staple
Avec cedram.org
logo JEP
Table des matières de ce volume | Article précédent | Article suivant
Yves Capdeboscq
On a counter-example to quantitative Jacobian bounds
(Sur un contre-exemple aux bornes quantitatives du jacobien)
Journal de l'École polytechnique — Mathématiques, 2 (2015), p. 171-178, doi: 10.5802/jep.21
Article PDF | TeX source
Class. Math.: 35J55, 35R30, 35B27
Mots clés: Théorème de Radó-Kneser-Choquet, problèmes inverses hybrides, tomographie d’impédance, homogénéisation

Résumé - Abstract

Cette note fournit un contre-exemple à la positivité locale du déterminant jacobien des solutions de l’équation de conduction en dimension $3$. On montre que le signe du déterminant ne peut pas être imposé par un choix a priori de données au bord dans $H^{{1}/{2}}(\partial \Omega )$ dépendant seulement des bornes inférieure et supérieure de la conductivité, même localement. L’argument utilise une conductivité scalaire à deux phases construite par Briane, Milton & Nesi [11, 10].

Bibliographie

[1] G. Alessandrini & V. Nesi, “Univalent $\sigma $-harmonic mappings”, Arch. Rational Mech. Anal. 158 (2001) no. 2, p. 155-171 Article |  MR 1838656 |  Zbl 0977.31006
[2] G. Alessandrini & V. Nesi, “Beltrami operators, non-symmetric elliptic equations and quantitative Jacobian bounds”, Ann. Acad. Sci. Fenn. Math. 34 (2009) no. 1, p. 47-67  MR 2489016 |  Zbl 1177.30019
[3] G. Alessandrini & V. Nesi, “Quantitative estimates on Jacobians for hybrid inverse problems”, arXiv:1501.03005, 2015
[4] H. Ammari, E. Bonnetier & Y. Capdeboscq, “Enhanced resolution in structured media”, SIAM J. Appl. Math. 70 (2009/10) no. 5, p. 1428-1452 Article |  MR 2578678 |  Zbl 1202.35343
[5] G. Bal, E. Bonnetier, F. Monard & F. Triki, “Inverse diffusion from knowledge of power densities”, Inverse Probl. Imaging 7 (2013) no. 2, p. 353-375 Article |  MR 3063538 |  Zbl 1267.35249
[6] G. Bal & G. Uhlmann, “Inverse diffusion theory of photoacoustics”, Inverse Problems 26 (2010) no. 8  MR 2658827 |  Zbl 1197.35311
[7] P. Bauman, A. Marini & V. Nesi, “Univalent solutions of an elliptic system of partial differential equations arising in homogenization”, Indiana Univ. Math. J. 50 (2001) no. 2, p. 747-757 Article |  MR 1871388
[8] M. F. Ben Hassen & E. Bonnetier, “An asymptotic formula for the voltage potential in a perturbed $\epsilon $-periodic composite medium containing misplaced inclusions of size $\epsilon $”, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006) no. 4, p. 669-700 Article |  MR 2250439 |  Zbl 1105.35011
[9] A. Bensoussan, J.-L. Lions & G. C. Papanicolaou, Asymptotic Analysis For Periodic Structures, North-Holland Publishing Co., Amsterdam, 1978  MR 2839402 |  Zbl 1229.35001
[10] M. Briane & G. W. Milton, “Homogenization of the three-dimensional Hall effect and change of sign of the Hall coefficient”, Arch. Rational Mech. Anal. 193 (2009) no. 3, p. 715-736 Article |  MR 2525116 |  Zbl 1170.74019
[11] M. Briane, G. W. Milton & V. Nesi, “Change of sign of the corrector’s determinant for homogenization in three-dimensional conductivity”, Arch. Rational Mech. Anal. 173 (2004) no. 1, p. 133-150  MR 2073507 |  Zbl 1118.78009
[12] A.-P. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), Soc. Brasil. Mat., 1980, p. 65–73  MR 590275
[13] P. Duren, Harmonic mappings in the plane, Cambridge Tracts in Mathematics 156, Cambridge University Press, Cambridge, 2004 Article |  MR 2048384 |  Zbl 1055.31001
[14] R. E. Greene & H. Wu, “Embedding of open Riemannian manifolds by harmonic functions”, Ann. Inst. Fourier (Grenoble) 25 (1975) no. 1, vii, p. 215-235 Cedram |  MR 382701 |  Zbl 0307.31003
[15] R. E. Greene & H. Wu, Whitney’s imbedding theorem by solutions of elliptic equations and geometric consequences, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973), American Mathematical Society, 1975, p. 287–296  MR 407908 |  Zbl 0322.31007
[16] M. Kadic, R. Schittny, T. Bückmann, Ch. Kern & M. Wegener, “Hall-Effect Sign Inversion in a Realizable 3D Metamaterial”, Phys. Rev. X 5 (2015) Article
[17] H. Koch & D. Tataru, “Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients”, Comm. Pure Appl. Math. 54 (2001) no. 3, p. 339-360 Article |  MR 1809741 |  Zbl 1033.35025
[18] R. S. Laugesen, “Injectivity can fail for higher-dimensional harmonic extensions”, Complex Variables Theory Appl. 28 (1996) no. 4, p. 357-369  MR 1700199 |  Zbl 0871.54020
[19] Y. Y. Li & L. Nirenberg, “Estimates for elliptic systems from composite material”, Comm. Pure Appl. Math. 56 (2003), p. 892-925  MR 1990481 |  Zbl 1125.35339
[20] Y. Y. Li & M. S. Vogelius, “Gradient estimates for solutions of divergence form elliptic equations with discontinuous coefficients”, Arch. Rational Mech. Anal. 153 (2000), p. 91-151  MR 1770682 |  Zbl 0958.35060
[21] R. Lipton & T. Mengesha, “Representation formulas for $L^\infty $ norms of weakly convergent sequences of gradient fields in homogenization”, ESAIM Math. Model. Numer. Anal. 46 (2012), p. 1121-1146 Numdam |  MR 2916375 |  Zbl 1273.35038
[22] F. Monard & G. Bal, “Inverse diffusion problems with redundant internal information”, Inverse Probl. Imaging 6 (2012) no. 2, p. 289-313 Article |  MR 2942741 |  Zbl 1302.35449
[23] G. Sylvester, “A global uniqueness theorem for an inverse boundary value problem”, Ann. of Math. (2) 125 (1987), p. 153-169  MR 873380 |  Zbl 0625.35078
[24] J. C. Wood, “Lewy’s theorem fails in higher dimensions”, Math. Scand. 69 (1991) no. 2  MR 1156423 |  Zbl 0711.31003