Partial data inverse problems for quasilinear conductivity equations. (English) Zbl 1512.35665
Authors’ abstract: We show that the knowledge of the Dirichlet-to-Neumann maps given on an arbitrary open non-empty portion of the boundary of a smooth domain in \(\mathbb{R}^n\), \(n\ge 2\), for classes of semilinear and quasilinear conductivity equations, determines the nonlinear conductivities uniquely. The main ingredient in the proof is a certain \(L^1\)-density result involving sums of products of gradients of harmonic functions which vanish on a closed proper subset of the boundary.
Reviewer: Giovanni S. Alberti (Genova)
MSC:
| 35R30 | Inverse problems for PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35J62 | Quasilinear elliptic equations |
References:
| [1] | Agranovich, M.: Sobolev Spaces, their Generalizations and Elliptic Problems in Smooth and Lipschitz Domains, Springer Monographs in Mathematics. Springer, Cham (2015) · Zbl 1322.46002 |
| [2] | Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext (2011), New York: Springer, New York · Zbl 1220.46002 · doi:10.1007/978-0-387-70914-7 |
| [3] | Browder, F.: Functional analysis and partial differential equations. II. Math. Ann. 145, 81-226 (1961/62) · Zbl 0103.31602 |
| [4] | Cârstea, C., Feizmohammadi, A.: A density property for tensor products of gradients of harmonic functions and applications, preprint, arXiv:2009.11217 |
| [5] | Cârstea, C.; Feizmohammadi, A., An inverse boundary value problem for certain anisotropic quasilinear elliptic equations, J. Differ. Equ., 284, 318-349 (2021) · Zbl 1484.35405 · doi:10.1016/j.jde.2021.02.044 |
| [6] | Cârstea, C., Feizmohammadi, A., Kian, Y., Krupchyk, K., Uhlmann, G.: The Calderón inverse problem for isotropic quasilinear conductivities. Adv. Math. 391, Paper No. 107956 (2021) · Zbl 1479.35943 |
| [7] | Cârstea, C.; Nakamura, G.; Vashisth, M., Reconstruction for the coefficients of a quasilinear elliptic partial differential equation, Appl. Math. Lett., 98, 121-127 (2019) · Zbl 1423.35437 · doi:10.1016/j.aml.2019.06.009 |
| [8] | Dos Santos Ferreira, D.; Kenig, C.; Sjöstrand, J.; Uhlmann, G., On the linearized local Calderón problem, Math. Res. Lett., 16, 6, 955-970 (2009) · Zbl 1198.31003 · doi:10.4310/MRL.2009.v16.n6.a4 |
| [9] | Eskin, G., Lectures on Linear Partial Differential Equations, Graduate Studies in Mathematics (2011), Providence: American Mathematical Society, Providence · Zbl 1228.35001 |
| [10] | Feizmohammadi, A.; Oksanen, L., An inverse problem for a semi-linear elliptic equation in Riemannian geometries, J. Differ. Equ., 269, 6, 4683-4719 (2020) · Zbl 1448.58016 · doi:10.1016/j.jde.2020.03.037 |
| [11] | Gilbarg, D.; Trudinger, N., Elliptic Partial Differential Equations of Second Order, Reprint of the Classics in Mathematics (2001), Berlin: Springer, Berlin · doi:10.1007/978-3-642-61798-0 |
| [12] | Glowinski, R.; Rappaz, J., Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology, ESAIM: Math. Modell. Numer. Anal., 37, 175-186 (2003) · Zbl 1046.76002 · doi:10.1051/m2an:2003012 |
| [13] | Grisvard, P., Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 24 (1985), Boston: Pitman (Advanced Publishing Program), Boston · Zbl 0695.35060 |
