On a counter-example to quantitative Jacobian bounds
Journal de l’École polytechnique — Mathématiques, Volume 2 (2015), pp. 171-178.

This note provides a counter-example to the local positivity of the Jacobian determinant for solutions of the conductivity equation in dimension 3. It shows that the sign of the determinant cannot be imposed by an a priori choice of boundary data in H 1/2 (Ω) depending only on the upper and lower bound of the conductivity, even locally. The argument uses a scalar two-phase conductivity constructed by Briane, Milton & Nesi [11, 10].

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 (Ω) 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].

Received:
Accepted:
DOI: 10.5802/jep.21
Classification: 35J55, 35R30, 35B27
Keywords: Radó-Kneser-Choquet Theorem, hybrid inverse problems, impedance tomography, homogenization
Mot clés : Théorème de Radó-Kneser-Choquet, problèmes inverses hybrides, tomographie d’impédance, homogénéisation

Yves Capdeboscq 1

1 Mathematical Institute, University of Oxford, Andrew Wiles Building Woodstock Road, Oxford OX2 6GG, UK
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Yves Capdeboscq. On a counter-example to quantitative Jacobian bounds. Journal de l’École polytechnique — Mathématiques, Volume 2 (2015), pp. 171-178. doi : 10.5802/jep.21. https://jep.centre-mersenne.org/articles/10.5802/jep.21/

[1] G. Alessandrini & V. Nesi - “Univalent σ-harmonic mappings”, Arch. Rational Mech. Anal. 158 (2001) no. 2, p. 155-171 | DOI | MR | Zbl

[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 | Zbl

[3] G. Alessandrini & V. Nesi - “Quantitative estimates on Jacobians for hybrid inverse problems” (2015), arXiv:1501.03005 | DOI | Zbl

[4] H. Ammari, E. Bonnetier & Y. Capdeboscq - “Enhanced resolution in structured media”, SIAM J. Appl. Math. 70 (2009/10) no. 5, p. 1428-1452 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[6] G. Bal & G. Uhlmann - “Inverse diffusion theory of photoacoustics”, Inverse Problems 26 (2010) no. 8, 085010 pages | MR | Zbl

[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 | DOI | MR | Zbl

[8] M. F. Ben Hassen & E. Bonnetier - “An asymptotic formula for the voltage potential in a perturbed ϵ-periodic composite medium containing misplaced inclusions of size ϵ, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006) no. 4, p. 669-700 | DOI | MR | Zbl

[9] A. Bensoussan, J.-L. Lions & G. C. Papanicolaou - Asymptotic Analysis For Periodic Structures, North-Holland Publishing Co., Amsterdam, 1978

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[12] A.-P. Calderón - “On an inverse boundary value problem”, in Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), Soc. Brasil. Mat., Rio de Janeiro, 1980, p. 65-73

[13] P. Duren - Harmonic mappings in the plane, Cambridge Tracts in Mathematics, vol. 156, Cambridge University Press, Cambridge, 2004 | DOI | MR | Zbl

[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 | DOI | Numdam | MR | Zbl

[15] R. E. Greene & H. Wu - “Whitney’s imbedding theorem by solutions of elliptic equations and geometric consequences”, in Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973), American Mathematical Society, Providence, R. I., 1975, p. 287-296 | Zbl

[16] M. Kadic, R. Schittny, T. Bückmann, C. Kern & M. Wegener - “Hall-Effect Sign Inversion in a Realizable 3D Metamaterial”, Phys. Rev. X 5 (2015), 021030 pages | DOI

[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 | DOI | MR | Zbl

[18] R. S. Laugesen - “Injectivity can fail for higher-dimensional harmonic extensions”, Complex Variables Theory Appl. 28 (1996) no. 4, p. 357-369 | MR | Zbl

[19] Y. Y. Li & L. Nirenberg - “Estimates for elliptic systems from composite material”, Comm. Pure Appl. Math. 56 (2003), p. 892-925 | DOI | MR | Zbl

[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 | DOI | MR

[21] R. Lipton & T. Mengesha - “Representation formulas for L norms of weakly convergent sequences of gradient fields in homogenization”, ESAIM Math. Model. Numer. Anal. 46 (2012), p. 1121-1146 | DOI | Numdam | MR | Zbl

[22] F. Monard & G. Bal - “Inverse diffusion problems with redundant internal information”, Inverse Probl. Imaging 6 (2012) no. 2, p. 289-313 | DOI | MR | Zbl

[23] G. Sylvester - “A global uniqueness theorem for an inverse boundary value problem”, Ann. of Math. (2) 125 (1987), p. 153-169 | DOI | MR | Zbl

[24] J. C. Wood - “Lewy’s theorem fails in higher dimensions”, Math. Scand. 69 (1991) no. 2, 166 (1992) pages | MR | Zbl

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