Quantitative estimates on Jacobians for hybrid inverse problems

G. Alessandrini ; V. Nesi

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 8 (2015) no. 3, pp. 25-41

Résumé

We consider $\sigma$-harmonic mappings, that is mappings $U$ whose components $u_i$ solve a divergence structure elliptic equation ${\rm div} (\sigma \nabla u_i)=0$, for $i=1,\ldots,n $. We investigate whether, with suitably prescribed Dirichlet data, the Jacobian determinant can be bounded away from zero. Results of this sort are required in the treatment of the so-called hybrid inverse problems, and also in the field of homogenization studying bounds for the effective properties of composite materials.

Détail

Keywords: elliptic equations; Beltrami operators; hybrid inverse problems; composite materials.

Bibliographie

[1] Seo J. K., Woo E. J., “Magnetic Resonance Electrical Impedance Tomography (MREIT)”, SIAM Rev., 53:1 (2011), 40–68 | DOI | MR | Zbl

[2] Ammari H., Bonnetier E., Capdeboscq Y., Tanter M., Fink M., “Electrical Impedance Tomography by Elastic Deformation”, SIAM J. Appl. Math., 68:6 (2008), 1557–1573 | DOI | MR | Zbl

[3] Honda N., McLaughlin J., Nakamura G., “Conditional Stability for a Single Interior Measurement”, Inverse Problems, 30:5 (2014), 055001 | DOI | MR | Zbl

[4] Mandache N., “Exponential Instability in an Inverse Problem for the {S}chrödinger Equation”, Inverse Problems, 17:5 (2001), 1435–1444 | DOI | MR | Zbl

[5] Kohn R. V., Vogelius M., “Identification of an Unknown Conductivity by Means of Measurements at the Boundary”, Inverse Problems (New York, 1983), SIAM-AMS Proc., 14, Amer. Math. Soc., Providence, RI, 1984, 113–123 | MR

[6] Monard F., Bal G., “Inverse Anisotropic Diffusion from Power Density Measurements in Two Dimensions”, Inverse Problems, 28:8 (2012), 084001, 20 pp. | DOI | MR | Zbl

[7] Monard F., Bal G., “Inverse Anisotropic Conductivity from Power Densities in Dimension”, Comm. Partial Differential Equations, 38:7 (2013), 1183–1207 | DOI | MR | Zbl

[8] Parravicini G., Giudici M., Morossi G., Ponzini G., “Minimal a Priori Assignment in a Direct Method for Determining Phenomenological Coefficients Uniquely”, Inverse Problems, 11:3 (1995), 611–629 | DOI | MR | Zbl

[9] Nesi V., “Bounds on the Effective Conductivity of Two-Dimensional Composites Made of $n\geq 3$ Isotropic Phases in Prescribed Volume Fraction: the Weighted Translation Method”, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 125:6 (1995), 1219–1239 | DOI | MR | Zbl

[10] Hashin Z., Shtrikman S., “A Variational Approach to the Theory of the Effective Magnetic Permeability of Multiphase Materials”, Journal of Applied Physics, 33:10 (1962), 3125–3131 | DOI | Zbl

[11] Murat F., Tartar L., “H-Convergence. Mathematical Modelling of Composite Materials”, Progr. Nonlinear Differential Equations Appl., 31 (1997), 21–43 | MR | Zbl

[12] Tartar L., “Estimation de Coefficients Homogénéisés”, Third International Symposium (December 5–9, 1977), Mathematics, 704, Springer, Berlin, 1979, 364–373 | MR

[13] Tartar L., The General Theory of Homogenization, Unione Matematica Italiana, 7, Springer-Verlag, Berlin; UMI, Bologna, 2009 | MR | Zbl

[14] Kohn R. V., Strang G., “Optimal Design and Relaxation of Variational Problems, II”, Communications on Pure and Applied Mathematics, 39:2 (1986), 139–182 | DOI | MR | Zbl

[15] Lurie K. A., Cherkaev A. V., “Exact Estimates of Conductivity of Composites Formed by Two Isotropically Conducting Media Taken in Prescribed Proportion”, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 99:1–2 (1984), 71–87 | DOI | MR | Zbl

[16] Bauman P., Marini A., Nesi V., “Univalent Solutions of an Elliptic System of Partial Differential Equations Arising in Homogenization”, Indiana Univ. Math. J., 50:2 (2001), 747–757 | DOI | MR | Zbl

[17] Albin N., Cherkaev A., Nesi V., “Multiphase Laminates of Extremal Effective Conductivity in Two Dimensions”, J. Mech. Phys. Solids, 55:7 (2007), 1513–1553 | DOI | MR | Zbl

[18] Albin N., Conti S., Nesi V., “Improved Bounds for Composites and Rigidity of Gradient Fields”, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463:2084 (2007), 2031–2048 | DOI | MR | Zbl

[19] Briane M., Milton G. W., “Homogenization of the Three-Dimensional Hall Effect and Change of Sign of the Hall Coefficient”, Arch. Ration. Mech. Anal., 193:3 (2009), 715–736 | DOI | MR | Zbl

[20] Briane M., Milton G. W., “An Antisymmetric Effective Hall Matrix”, SIAM J. Appl. Math., 70:6 (2010), 1810–1820 | DOI | MR | Zbl

[21] Briane M., Milton G. W., Treibergs A., “Which Electric Fields are Realizable in Conducting Materials”, ESAIM: Mathematical Modelling and Numerical Analysis, 48:3 (2014), 307–323 | DOI | MR | Zbl

