN × Rn−N and π : Rn → RN denotes the orthogonal projection onto RN , then f = σ ◦ π ◦ R, where R : Rn → Rn is orthogonal and σ := (f f T )1/2 : RN → RN is an isomorphism The invariance of the measure with respect to orthogonal transformations and Fubini’s theorem then yield Hn (A) = Hn (R(A)) = Hn−N R(A) ∩ π −1 (z) dLN (z) Moreover, changing variables in the integral with z = σ(y), we get dy = | det σ| dz, consequently, Hn−N R(A) ∩ π −1 (z) dLN (z) = | det σ| Hn−N R(A) ∩ π −1 σ−1 (y) dLN (y) and, since R is orthogonal and R(A) ∩ π −1 σ− 1(y)) = R(A ∩ f −1 (y)), for any y ∈ RN Hn−N R(A) ∩ π −1 σ−1 (y) = Hn−N A ∩ f −1 (y) , that is, | det σ| Hn (A) = Hn−N A ∩ f −1 (y) dLN (y) The claim then follows, since from RRT = Id and ππ T = IdRN , we have J(Df )2 = det(f f T ) = | det σ|2 6.4 Area and Coarea Formulas 389 We now prove the theorem for f ∈ C (Ω) As for the area formula, it suffices to prove the theorem for A ⊂⊂ Ω, and we may assume f Lipschitz in A, i.e., |f (x) − f (y)| ≤ L|x − y| ∀x, y ∈ A for some L > Step f −1 (y) is closed in Ω, hence a Borel set, consequently a Hn−N -measurable set Step f (A) is LN -measurable This may be proved as in Step of the proof of Theorem 6.80 Step We now prove that ∗ RN Hn−N (A ∩ f −1 (y)) dLN (y) ≤ ωn−N ωN N n L L (A) ωn ∗ This proves the coarea formula when A is a null set The symbol When ϕ : RN → R, we define ∗ (6.49) deserves a definition h(x) dx h measurable, h ≥ ϕ ϕ(x) dLN (x) := inf Trivially, ∗ ϕ dx ≤ ∗ g(x) dx if ϕ ≤ g and ∗ ϕ dx = ϕ dx if ϕ is LN -measurable j For j = 1, 2, we choose a family of closed ball {Bij }i such that A ⊂ ∪∞ i=1 Bi , j j j ∞ diam Bi ≤ 1/j, Bi ⊂ Ω and i=1 |Bi | ≤ |A| + 1/j, and we set diam Bij gij (y) := ωn−N gij n−N χf (Bj ) (y) i LN -measurable According to Step 2, the functions are Moreover, for each j, by choosing as covering of A ∩ f −1 (y) the balls Bij that cover f −1 (y), i.e., those for which y ∈ f (Bij ), from the definition of Hausdorff measure we get diam Bij (A ∩ f −1 (y)) ≤ ωn−N Hn−N 1/j j y∈f (Bi ) ∞ n−N = gij (y) i=1 Since the functions gij are LN -measurable, ∗ RN Hn−N (A ∩ f −1 (y)) dy = ≤ ∗ lim inf Hn−N (A ∩ f −1 (y)) dy 1/j RN j→∞ ∗ ∞ lim inf RN j→∞ i=1 gij dy = ∞ lim inf RN j→∞ i=1 (6.50) gij dy and, applying Fatou’s lemma, ∞ lim inf RN j→∞ i=1 gij dy ≤ lim inf j→∞ ∞ i=1 ∞ = lim inf j→∞ RN gij dy ωn−N i=1 diam Bij (6.51) n−N HN (f (Bij )) On the other hand, by using the isodiametric inequality, see [GM4], we get HN (f (Bji )) ≤ LN LN (Bij ) ≤ ωN hence, joining with (6.50) and (6.51), we conclude diam(Bij ) N , 390 Hausdorff and Radon Measures ∗ RN Hn−N (A ∩ f −1 (y)) dy ≤ LN lim inf j→∞ ∞ ωn−N ωN i=1 diam Bij n ∞ ωn−N ωN = LN lim inf Ln (Bij ) j→∞ ωn i=1 ≤ LN ωn−N ωN n L (A) ωn Step If A is Ln -measurable, the map y → Hn−N (A ∩ f −1 (y)) is Ln -measurable (i) If A is compact or open, then y → Hn−N (A ∩ f −1 (y)) is a Borel, hence measurable, map If A is compact and yh → y, then A ∩ f −1 (y) contains all limit points of A ∩ f −1 (yh ) Consequently, every open covering of A ∩ f −1 (y) necessarily covers A ∩ f −1 (yh ) for h sufficiently large, therefore, for any δ > (A ∩ f −1 (yh )) ≤ Hn−N (A ∩ f −1 (y)) lim sup Hn−N δ δ h→∞ It follows that for any t ∈ R the set y ∈ Rn Hn−N (A ∩ f −1 (y)) ≥ t (A ∩ f −1 (y)) is Borel, actually upper is closed, i.e., that the map y → Hn−N δ semicontinuous ∀δ > If A is open, then A = ∪h Kh , Kh ⊂ Kh+1 , Kh compact The claim then follows from the compact case since Hn−N (A ∩ f −1 (y)) = lim Hn−N (Kh ∩ f −1 (y)) h→∞ (ii) If A is an Ln null set, then y → Hn−N (A ∩ f −1 (y)) = for LN a.e y ∈ RN Consider a sequence of open sets Aj such that A ⊂ Aj , Aj+1 ⊂ Aj ∀j and Ln (Aj ) < 1/j From Step and Step we infer ∗ Hn−N (A ∩ f −1 (y)) dy ≤ and, for j → ∞, ∗ ∗ Hn−N (Aj ∩ f −1 (y)) dy ≤ C Ln (Aj ) ≤ C j Hn−N (A ∩ f −1 (y)) dy = 0, that is, Hn−N (A ∩ f −1 (y)) = for LN -a.e y ∈ RN (iii) If A is Ln -measurabile, the map y → Hn−N (A ∩ f −1 (y)) is LN -measurable It suffices to decompose A as an at most denumerable union of compact sets and of Ln -null sets and apply (i) and (ii) to each term of the decomposition Step We prove the coarea formula when A ⊂ B where B := x ∈ Rn J(Df )(x) > Since A is foliated by the smooth submanifods f −1 (y), we try to reduce to the coarea formula for linear maps Let I(n − N, n) denote the set of (n − N )-tuples of increasing numbers between and n For each λ = (λ1 , λ2 , , λn−N ) ∈ I(n − N, n) let πλ be the orthogonal projection onto the n − N -coordinate plane of coordinates xλ1 , , xλn−N , πλ (x1 , , xn ) = (xλ , , xλ n−N ) Since Df (x0 ) has maximal rank, for every x0 ∈ B there exists λ ∈ I(n − N, n) such that Tx0 := (Df (x0 ), Dπλ )T is invertible According to the local invertibility 6.4 Area and Coarea Formulas 391 theorem, see [GM4], there is an open neighborhood Ux0 of x0 , such that the function hλ : Ux0 → Rn given by hλ (x) := (f (x), πλ (x)) is a diffeomorphism with open image, consequently, gλ := Tx−1 ◦ hλ : Ux0 → Rn is a diffeomorphism Moreover, Dgλ (x0 ) = Id since Dhλ (x0 ) = Tx0 Therefore, for t > 1, −1 Lip(gλ )≤t Lip(gλ ) ≤ t, and t−N ≤ J(Df )(x) ≤ tN J(Df )(x0 )J(Dgλ )(x) ∀x ∈ Ux0 , possibly taking a smaller neighborhood Ux0 Therefore, see Step of the proof of Theorem 6.80, for any t > there exist a disjoint family of Borel sets {Bk }, orthogonal projections πk onto n − N dimensional coordinate planes and linear isomorphisms Tk : Rn → Rn of the form (Lk (x), πk (x)) with Lk : Rn → RN of maximal rank such that the following hold: (i) B = ∪k Bk , where {Bk } is a family of Borel sets, (ii) for any k the map x → hk (x) := (f (x)), πk (x)) is a diffeomorphism in a neighborhood of Bk (iii) For any k the map gk := Tk−1 ◦ hλk is a diffeomorphism in an open neighborhood of Bk with Lip(gk ) ≤ t, Lip(gk−1 ) ≤ t and t−N J(Lk )J(Dgk )(x) ≤ J(Df )(x) ≤ tN J(Lk )J(Dgk )(x) ∀x ∈ Bk (6.52) Set Ak := A ∩ Bk , From the area formula and (6.52), trivially, t−N J(Lk ) Ln (gk (Ak )) ≤ J(Df )(x) dx ≤ tN J(Lk ) Ln (gk (Ak )), (6.53) Ak while, since f (x) = Lk ◦ gk (x), we infer by Step that J(Lk ) Ln (gk (Ak )) = RN = RN since gk (Ak ) ∩ L−1 k (y) n−N Hn−N gk (Ak ) ∩ L−1 (y) k (y) dL (6.54) Hn−N gk (Ak ∩ f −1 (y)) dLn−N (y) = gk (Ak ∩ f −1 (y)) On the other hand, for any y ∈ RN tN−n Hn−N Ak ∩ f −1 (y) ≤ Hn−N g(Ak ∩ f −1 (y)) ≤ tn−N Hn−N Ak ∩ f −1 (y) , (6.55) since Lip(gk ), Lip(gk−1 ) ≤ t By