Previous |  Up |  Next

Article

Full entry | Fulltext not available (moving wall 24 months)      Feedback
Keywords:
stability estimate; Dirichlet-to-Neumann map; Carleman estimate; polyharmonic operator
Summary:
We establish a $\ln $-$\ln $-type estimate for the electric potential associated to the zeroth order perturbation of the polyharmonic operator when the Dirichlet-to-Neumann map is taken only on a subset.
References:
[1] Alessandrini, G.: Stable determination of conductivity by boundary measurements. Appl. Anal. 27 (1988), 153-172. DOI 10.1080/00036818808839730 | MR 0922775 | Zbl 0616.35082
[2] Aroua, N., Bellassoued, M.: Stable determination of a second order perturbation of the polyharmonic operator by boundary measurements. J. Math. Anal. Appl. 522 (2023), Article ID 126965, 41 pages. DOI 10.1016/j.jmaa.2022.126965 | MR 4530920 | Zbl 1510.35392
[3] Ashbaugh, M. S.: On universal inequalities for the low eigenvalues of the buckling problem. Partial Differential Equations and Inverse Problems Contemporary Mathematics 362. AMS, Providence (2004), 13-31. MR 2091488 | Zbl 1062.35050
[4] Assylbekov, Y. M.: Inverse problems for the perturbed polyharmonic operator with coefficients in Sobolev spaces with non-positive order. Inverse Problems 32 (2016), Article ID 105009, 22 pages. DOI 10.1088/0266-5611/32/10/105009 | MR 3627033 | Zbl 1369.35112
[5] Assylbekov, Y. M., Iyer, K.: Determining rough first order perturbations of the polyharmonic operator. Inverse Probl. Imaging 13 (2019), 1045-1066. DOI 10.3934/ipi.2019047 | MR 4027047 | Zbl 1423.35433
[6] Assylbekov, Y. M., Yang, Y.: Determining the first order perturbation of a polyharmonic operator on admissible manifolds. J. Differ. Equations 262 (2017), 590-614. DOI 10.1016/j.jde.2016.09.039 | MR 3567495 | Zbl 1354.58014
[7] Bhattacharyya, S.: An inverse problem for the magnetic Schrödinger operator on Riemannian manifolds from partial boundary data. Inverse Probl. Imaging 12 (2018), 801-830. DOI 10.3934/ipi.2018034 | MR 3810180 | Zbl 1432.58017
[8] Bhattacharyya, S., Ghosh, T.: Inverse boundary value problem of determining up to a second order tensor appear in the lower order perturbation of a polyharmonic operator. J. Fourier Anal. Appl. 25 (2019), 661-683. DOI 10.1007/s00041-018-9625-3 | MR 3953481 | Zbl 1414.35261
[9] Bhattacharyya, S., Ghosh, T.: An inverse problem on determining second order symmetric tensor for perturbed biharmonic operator. Math. Ann. 384 (2022), 457-489. DOI 10.1007/s00208-021-02276-6 | MR 4476229 | Zbl 1497.35511
[10] Bhattacharyya, S., Kumar, P.: Local data inverse problem for the polyharmonic operator with anisotropic perturbations. Inverse Probl. 40 (2024), Article ID 055004, 22 pages. DOI 10.1088/1361-6420/ad3164 | MR 4723844 | Zbl 1548.35295
[11] Bukhgeim, A. L., Uhlmann, G.: Recovering a potential from partial Cauchy data. Commun. Partial Differ. Equations 27 (2002), 653-668. DOI 10.1081/PDE-120002868 | MR 1900557 | Zbl 0998.35063
[12] Calderón, A. P.: On an inverse boundary value problem. Seminar on Numerical Analysis and its Applications to Continuum Physics Sociedade Brasileira de Matemática, Rio de Janeiro (1980), 65-73. MR 0590275
[13] Choudhury, A. P., Heck, H.: Stability of the inverse boundary value problem for the biharmonic operator: Logarithmic estimates. J. Inverse Ill-Posed Probl. 25 (2017), 251-263. DOI 10.1515/jiip-2016-0019 | MR 3630136 | Zbl 1368.31003
[14] Choudhury, A. P., Krishnan, V. P.: Stability estimates for the inverse boundary value problem for the biharmonic operator with bounded potentials. J. Math. Anal. Appl. 431 (2015), 300-316. DOI 10.1016/j.jmaa.2015.05.054 | MR 3357587 | Zbl 1325.35276
[15] Ferreira, D. dos Santos, Kenig, C. E., Sjöstrand, J., Uhlmann, G.: Determining a magnetic Schrödinger operator from partial Cauchy data. Commun. Math. Phys. 271 (2007), 467-488. DOI 10.1007/s00220-006-0151-9 | MR 2287913 | Zbl 1148.35096
[16] Gazzola, F., Grunau, H.-C., Sweers, G.: Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains. Lecture Notes in Mathematics 1991. Springer, Berlin (2010). DOI 10.1007/978-3-642-12245-3 | MR 2667016 | Zbl 1239.35002