[22] Briane M., Nesi V., Is It Wise to Keep Laminating?, ESAIM Control Optim. Calc. Var., 10:4 (2004), 452–477 | DOI | MR | Zbl

[23] Briane M., Milton G. W., Nesi V., “Change of Sign of the Corrector's Determinant for Homogenization in Three-Dimensional Conductivity”, Arch. Ration. Mech. Anal., 173:1 (2004), 133–150 | DOI | MR | Zbl

[24] Bers L., Nirenberg L., “On a Representation Theorem for Linear Elliptic Systems with Discontinuous Coefficients and Its Applications”, Convegno Internazionale sulle Equazioni Lineari alle Derivate Parziali (Trieste, 1954), Edizioni Cremonese, Roma, 1955, 111–140 | MR

[25] Bojarski B., D'Onofrio L., Iwaniec T., Sbordone C., “G-Closed Classes of Elliptic Operators in the Complex Plane”, Ricerche Mat., 54:2 (2005), 403–432 | MR | Zbl

[26] Alessandrini G., Nesi V., “Univalent $\sigma$-Harmonic Mappings”, Arch. Ration. Mech. Anal., 158:2 (2001), 155–171 | DOI | MR | Zbl

[27] Alessandrini G., Nesi V., “Beltrami Operators, Non-Symmetric Elliptic Equations and Quantitative Jacobian Bounds”, Ann. Acad. Sci. Fenn. Math., 34:1 (2009), 47–67 | MR | Zbl

[28] Coifman R. R., Fefferman C., “Weighted Norm Inequalities for Maximal Functions and Singular Integrals”, Studia Math., 51 (1974), 241–250 | MR | Zbl

[29] Lewy H., “On the Non-Vanishing of the Jacobian in Certain One-To-One Mappings”, Bull. Amer. Math. Soc., 42:10 (1936), 689–692 | DOI | MR

[30] Choquet G., “Sur un type de transformation analytique generalisant la representation conforme et definie au moyen de fonctions harmoniques”, Bull. Sci. Math. (2), 69 (1945), 156–165 | MR | Zbl

[31] Alessandrini G., Nesi V., “Invertible Harmonic Mappings, Beyond Kneser”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. Series 5, 8:3 (2009), 451–468 | MR | Zbl

[32] Kneser H., “Lösung der aufgabe 41”, Jahresber. deutsch. math.-verein., 35 (1926), 123–124

[33] Radó T., “Aufgabe 41”, Jahresber. deutsch. math.-verein., 35 (1926), 49

[34] Leonetti F., Nesi V., “Quasiconformal Solutions to Certain First Order Systems and the Proof of a Conjecture of G. W. Milton”, J. Math. Pures Appl., 76:2 (1997), 109–124 | DOI | MR | Zbl

[35] Alessandrini G., Magnanini R., “Elliptic Equations in Divergence Form, Geometric Critical Points of Solutions, and Stekloff Eigenfunctions”, SIAM J. Math. Anal., 25:5 (1994), 1259–1268 | DOI | MR | Zbl

[36] Nachman A., Tamasan A., Timonov A., “Conductivity Imaging with a Single Measurement of Boundary and Interior Data”, Inverse Problems, 23:6 (2007), 2551–2563 | DOI | MR | Zbl

[37] Gilbarg D., Trudinger N. S., Elliptic Partial Differential Equations of Second Order, Grundlehren der mathematischen wissenschaften, 224, Springer-Verlag, Berlin, 1983 | DOI | MR | Zbl

[38] Finn R., Gilbarg D., “Asymptotic Behavior and Uniqueness of Plane Subsonic Flows”, Comm. Pure Appl. Math., 10:1 (1957), 23–63 | DOI | MR | Zbl

[39] Alberti G. S., Enforcing Local Non-Zero Constraints in PDEs and Applications to Hybrid Imaging Problems, 2014, arXiv: 1406.3248 | MR

[40] Alessandrini G., “Critical Points of Solutions of Elliptic Equations in Two Variables”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. Series 4, 14:2 (1987), 229–256 | MR | Zbl

[41] Alessandrini G., “The Length of Level Lines of Solutions of Elliptic Equations in the Plane”, Arch. Rational Mech. Anal., 102:2 (1988), 183–191 | DOI | MR | Zbl

[42] Alessandrini G., Nesi V., “Estimates for the Dilatation of $\sigma$-Harmonic Mappings”, Rendiconti di Matematica e delle sue applicazioni, 35:3–4 (2014), 215–225 | MR

[43] Duren P., Harmonic Mappings in the Plane, Cambridge University Press, 2004 | DOI | MR | Zbl

[44] Meyers N. G., “An $L^p$-Estimate for the Gradient of Solutions of Second Order Elliptic Divergence Equations”, Ann. Scuola Norm. Sup. Pisa (3), 17 (1963), 189–206 | MR | Zbl

[45] Wood J. C., “Lewy's Theorem Fails in Higher Dimensions”, Math. Scand., 69:2 (1991), 166–166 | DOI | MR | Zbl

[46] Melas A. D., “An Example of a Harmonic map between Euclidean Balls”, Proc. Amer. Math. Soc., 117:3 (1993), 857–859 | DOI | MR | Zbl

[47] Laugesen R. S., “Injectivity Can Fail for Higher-Dimensional Harmonic Extensions”, Complex Variables Theory Appl., 28:4 (1996), 357–369 | DOI | MR | Zbl

[48] Jin Z. R., Kazdan J. L., “On the Rank of Harmonic Maps”, Math. Z., 207:4 (1991), 535–537 | DOI | MR | Zbl