# zbMATH — the first resource for mathematics

On a counter-example to quantitative Jacobian bounds. (Sur un contre-exemple aux bornes quantitatives du jacobien.) (English. French summary) Zbl 1327.35432
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 $$H^{1/2}(\partial\Omega)$$ depending only on the upper and lower bound of the conductivity, even locally.

##### MSC:
 35R30 Inverse problems for PDEs 35Q60 PDEs in connection with optics and electromagnetic theory 78M40 Homogenization in optics and electromagnetic theory 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35J57 Boundary value problems for second-order elliptic systems
Full Text:
##### References:
 [1] Alessandrini, G.; Nesi, V., Univalent $$σ$$-harmonic mappings, Arch. Rational Mech. Anal., 158, 2, 155-171, (2001) · Zbl 0977.31006 [2] Alessandrini, G.; Nesi, V., Beltrami operators, non-symmetric elliptic equations and quantitative Jacobian bounds, Ann. Acad. Sci. Fenn. Math., 34, 1, 47-67, (2009) · Zbl 1177.30019 [3] Alessandrini, G.; Nesi, V., Quantitative estimates on Jacobians for hybrid inverse problems, (2015) · Zbl 1341.31005 [4] Ammari, H.; Bonnetier, E.; Capdeboscq, Y., Enhanced resolution in structured media, SIAM J. Appl. Math., 70, 5, 1428-1452, (200910) · Zbl 1202.35343 [5] Bal, G.; Bonnetier, E.; Monard, F.; Triki, F., Inverse diffusion from knowledge of power densities, Inverse Probl. Imaging, 7, 2, 353-375, (2013) · Zbl 1267.35249 [6] Bal, G.; Uhlmann, G., Inverse diffusion theory of photoacoustics, Inverse Problems, 26, 8, 085010 pp., (2010) · Zbl 1197.35311 [7] 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, 747-757, (2001) · Zbl 1330.35121 [8] Ben Hassen, M. F.; Bonnetier, E., 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, 4, 669-700, (2006) · Zbl 1105.35011 [9] Bensoussan, A.; Lions, J.-L.; Papanicolaou, G. C., Asymptotic Analysis For Periodic Structures, xxiv+700 pp., (1978), North-Holland Publishing Co., Amsterdam · Zbl 1229.35001 [10] Briane, M.; Milton, G. W., Homogenization of the three-dimensional Hall effect and change of sign of the Hall coefficient, Arch. Rational Mech. Anal., 193, 3, 715-736, (2009) · Zbl 1170.74019 [11] Briane, M.; Milton, G. W.; Nesi, V., Change of sign of the corrector’s determinant for homogenization in three-dimensional conductivity, Arch. Rational Mech. Anal., 173, 1, 133-150, (2004) · Zbl 1118.78009 [12] Calderón, A.-P., Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), On an inverse boundary value problem, 65-73, (1980), Soc. Brasil. Mat., Rio de Janeiro [13] Duren, P., Harmonic mappings in the plane, 156, xii+212 pp., (2004), Cambridge University Press, Cambridge · Zbl 1055.31001 [14] Greene, R. E.; Wu, H., Embedding of open Riemannian manifolds by harmonic functions, Ann. Inst. Fourier (Grenoble), 25, 1, 215-235, (1975) · Zbl 0307.31003 [15] Greene, R. E.; Wu, H., Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973), Whitney’s imbedding theorem by solutions of elliptic equations and geometric consequences, 287-296, (1975), American Mathematical Society, Providence, R. I. · Zbl 0322.31007 [16] Kadic, M.; Schittny, R.; Bückmann, T.; Kern, Ch.; Wegener, M., Hall-effect sign inversion in a realizable 3D metamaterial, Phys. Rev. X, 5, 021030 pp., (2015) [17] Koch, H.; Tataru, D., Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients, Comm. Pure Appl. Math., 54, 3, 339-360, (2001) · Zbl 1033.35025 [18] Laugesen, R. S., Injectivity can fail for higher-dimensional harmonic extensions, Complex Variables Theory Appl., 28, 4, 357-369, (1996) · Zbl 0871.54020 [19] Li, Y. Y.; Nirenberg, L., Estimates for elliptic systems from composite material, Comm. Pure Appl. Math., 56, 892-925, (2003) · Zbl 1125.35339 [20] Li, Y. Y.; Vogelius, M. S., Gradient estimates for solutions of divergence form elliptic equations with discontinuous coefficients, Arch. Rational Mech. Anal., 153, 91-151, (2000) · Zbl 0958.35060 [21] Lipton, R.; Mengesha, T., Representation formulas for $$L^∞$$ norms of weakly convergent sequences of gradient fields in homogenization, ESAIM Math. Model. Numer. Anal., 46, 1121-1146, (2012) · Zbl 1273.35038 [22] Monard, F.; Bal, G., Inverse diffusion problems with redundant internal information, Inverse Probl. Imaging, 6, 2, 289-313, (2012) · Zbl 1302.35449 [23] Sylvester, G. J. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2), 125, 153-169, (1987) · Zbl 0625.35078 [24] Wood, J. C., Lewy’s theorem fails in higher dimensions, Math. Scand., 69, 2, 166 (1992) pp., (1991) · Zbl 0711.31003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.