# American Institute of Mathematical Sciences

## Quantitative jacobian determinant bounds for the conductivity equation in high contrast composite media

 1 Université de Paris and Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions (LJLL), F-75006 Paris, France 2 Mathematical Institute, University of Oxford, OX2 6GG, UK

Received  June 2019 Revised  April 2020 Published  July 2020

We consider the conductivity equation in a bounded domain in $\mathbb{R}^{d}$ with $d\geq3$. In this study, the medium corresponds to a very contrasted two phase homogeneous and isotropic material, consisting of a unit matrix phase, and an inclusion with high conductivity. The geometry of the inclusion phase is so that the resulting Jacobian determinant of the gradients of solutions $DU$ takes both positive and negatives values. In this work, we construct a class of inclusions $Q$ and boundary conditions $\phi$ such that the determinant of the solution of the boundary value problem satisfies this sign-changing constraint. We provide lower bounds for the measure of the sets where the Jacobian determinant is greater than a positive constant (or lower than a negative constant). Different sign changing structures where introduced in [9], where the existence of such media was first established. The quantitative estimates provided here are new.

Citation: Yves Capdeboscq, Haun Chen Yang Ong. Quantitative jacobian determinant bounds for the conductivity equation in high contrast composite media. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020228
##### References:
 [1] G. S. Alberti and Y. Capdeboscq, Lectures on Elliptic Methods for Hybrid Inverse Problems, vol. 25 of Cours Spécialisés, Société Mathématique de France, 2018.  Google Scholar [2] G. Alessandrini and R. Magnanini, Elliptic equations in divergence form, geometric critical points of solutions, and {S}tekloff eigenfunctions, SIAM J. Math. Anal., 25 (1994), 1259-1268.  doi: 10.1137/S0036141093249080.  Google Scholar [3] G. Alessandrini and V. Nesi, Quantitative estimates on jacobians for hybrid inverse problems, Vestnik YuUrGU. Ser. Mat. Model. Progr., 8 (2015), 25-41.  doi: 10.14529/mmp150302.  Google Scholar [4] G. Alessandrini and V. Nesi, Univalent $\sigma$-harmonic mappings, Arch. Ration. Mech. Anal., 158 (2001), 155-171.  doi: 10.1007/PL00004242.  Google Scholar [5] H. Ammari, E. Bonnetier, F. Triki and M. Vogelius, Elliptic estimates in composite media with smooth inclusions: An integral equation approach, Ann. Sci. Éc. Norm. Supér. (4), 48 (2015), 453–495. doi: 10.24033/asens.2249.  Google Scholar [6] H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, vol. 1846 of Lecture Notes in Mathematics, Springer, 2004. doi: 10.1007/b98245.  Google Scholar [7] E. S. Bao, Y. Y. Li and B. Yin, Gradient estimates for the perfect conductivity problem, Archive for Rational Mechanics and Analysis, 193 (2009), 195-226.  doi: 10.1007/s00205-008-0159-8.  Google Scholar [8] L. Berlyand and H. Owhadi, Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast, Archive for Rational Mechanics and Analysis, 198 (2010), 677-721.  doi: 10.1007/s00205-010-0302-1.  Google Scholar [9] M. Briane and G. W. Milton, Change of sign of the correctors determinant for homogenization in three-dimensional conductivity, Archive for Rational Mechanics and Analysis, 173 (2004), 133-150.  doi: 10.1007/s00205-004-0315-8.  Google Scholar [10] Y. Capdeboscq, On a counter-example to quantitative jacobian bounds, Journal de l'École polytechnique - Mathématiques, 2 (2015), 171–178. doi: 10.5802/jep.21.  Google Scholar [11] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences, 2nd edition, Springer Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03537-5.  Google Scholar [12] S. Friedland, Variation of tensor powers and spectra, Linear and Multilinear Algebra, 12 (1982/83), 81-98.  doi: 10.1080/03081088208817475.  Google Scholar [13] P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985.  Google Scholar [14] H. Kang, K. Kim, H. Lee, J. Shin and S. Yu, Spectral properties of the Neumann-Poincaré operator and uniformity of estimates for the conductivity equation with complex coefficients, J. Lond. Math. Soc. (2), 93 (2016), 519–545. doi: 10.1112/jlms/jdw003.  Google Scholar [15] R. S. Laugesen, Injectivity can fail for higher-dimensional harmonic extensions, Complex Variables Theory Appl., 28 (1996), 357-369.  doi: 10.1080/17476939608814865.  Google Scholar [16] Y. Y. Li and M. S. Vogelius, Gradient estimates for solutions of divergence form elliptic equations with discontinuous coefficients, Arch. Rational Mech. Anal, 153 (2000), 91-151.  doi: 10.1007/s002050000082.  Google Scholar [17] A. D. Melas, An example of a harmonic map between Euclidean balls, Proc. Amer. Math. Soc., 117 (1993), 857-859.  doi: 10.1090/S0002-9939-1993-1112497-9.  Google Scholar [18] K.-O. Widman, Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations, Math. Scand., 21 (1967), 17–37 (1968). doi: 10.7146/math.scand.a-10841.  Google Scholar [19] J. C. Wood, Lewy's theorem fails in higher dimensions, Math. Scand., 69 (1991), 166 (1992). doi: 10.7146/math.scand.a-12375.  Google Scholar

