Article
| Title: | Stability estimates for polyharmonic operator from partial boundary measurements (English) |
| Author: | Ma, Yanjun |
| Language: | English |
| Journal: | Applications of Mathematics |
| ISSN: | 0862-7940 (print) |
| ISSN: | 1572-9109 (online) |
| Volume: | 71 |
| Issue: | 3 |
| Year: | 2026 |
| Pages: | 409-426 |
| Summary lang: | English |
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| Category: | math |
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| 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. (English) |
| Keyword: | stability estimate |
| Keyword: | Dirichlet-to-Neumann map |
| Keyword: | Carleman estimate |
| Keyword: | polyharmonic operator |
| MSC: | 35B20 |
| MSC: | 35J40 |
| MSC: | 35R30 |
| DOI: | 10.21136/AM.2026.0194-24 |
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| Date available: | 2026-07-02T06:53:08Z |
| Last updated: | 2026-08-19 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153675 |
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| Reference: | [1] Alessandrini, G.: Stable determination of conductivity by boundary measurements.Appl. Anal. 27 (1988), 153-172. Zbl 0616.35082, MR 0922775, 10.1080/00036818808839730 |
| Reference: | [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. Zbl 1510.35392, MR 4530920, 10.1016/j.jmaa.2022.126965 |
| Reference: | [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. Zbl 1062.35050, MR 2091488 |
| Reference: | [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. Zbl 1369.35112, MR 3627033, 10.1088/0266-5611/32/10/105009 |
| Reference: | [5] Assylbekov, Y. M., Iyer, K.: Determining rough first order perturbations of the polyharmonic operator.Inverse Probl. Imaging 13 (2019), 1045-1066. Zbl 1423.35433, MR 4027047, 10.3934/ipi.2019047 |
| Reference: | [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. Zbl 1354.58014, MR 3567495, 10.1016/j.jde.2016.09.039 |
| Reference: | [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. Zbl 1432.58017, MR 3810180, 10.3934/ipi.2018034 |
| Reference: | [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. Zbl 1414.35261, MR 3953481, 10.1007/s00041-018-9625-3 |
| Reference: | [9] Bhattacharyya, S., Ghosh, T.: An inverse problem on determining second order symmetric tensor for perturbed biharmonic operator.Math. Ann. 384 (2022), 457-489. Zbl 1497.35511, MR 4476229, 10.1007/s00208-021-02276-6 |
| Reference: | [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. Zbl 1548.35295, MR 4723844, 10.1088/1361-6420/ad3164 |
| Reference: | [11] Bukhgeim, A. L., Uhlmann, G.: Recovering a potential from partial Cauchy data.Commun. Partial Differ. Equations 27 (2002), 653-668. Zbl 0998.35063, MR 1900557, 10.1081/PDE-120002868 |
| Reference: | [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 |
| Reference: | [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. Zbl 1368.31003, MR 3630136, 10.1515/jiip-2016-0019 |
| Reference: | [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. Zbl 1325.35276, MR 3357587, 10.1016/j.jmaa.2015.05.054 |
| Reference: | [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. Zbl 1148.35096, MR 2287913, 10.1007/s00220-006-0151-9 |
| Reference: | [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). Zbl 1239.35002, MR 2667016, 10.1007/978-3-642-12245-3 |
| Reference: | [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. Zbl 1328.35307, MR 3405369, 10.1088/0266-5611/31/10/105009 |
| Reference: | [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. Zbl 1432.35253, MR 3546596, 10.1080/00036811.2015.1092522 |
| Reference: | [19] Grubb, G.: Distributions and Operators.Graduate Texts in Mathematics 252. Springer, New York (2009). Zbl 1171.47001, MR 2453959, 10.1007/978-0-387-84895-2 |
| Reference: | [20] Heck, H., Wang, J.-N.: Stability estimates for the inverse boundary value problem by partial Cauchy data.Inverse Probl. 22 (2006), 1787-1796. Zbl 1106.35133, MR 2261266, 10.1088/0266-5611/22/5/015 |
| Reference: | [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. Zbl 0770.35078, MR 1127214, 10.1016/0898-1221(91)90131-M |
| Reference: | [22] Isakov, V.: Completeness of products of solutions and some inverse problems for PDE.J. Differ. Equations 92 (1991), 305-316. Zbl 0728.35141, MR 1120907, 10.1016/0022-0396(91)90051-A |
| Reference: | [23] Kenig, C. E., Sjöestrand, J., Uhlmann, G.: The Calderón problem with partial data.Ann. Math. (2) 165 (2007), 567-591. Zbl 1127.35079, MR 2299741, 10.4007/annals.2007.165.567 |
| Reference: | [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. Zbl 1239.35184, MR 2873860, 10.1016/j.jfa.2011.11.021 |
| Reference: | [25] Krupchyk, K., Lassas, M., Uhlmann, G.: Inverse boundary value problems for the perturbed polyharmonic operator.Trans. Am. Math. Soc. 366 (2014), 95-112. Zbl 1317.35295, MR 3118392, 10.1090/S0002-9947-2013-05713-3 |
| Reference: | [26] Krupchyk, K., Uhlmann, G.: Inverse boundary problems for polyharmonic operators with unbounded potentials.J. Spectr. Theory 6 (2016), 145-183. Zbl 1345.35139, MR 3484382, 10.4171/JST/122 |
| Reference: | [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. Zbl 1542.35446, MR 4736034, 10.1088/1361-6420/ad3be6 |
| Reference: | [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. Zbl 1572.35329, MR 4925885, 10.1016/j.jde.2025.113575 |
| Reference: | [29] Mandache, N.: Exponential instability in an inverse problem for the Schrödinger equation.Inverse Probl. 17 (2001), 1435-1444. Zbl 0985.35110, MR 1862200, 10.1088/0266-5611/17/5/313 |
| Reference: | [30] Salo, M., Tzou, L.: Carleman estimates and inverse problems for Dirac operators.Math. Ann. 344 (2009), 161-184. Zbl 1169.35063, MR 2481057, 10.1007/s00208-008-0301-9 |
| Reference: | [31] Serov, V. S.: Borg-Levinson theorem for perturbations of the bi-harmonic operator.Inverse Probl. 32 (2016), Article ID 045002, 19 pages. Zbl 1380.47037, MR 3476498, 10.1088/0266-5611/32/4/045002 |
| Reference: | [32] Sylvester, J., Uhlmann, G.: A global uniqueness theorem for an inverse boundary value problem.Ann. Math. (2) 125 (1987), 153-169. Zbl 0625.35078, MR 0873380, 10.2307/1971291 |
| Reference: | [33] Vessella, S.: A continuous dependence result in the analytic continuation problem.Forum Math. 11 (1999), 695-703. Zbl 0933.35192, MR 1724631, 10.1515/form.1999.020 |
| Reference: | [34] Yan, L.: Inverse boundary problems for biharmonic operators in transversally anisotropic geometries.SIAM J. Math. Anal. 53 (2021), 6617-6653. Zbl 1479.35903, MR 4344435, 10.1137/21M1391419 |
| Reference: | [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. Zbl 1298.35255, MR 3260235, 10.1016/j.jde.2014.07.003 |
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