integrating (6.55) in y with respect to LN , we then infer tN−n RN Hn−N (Ak ∩ f −1 (y)) dy ≤ J(Lk ) J(Dgk )(x) dx Ak ≤ tn−N RN (6.56) Hn−N (Ak ∩ f −1 (y)) dy The proof is then concluded from (6.53) and (6.56) if we sum in k and let t → Step We now prove the coarea formula for sets A ⊂ B c := {x | J(Df )(x) = 0} For > set g : Rn × RN → RN , π : R ×R n N →R , N x ∈ Rn , w ∈ RN g(x, w) := f (x) + w, π(x, w) := w, x ∈ R , w ∈ RN n 392 Hausdorff and Radon Measures so that Dg(x, w) = Df + Id and J(Dg) ≤ C Since for any w ∈ RN RN Hn−N (A ∩ f −1 (y)) dy = RN Hn−N (A ∩ f −1 (y − w)) dy, we infer ωN Hn−N (A ∩ f −1 (y)) dy = RN dw RN B(0,1) Hn−N (A ∩ f −1 (y − w)) dy (6.57) Next, we notice that if C := A × B(0, 1), we have ⎧ ⎨∅ −1 −1 C ∩ g (y) ∩ π (w) = ⎩(A ∩ f −1 (y − w)) × {w} if w ∈ / B(0, 1), if w ∈ B(0, 1) (6.58) for all y, w ∈ RN In fact, (x, z) ∈ C ∩ g −1 (y) ∩ π −1 (w) if and only if x ∈ A, z ∈ B(0, 1), f (x) + z = y, z = w, i.e., if and only if x ∈ A, z = w ∈ B(0, 1), f (x) = y − z, or if and only if (x, z) ∈ (A ∩ f −1 (y − w)) × {w} w ∈ B(0, 1), Returning to (6.57), we now use Fubini’s theorem and Step to get RN Hn−N (A ∩ f −1 (y)) dy = ωN = RN dw RN RN Hn−N (C ∩ g −1 (y) ∩ π −1 (w)) dy Hn (C ∩ g −1 (y)) dy = J(Dg) dx dz C n ≤ sup J(Dg)Hn+N (C) ≤ C H (A) C For → we conclude that RN Hn−N (A ∩ f −1 (y)) dy = = J(Df ) dx A Step In the general case, we write A = (A∩B)∪(A∩B c ), where B := {x | J(Df )(x) > 0} Applying Step to A ∩ B and Step to A ∩ B c , we conclude the proof 6.5 Exercises 6.85 ¶ Stereographic projection Let S n = {(x, z) ∈ Rn × R | |x|2 + z = 1} be the unit sphere in Rn+1 The stereographic projection of the sphere onto Rn from the South Pole PS = ((0, , 0), 1) is the map σ : S n → Rn given by σ((x, z)) := x/(1 + z), (x, z) ∈ S n Show that the inverse of the stereographic projection is the map u : Rn → S n ⊂ Rn+1 given by u(x) := − |x|2 , x, + |x| + |x|2 x ∈ Rn If A1 (x), , An (x) denote the columns of Du(x), show that Ai (x) • Aj (x) = λ2 (x)δij Infer that 6.5 Exercises Ju2 (x) = 393 1 |Du|2n = n nn n (1 + |x|2 )n and conclude that Hn (S n ) = nn/2 Rn 1 dx = n/2 (1 + |x|2 )n n Rn |Du|n dx 6.86 ¶ Kantorovich’s inequality Let μ be a probability measure in [0, 1], let f : [0, 1] → R be a continuous function and let m := inf [0,1] f (x) and M := sup[0,1] Show that if < m ≤ M < +∞, then 1 f dμ 0 dμ f ≤ (m + M )2 4mM with equality if and only if μ is concentrated on the sets {x | f (x) = m} and {x | f (x) = M } with μ {x | f (x) = m} = μ {x | f (x) = M } = [Hint Notice that 1 f dμ for all λ ∈ [0, 1].] dμ f ≤ λf + λf dμ ... Laplace’s operator and the solutions of Δu = are called harmonic functions M Giaquinta and G Modica, Mathematical Analysis, Foundations and Advanced Techniques for Functions of Several Variables,... method of separation of variables in the study of partial differential equations Then we introduce Lebesgue’s spaces of psummable functions and we continue with some elements of the theory of Sobolev... method of separation of variables, we begin by seeking nonzero solutions of Laplace’s equations in the disk of the form u(r, θ) = R(r)Θ(θ), finding for R and Θ r2 R rR Θ + =− R R Θ Therefore,