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##### References:
 [1] G. S. Alberti and Y. Capdeboscq, Lectures on Elliptic Methods for Hybrid Inverse Problems, vol. 25 of Cours Spécialisés, Société Mathématique de France, 2018.  Google Scholar [2] G. Alessandrini and R. Magnanini, Elliptic equations in divergence form, geometric critical points of solutions, and {S}tekloff eigenfunctions, SIAM J. Math. Anal., 25 (1994), 1259-1268.  doi: 10.1137/S0036141093249080.  Google Scholar [3] G. Alessandrini and V. Nesi, Quantitative estimates on jacobians for hybrid inverse problems, Vestnik YuUrGU. Ser. Mat. Model. Progr., 8 (2015), 25-41.  doi: 10.14529/mmp150302.  Google Scholar [4] G. Alessandrini and V. Nesi, Univalent $\sigma$-harmonic mappings, Arch. Ration. Mech. Anal., 158 (2001), 155-171.  doi: 10.1007/PL00004242.  Google Scholar [5] H. Ammari, E. Bonnetier, F. Triki and M. Vogelius, Elliptic estimates in composite media with smooth inclusions: An integral equation approach, Ann. Sci. Éc. Norm. Supér. (4), 48 (2015), 453–495. doi: 10.24033/asens.2249.  Google Scholar [6] H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, vol. 1846 of Lecture Notes in Mathematics, Springer, 2004. doi: 10.1007/b98245.  Google Scholar [7] E. S. Bao, Y. Y. Li and B. Yin, Gradient estimates for the perfect conductivity problem, Archive for Rational Mechanics and Analysis, 193 (2009), 195-226.  doi: 10.1007/s00205-008-0159-8.  Google Scholar [8] L. Berlyand and H. Owhadi, Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast, Archive for Rational Mechanics and Analysis, 198 (2010), 677-721.  doi: 10.1007/s00205-010-0302-1.  Google Scholar [9] M. Briane and G. W. Milton, Change of sign of the correctors determinant for homogenization in three-dimensional conductivity, Archive for Rational Mechanics and Analysis, 173 (2004), 133-150.  doi: 10.1007/s00205-004-0315-8.  Google Scholar [10] Y. Capdeboscq, On a counter-example to quantitative jacobian bounds, Journal de l'École polytechnique - Mathématiques, 2 (2015), 171–178. doi: 10.5802/jep.21.  Google Scholar [11] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences, 2nd edition, Springer Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03537-5.  Google Scholar [12] S. Friedland, Variation of tensor powers and spectra, Linear and Multilinear Algebra, 12 (1982/83), 81-98.  doi: 10.1080/03081088208817475.  Google Scholar [13] P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985.  Google Scholar [14] H. Kang, K. Kim, H. Lee, J. Shin and S. Yu, Spectral properties of the Neumann-Poincaré operator and uniformity of estimates for the conductivity equation with complex coefficients, J. Lond. Math. Soc. (2), 93 (2016), 519–545. doi: 10.1112/jlms/jdw003.  Google Scholar [15] R. S. Laugesen, Injectivity can fail for higher-dimensional harmonic extensions, Complex Variables Theory Appl., 28 (1996), 357-369.  doi: 10.1080/17476939608814865.  Google Scholar [16] Y. Y. Li and M. S. Vogelius, Gradient estimates for solutions of divergence form elliptic equations with discontinuous coefficients, Arch. Rational Mech. Anal, 153 (2000), 91-151.  doi: 10.1007/s002050000082.  Google Scholar [17] A. D. Melas, An example of a harmonic map between Euclidean balls, Proc. Amer. Math. Soc., 117 (1993), 857-859.  doi: 10.1090/S0002-9939-1993-1112497-9.  Google Scholar [18] K.-O. Widman, Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations, Math. Scand., 21 (1967), 17–37 (1968). doi: 10.7146/math.scand.a-10841.  Google Scholar [19] J. C. Wood, Lewy's theorem fails in higher dimensions, Math. Scand., 69 (1991), 166 (1992). doi: 10.7146/math.scand.a-12375.  Google Scholar
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