| [14] | Grubb, G., Distributions and Operators, Graduate Texts in Mathematics, 252 (2009), New York: Springer, New York · Zbl 1171.47001 |
| [15] | Hervas, D.; Sun, Z., An inverse boundary value problem for quasilinear elliptic equations, Commun. Partial Differ. Equ., 27, 11-12, 2449-2490 (2002) · Zbl 1013.35084 · doi:10.1081/PDE-120016164 |
| [16] | Hörmander, L., The boundary problems of physical geodesy, Arch. Rational Mech. Anal., 62, 1, 1-52 (1976) · Zbl 0331.35020 · doi:10.1007/BF00251855 |
| [17] | Imanuvilov, O.; Uhlmann, G.; Yamamoto, M., The Calderón problem with partial data in two dimensions, J. Am. Math. Soc., 23, 3, 655-691 (2010) · Zbl 1201.35183 · doi:10.1090/S0894-0347-10-00656-9 |
| [18] | Imanuvilov, O.; Yamamoto, M., Unique determination of potentials and semilinear terms of semilinear elliptic equations from partial Cauchy data, J. Inverse Ill-Posed Probl., 21, 1, 85-108 (2013) · Zbl 1273.35318 · doi:10.1515/jip-2012-0033 |
| [19] | Isakov, V., Uniqueness of recovery of some quasilinear partial differential equations, Commun. Partial Differ. Equ., 26, 11-12, 1947-1973 (2001) · Zbl 0997.35018 · doi:10.1081/PDE-100107813 |
| [20] | Isakov, V.; Nachman, A., Global uniqueness for a two-dimensional semilinear elliptic inverse problem, Trans. Am. Math. Soc., 347, 9, 3375-3390 (1995) · Zbl 0849.35148 · doi:10.1090/S0002-9947-1995-1311909-1 |
| [21] | Isakov, V.; Sylvester, J., Global uniqueness for a semilinear elliptic inverse problem, Commun. Pure Appl. Math., 47, 10, 1403-1410 (1994) · Zbl 0817.35126 · doi:10.1002/cpa.3160471005 |
| [22] | Kang, K.; Nakamura, G., Identification of nonlinearity in a conductivity equation via the Dirichlet-to-Neumann map, Inverse Problems, 18, 4, 1079-1088 (2002) · Zbl 1043.35146 · doi:10.1088/0266-5611/18/4/309 |
| [23] | Kapanadze, D.; Mishuris, G.; Pesetskaya, E., Exact solution of a nonlinear heat conduction problem in a doubly periodic 2D composite material, Arch. Mech., 67, 2, 157-178 (2015) · Zbl 1329.80005 |
| [24] | Krupchyk, K.; Uhlmann, G., Partial data inverse problems for semilinear elliptic equations with gradient nonlinearities, Math. Res. Lett., 27, 1801-1824 (2020) · Zbl 1459.35399 · doi:10.4310/MRL.2020.v27.n6.a10 |
| [25] | Krupchyk, K.; Uhlmann, G., A remark on partial data inverse problems for semilinear elliptic equations, Proc. Am. Math. Soc., 148, 2, 681-685 (2020) · Zbl 1431.35246 · doi:10.1090/proc/14844 |
| [26] | Krupchyk, K., Uhlmann, G.: Inverse problems for nonlinear magnetic Schrödinger equations on conformally transversally anisotropic manifolds, preprint, arXiv:2009.05089 (to appear in Analysis and PDE) |
| [27] | Kurylev, Y.; Lassas, M.; Uhlmann, G., Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations, Invent. Math., 212, 3, 781-857 (2018) · Zbl 1396.35074 · doi:10.1007/s00222-017-0780-y |
| [28] | Lai, R.-Y., Zhou, T.: Partial Data Inverse Problems for Nonlinear Magnetic Schrödinger Equations, preprint arXiv:2007.02475 (to appear in Mathematical Research Letters) |