[17] Ghosh, T.: An inverse problem on determining upto first order perturbations of a fourth order operator with partial boundary data. Inverse Probl. 31 (2015), Article ID 105009, 19 pages. DOI 10.1088/0266-5611/31/10/105009 | MR 3405369 | Zbl 1328.35307
[18] Ghosh, T., Krishnan, V. P.: Determination of lower order perturbations of the polyharmonic operator from partial boundary data. Appl. Anal. 95 (2016), 2444-2463. DOI 10.1080/00036811.2015.1092522 | MR 3546596 | Zbl 1432.35253
[19] Grubb, G.: Distributions and Operators. Graduate Texts in Mathematics 252. Springer, New York (2009). DOI 10.1007/978-0-387-84895-2 | MR 2453959 | Zbl 1171.47001
[20] Heck, H., Wang, J.-N.: Stability estimates for the inverse boundary value problem by partial Cauchy data. Inverse Probl. 22 (2006), 1787-1796. DOI 10.1088/0266-5611/22/5/015 | MR 2261266 | Zbl 1106.35133
[21] Ikehata, M.: A special Green's function for the biharmonic operator and its application to an inverse boundary value problem. Comput. Math. Appl. 22 (1991), 53-66. DOI 10.1016/0898-1221(91)90131-M | MR 1127214 | Zbl 0770.35078
[22] Isakov, V.: Completeness of products of solutions and some inverse problems for PDE. J. Differ. Equations 92 (1991), 305-316. DOI 10.1016/0022-0396(91)90051-A | MR 1120907 | Zbl 0728.35141
[23] Kenig, C. E., Sjöestrand, J., Uhlmann, G.: The Calderón problem with partial data. Ann. Math. (2) 165 (2007), 567-591. DOI 10.4007/annals.2007.165.567 | MR 2299741 | Zbl 1127.35079
[24] Krupchyk, K., Lassas, M., Uhlmann, G.: Determining a first order perturbation of the biharmonic operator by partial boundary measurements. J. Funct. Anal. 262 (2012), 1781-1801. DOI 10.1016/j.jfa.2011.11.021 | MR 2873860 | Zbl 1239.35184
[25] Krupchyk, K., Lassas, M., Uhlmann, G.: Inverse boundary value problems for the perturbed polyharmonic operator. Trans. Am. Math. Soc. 366 (2014), 95-112. DOI 10.1090/S0002-9947-2013-05713-3 | MR 3118392 | Zbl 1317.35295
[26] Krupchyk, K., Uhlmann, G.: Inverse boundary problems for polyharmonic operators with unbounded potentials. J. Spectr. Theory 6 (2016), 145-183. DOI 10.4171/JST/122 | MR 3484382 | Zbl 1345.35139
[27] Liu, B.: Stability estimates for an inverse boundary value problem for biharmonic operators with first order perturbation from partial data. Inverse Probl. 40 (2024), Article ID 065001, 25 pages. DOI 10.1088/1361-6420/ad3be6 | MR 4736034 | Zbl 1542.35446
[28] Liu, B., Selim, S.: Stable determination of the first order perturbation of the biharmonic operator from partial data. J. Differ. Equations 445 (2025), Article ID 113575, 30 pages. DOI 10.1016/j.jde.2025.113575 | MR 4925885 | Zbl 1572.35329
[29] Mandache, N.: Exponential instability in an inverse problem for the Schrödinger equation. Inverse Probl. 17 (2001), 1435-1444. DOI 10.1088/0266-5611/17/5/313 | MR 1862200 | Zbl 0985.35110
[30] Salo, M., Tzou, L.: Carleman estimates and inverse problems for Dirac operators. Math. Ann. 344 (2009), 161-184. DOI 10.1007/s00208-008-0301-9 | MR 2481057 | Zbl 1169.35063
[31] Serov, V. S.: Borg-Levinson theorem for perturbations of the bi-harmonic operator. Inverse Probl. 32 (2016), Article ID 045002, 19 pages. DOI 10.1088/0266-5611/32/4/045002 | MR 3476498 | Zbl 1380.47037
[32] Sylvester, J., Uhlmann, G.: A global uniqueness theorem for an inverse boundary value problem. Ann. Math. (2) 125 (1987), 153-169. DOI 10.2307/1971291 | MR 0873380 | Zbl 0625.35078
[33] Vessella, S.: A continuous dependence result in the analytic continuation problem. Forum Math. 11 (1999), 695-703. DOI 10.1515/form.1999.020 | MR 1724631 | Zbl 0933.35192
[34] Yan, L.: Inverse boundary problems for biharmonic operators in transversally anisotropic geometries. SIAM J. Math. Anal. 53 (2021), 6617-6653. DOI 10.1137/21M1391419 | MR 4344435 | Zbl 1479.35903
[35] Yang, Y.: Determining the first order perturbation of a bi-harmonic operator on bounded and unbounded domains from partial data. J. Differ. Equations 257 (2014), 3607-3639. DOI 10.1016/j.jde.2014.07.003 | MR 3260235 | Zbl 1298.35255
Partner of
EuDML logo