×

Carleman’s formula of a solutions of the Poisson equation in bounded domain. (English) Zbl 1486.35155

Summary: We suggest an explicit continuation formula for a solution to the Cauchy problem for the Poisson equation in a domain from its values and values of its normal derivative on a part of the boundary. We construct the continuation formula of this problem based on the Carleman-Yarmuhamedov function method.

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J25 Boundary value problems for second-order elliptic equations
PDF BibTeX XML Cite
Full Text: DOI MNR

References:

[1] Aizenberg L. A., Tarkhanov N. N., “An abstract Carleman formula”, Dokl. Math., 37:1 (1988), 235-238 · Zbl 0780.35030
[2] Aizenberg L. A., Carleman’s Formulas in Complex Analysis. Theory and Applications, Math. Applications, 244, Springer, Dordrecht, 1993, 299 pp.
[3] Alessandrini G., Rondi L., Rosset E., and Vessella S., “The stability for the Cauchy problem for elliptic equations”, Inverse problems, 25:12 (2009), 123004 · Zbl 1190.35228
[4] Arbuzov E. V., Bukhgeim A. L., “The Carleman”s formula for the Maxwell’s equations on a plane”, Sib. Elektron. Mat. Izv., 2008, no. 5, 448-455 (in Russian) · Zbl 1299.35109
[5] {Belgacem F. B.} Why is the Cauchy problem severely ill-posed?, Inverse problems, 23:2 (2007), 823-836 · Zbl 1118.35060
[6] Bers L., John F., Schechter M., Partial Differential Equations, Interscience, NY, 1964, 343 pp. · Zbl 0126.00207
[7] Blum J., Numerical Simulation and Optimal Control in Plasma Physics With Applications to Tokamaks, John Wiley and Sons Inc., NY, 1989, 363 pp. · Zbl 0717.76009
[8] Carleman T., Les Founctions Quasi Analitiques, Gauthier-Villars, Paris, 1926, 115 pp. (in French)
[9] Chen G., Zhou J., Boundary Element Methods., Academic Press, London etc., 1992, 646 pp. · Zbl 0842.65071
[10] Colli-Franzone P., Guerri L., Tentoni S., Viganotti C., Baruffi S., Spaggiari S., Taccardi B., “A mathematical procedure for solving the inverse potential problem of electrocardiography. analysis of the time-space accuracy from in vitro experimental data”, Math. Biosci., 77:1-2 (1985), 353-396 · Zbl 0578.92006
[11] Dzhrbashyan M. M., {Integral’nye preobrazovaniya i predstavleniya funkcij v kompleksnoj oblasti} [Integral Transformations and Representations of Functions in a Compkjoplex Domain], Nauka, M., 1966, 670 pp. (in Russian)
[12] Ermamatova Z. E., “Carleman”s formula of a solution of the Poisson equation”, Int. J. Integrated Education, 4:7 (2021), 112-117
[13] Fasino D., Inglese G., “An inverse Robin problem for Laplace”s equation: theoretical results and numerical methods”, Inverse problems, 15:1 (1999), 41-48. · Zbl 0922.35188
[14] Fok V. A., Kuni F. M., “On the introduction of a ”suppressing’ function in dispersion relations”, Dokl. Akad. Nauk SSSR, 127:6 (1959), 1195-1198 (in Russian) · Zbl 0092.45501
[15] Gilbarg D., Trudinger N. S., Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-Heidelberg-NY-Tokyo, 1983, 513 pp. · Zbl 0562.35001
[16] Goluzin G. M., Krylov V. I., “A generalized Carleman formula and its application to analytic continuation of functions”, Mat. Sb., 40:2 (1933), 144-149 (in Russian)
[17] Hadamard J., Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Yale University Press, New Haven, 1923, 334 pp. · JFM 49.0725.04
[18] Hu X., Mu L., Ye X., “A simple finite element method of the Cauchy problem for Poisson equation”, Int. J. Numer. Anal. Model., 14:4-5 (2017), 591-603 · Zbl 1380.65372
[19] Inglese G., “An inverse problem in corrosion detection”, Inverse problems, 13:4 (1997), 977-994 · Zbl 0882.35133
[20] Ivanov V. K., “Cauchy problem for the Laplace equation in an infinite strip”, Differ. Uravn., 1:1 (1965), 131-136 (in Russian) · Zbl 0192.20101
[21] Kabanikhin S. I., Inverse and Ill-Posed Problems: Theory and Applications, De Gruyter, Berlin, 2011, 459 pp. · Zbl 1247.65077
[22] Lattès R., Lions J.-L., The Method of Quasi-Reversibility: Applications to Partial Differential Equations, Modern Analytic and Computational Methods in Science and Mathematics, 18, American Elsevier Pub. Co., NY, 1969, 388 pp. · Zbl 1220.65002
[23] Lavrentiev M. M., Some Improperly Posed Problems of Mathematical Physics, Berlin, Springer, 1967, 76 pp. · Zbl 0149.41902