| [29] | Lassas, M.; Liimatainen, T.; Lin, Y-H; Salo, M., Inverse problems for elliptic equations with power type nonlinearities, J. Math. Pures Appl., 145, 44-82 (2021) · Zbl 1460.35395 · doi:10.1016/j.matpur.2020.11.006 |
| [30] | Lassas, M.; Liimatainen, T.; Lin, Y-H; Salo, M., Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations, Revista Matemática Iberoamericana, 37, 1553-1580 (2021) · Zbl 1473.35653 · doi:10.4171/rmi/1242 |
| [31] | McLean, W., Strongly Elliptic Systems and Boundary Integral Equations (2000), Cambridge: Cambridge University Press, Cambridge · Zbl 0948.35001 |
| [32] | Medková, D., The Laplace Equation, Boundary Value Problems on Bounded and Unbounded Lipschitz Domains (2018), Cham: Springer, Cham · Zbl 1457.35002 |
| [33] | Mironescu, P.: Note on Gagliardo’s theorem “\( \text{tr}\, W^{1,1}=L^1\)”. Ann. Univ. Buchar. Math. Ser. 6(LXIV)(1), 99-103 (2015) · Zbl 1389.46033 |
| [34] | Muñoz, C., Uhlmann, G.: The Calderón problem for quasilinear elliptic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 37(5), 1143-1166 (2020) · Zbl 1457.35093 |
| [35] | Nachman, A., Global uniqueness for a two-dimensional inverse boundary value problem, Ann. Math. (2), 143, 1, 71-96 (1996) · Zbl 0857.35135 · doi:10.2307/2118653 |
| [36] | Ponte Castañeda, P.; Kailasam, M., Nonlinear electrical conductivity in heterogeneous media, Proc. R. Soc. Lond. Ser. A, 453, 1959, 793-816 (1997) · Zbl 0886.73039 · doi:10.1098/rspa.1997.0044 |
| [37] | Pöschel, J.; Trubowitz, E., Inverse Spectral Theory, Pure and Applied Mathematics, 130 (1987), Boston: Academic Press Inc, Boston · Zbl 0623.34001 |
| [38] | Triebel, H., Interpolation theory, function spaces, differential operators (1978), Amsterdam: North-Holland, Amsterdam · Zbl 0387.46032 |
| [39] | Triebel, H., Spaces of Besov-Hardy-Sobolev type on complete Riemannian manifolds, Ark. Mat., 24, 2, 299-337 (1986) · Zbl 0664.46026 |
| [40] | Schlag, W., A Course in Complex Analysis and Riemann Surfaces, Graduate Studies in Mathematics, 154 (2014), Providence: American Mathematical Society, Providence · Zbl 1326.30003 · doi:10.1090/gsm/154 |
| [41] | Shankar, R., Recovering a quasilinear conductivity from boundary measurements, Inverse Problems, 27, 015014 (2020) |
| [42] | Sun, Z., On a quasilinear inverse boundary value problem, Math. Z., 221, 2, 293-305 (1996) · Zbl 0843.35137 · doi:10.1007/PL00022738 |
| [43] | Sun, Z., Inverse boundary value problems for a class of semilinear elliptic equations, Adv. Appl. Math., 32, 4, 791-800 (2004) · Zbl 1059.35176 · doi:10.1016/j.aam.2003.06.001 |
| [44] | Sun, Z., Conjectures in inverse boundary value problems for quasilinear elliptic equations, Cubo, 7, 3, 65-73 (2005) · Zbl 1100.35123 |
| [45] | Sun, Z., An inverse boundary-value problem for semilinear elliptic equations, Electron. J. Differ. Equ., 2010, 37, 5 (2010) · Zbl 1189.35380 |
| [46] | Sun, Z.; Uhlmann, G., Inverse problems in quasilinear anisotropic media, Am. J. Math., 119, 4, 771-797 (1997) · Zbl 0886.35176 · doi:10.1353/ajm.1997.0027 |
| [47] | Sylvester, J.; Uhlmann, G., A global uniqueness theorem for an inverse boundary value problem, Ann. Math. (2), 125, 1, 153-169 (1987) · Zbl 0625.35078 · doi:10.2307/1971291 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