[24] Lavrentiev M. M., “On Cauchy problem for linear elliptical equations of the second order”, Dokl. Akad. Nauk SSSR, 112:2 (1957), 195-197 (in Russian) · Zbl 0077.30001
[25] Makhmudov O. I., “The Cauchy problem for a system of equations of the theory of elasticity and thermoelasticity in space”, Russian Math. (Iz. VUZ), 48:2 (2004), 40-50 · Zbl 1084.35114
[26] Makhmudov O., Niyozov I., Tarkhanov N., “The Cauchy problem of couple-stress elasticity”, Contemporary Math., 455 (2008), 297-310 · Zbl 1149.74011
[27] Makhmudov K. O., Makhmudov O. I., Tarkhanov N., “Equations of Maxwell type”, J. Math. Anal. App., 378:1 (2011), 64-75 · Zbl 1211.35256
[28] Mergelyan S. N., “Harmonic approximation and approximate solution of the Cauchy problem for the Laplace equation”, Uspekhi Mat. Nauk, 11:5 (1956), 3-26 · Zbl 0074.09003
[29] Miranda C., Partial Differential Equations of Elliptic Type, Springer, Berlin, Heidelberg, NY, 1970, 370 pp. · Zbl 0198.14101
[30] Peicheva A. S., “Regularization of the Cauchy problem for elliptic operators”, J. Sibirian Federal University. Math. Phy., 2 (2018), 191-193 · Zbl 07325405
[31] Sattorov E. N., “Regularization of the solution of the Cauchy problem for the generalized Moisil-Theodoresco system”, Diff. Equat., 44 (2008), 1136—1146 · Zbl 1187.35053
[32] Sattorov E. N., “Continuation of a solution to a homogeneous system of Maxwell equations”, Russian Math. (Iz. VUZ), 52:8 (2008), 65—69 · Zbl 1178.35359
[33] Sattorov E. N., “On the continuation of the solutions of a generalized Cauchy-Riemann system in space”, Math. Notes, 85:5 (2009), 733-745 · Zbl 1176.35043
[34] Sattorov E. N., “Regularization of the solution of the Cauchy problem for the system of Maxwell equations in an unbounded domain”, Math. Notes, 86:6 (2009), 422-431 · Zbl 1181.35281
[35] Sattorov E. N., “Reconstruction of solutions to a generalized Moisil-Teodorescu system in a spatial domain from their values on a part of the boundary”, Russian Math., 55:1 (2011), 62—73 · Zbl 1231.35294
[36] Sattorov E. N., Ermamatova Z. E., “Recovery of solutions to homogeneous system of Maxwell”s equations with prescribed values on a part of the boundary of domain”, Russian Math., 63:2 (2019), 35—43 · Zbl 1442.65330
[37] Sattorov E. N., Ermamatova F. E., “Carleman”s formula for solutions of the generalized Cauchy-Riemann system in multidimensional spatial domain”, Contem. Problems in Mathematics and Physics, 65:1 (2019), 95-108
[38] Sattorov E. N., Ermamatova F. E., “On continuation of solutions of generalized Cauchy-Riemann system in an unbounded subdomain of multidimensional space”, Russian Math., 65:2 (2021), 22-38 · Zbl 1468.30078
[39] Sattorov E. N., Ermamatova F. E., “Cauchy problem for a generalized Cauchy-Riemann system in a multidimensional bounded spatial domain”, Diff. Equat., 57:1 (2021), 86-99 · Zbl 07314330
[40] Sattorov E. N., Mardanov D. A., “The Cauchy problem for the system of Maxwell equations”, Siberian Math. J., 44:4 (2003), 671-679 · Zbl 1028.35033
[41] Tarkhanov N. N., “The Carleman matrix for elliptic systems”, Dokl. Akad. Nauk SSSR, 284:2 (1985), 294-297 (in Russian) · Zbl 0601.35030
[42] Tarkhanov N. N., The Cauchy Problem for Solutions of Elliptic Equations, Akad. Verl., Berlin, 1995, 479 pp. · Zbl 0831.35001
[43] Tikhonov A. N., “On the solution of ill-posed problems and the method of regularization”, Dokl. Akad. Nauk SSSR, 151:3 (1963), 501-504 (in Russian) · Zbl 0141.11001
[44] Tikhonov A. N., Arsenin V. Y., Solutions of Ill-Posed Problems, John Wiley & Sons, NY, 1977 · Zbl 0354.65028
[45] Yarmukhamedov Sh., “On the Cauchy problem for Laplace”s equation”, Dokl. Akad. Nauk SSSR, 235:2 (1977), 281—283 (in Russian) · Zbl 0387.35022
[46] Yarmukhamedov Sh., “Continuing solutions to the Helmholtz equation”, Doklady Math., 56:3 (1997), 887-890 · Zbl 0969.35041
[47] Yarmukhamedov Sh., “A Carleman function and the Cauchy problem for the Laplace equation”, Sib. Math. J., 45:3 (2004), 580-595 · Zbl 1051.31002
[48] Yarmukhamedov Sh., “Representation of Harmonic Functions as Potentials and the Cauchy Problem”, Math. Notes, 83:5 (2008), 693-706 · Zbl 1161.31004
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.