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Mixed impedance boundary value problems for the Laplace-Beltrami equation. (English) Zbl 1454.35128
The paper under review is concerned with the analysis of the mixed impedance-Neumann-Dirichlet boundary value problem for the Laplace-Beltrami equation on a compact smooth surface with smooth boundary. In the first part of this paper, by using the Lax-Milgram Lemma, the authors establish that this problem has a unique solution in the classical weak setting. Next, the authors consider the mixed impedance-Neumann-Dirichlet boundary value problem in a nonclassical setting of the Bessel potential space. The basic tool for this analysis is a quasilocalization technique. This allows to obtain a necessary and sufficient condition for the Fredholmness of the mixed impedance-Neumann-Dirichlet boundary value problem and to indicate a large set of the space parameters for which the initial BVP is uniquely solvable in the nonclassical setting. As a consequence, the authors prove that the mixed impedance-Neumann-Dirichlet boundary value problem has a unique solution in the classical weak setting for arbitrary complex values of the nonzero constant in the impedance condition.

MSC:
35J57 Boundary value problems for second-order elliptic systems
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
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[1] A.-S. Bonnet-Ben Dhia, L. Chesnel, and P. Ciarlet, Jr., “\(T\)-coercivity for scalar interface problems between dielectrics and metamaterials”, ESAIM Math. Model. Numer. Anal. 46:6 (2012), 1363-1387. Mathematical Reviews (MathSciNet): MR2996331
Zentralblatt MATH: 1276.78008
Digital Object Identifier: doi:10.1051/m2an/2012006
· Zbl 1276.78008
[2] A.-S. Bonnet-Bendhia and A. Tillequin, “A limiting absorption principle for scattering problems with unbounded obstacles”, Math. Methods Appl. Sci. 24:14 (2001), 1089-1111. Mathematical Reviews (MathSciNet): MR1855301
Zentralblatt MATH: 0987.78014
Digital Object Identifier: doi:10.1002/mma.259
· Zbl 0987.78014
[3] T. Buchukuri, O. Chkadua, R. Duduchava, and D. Natroshvili, “Interface crack problems for metallic-piezoelectric composite structures”, Mem. Differ. Equ. Math. Phys. 55 (2012), 1-150. Mathematical Reviews (MathSciNet): MR3026952
Zentralblatt MATH: 1407.74075
· Zbl 1407.74075
[4] L. P. Castro and D. Kapanadze, “On wave diffraction by a half-plane with different face impedances”, Math. Methods Appl. Sci. 30:5 (2007), 513-527. Mathematical Reviews (MathSciNet): MR2298679
Zentralblatt MATH: 1128.78005
Digital Object Identifier: doi:10.1002/mma.794
· Zbl 1128.78005
[5] L. P. Castro and D. Kapanadze, “Dirichlet-Neumann-impedance boundary value problems arising in rectangular wedge diffraction problems”, Proc. Amer. Math. Soc. 136:6 (2008), 2113-2123. Mathematical Reviews (MathSciNet): MR2383517
Zentralblatt MATH: 1185.35055
Digital Object Identifier: doi:10.1090/S0002-9939-08-09288-5
· Zbl 1185.35055
[6] L. P. Castro and D. Kapanadze, “The impedance boundary-value problem of diffraction by a strip”, J. Math. Anal. Appl. 337:2 (2008), 1031-1040. Mathematical Reviews (MathSciNet): MR2386353
Zentralblatt MATH: 1151.35015
Digital Object Identifier: doi:10.1016/j.jmaa.2007.04.037
· Zbl 1151.35015
[7] L. P. Castro and D. Kapanadze, “Wave diffraction by a strip with first and second kind boundary conditions: the real wave number case”, Math. Nachr. 281:10 (2008), 1400-1411. Mathematical Reviews (MathSciNet): MR2454942
Zentralblatt MATH: 1180.35182
Digital Object Identifier: doi:10.1002/mana.200510687
· Zbl 1180.35182
[8] L. P. Castro and D. Kapanadze, “Exterior wedge diffraction problems with Dirichlet, Neumann and impedance boundary conditions”, Acta Appl. Math. 110:1 (2010), 289-311. Mathematical Reviews (MathSciNet): MR2601658
Zentralblatt MATH: 1189.35396
Digital Object Identifier: doi:10.1007/s10440-008-9408-y
· Zbl 1189.35396
[9] L. P. Castro and D. Kapanadze, “Wave diffraction by a half-plane with an obstacle perpendicular to the boundary”, J. Differential Equations 254:2 (2013), 493-510. Mathematical Reviews (MathSciNet): MR2990040
Zentralblatt MATH: 1259.35071
Digital Object Identifier: doi:10.1016/j.jde.2012.08.030
· Zbl 1259.35071
[10] L. P. Castro and D. Kapanadze, “Wave diffraction by wedges having arbitrary aperture angle”, J. Math. Anal. Appl. 421:2 (2015), 1295-1314. Mathematical Reviews (MathSciNet): MR3258320
Zentralblatt MATH: 1300.35005
Digital Object Identifier: doi:10.1016/j.jmaa.2014.07.080
· Zbl 1300.35005
[11] L. P. Castro, R. Duduchava, and F.-O. Speck, “Localization and minimal normalization of some basic mixed boundary value problems”, pp. 73-100 in Factorization, singular operators and related problems: proceedings of the conference in honour of professor Georgii Litvinchuk (Funchal), edited by S. Samko et al., Kluwer, Dordrecht, 2003. Mathematical Reviews (MathSciNet): MR2001593
[12] L. P. Castro, F.-O. Speck, and F. S. Teixeira, “Explicit solution of a Dirichlet-Neumann wedge diffraction problem with a strip”, J. Integral Equations Appl. 15:4 (2003), 359-383. Mathematical Reviews (MathSciNet): MR2058809
Zentralblatt MATH: 1069.47073
Digital Object Identifier: doi:10.1216/jiea/1181074982
Project Euclid: euclid.jiea/1181074982
· Zbl 1069.47073
[13] L. P. Castro, F.-O. Speck, and F. S. Teixeira, “On a class of wedge diffraction problems posted by Erhard Meister”, pp. 213-240 in Operator theoretical methods and applications to mathematical physics, edited by I. Gohberg et al., Operator Theory: Advances and Applications 147, Birkhäuser, Basel, 2004. Mathematical Reviews (MathSciNet): MR2053691
Zentralblatt MATH: 1187.35029
· Zbl 1187.35029
[14] L. P. Castro, F.-O. Speck, and F. S. Teixeira, “Mixed boundary value problems for the Helmholtz equation in a quadrant”, Integral Equations Operator Theory 56:1 (2006), 1-44. Mathematical Reviews (MathSciNet): MR2256995
Zentralblatt MATH: 1206.35089
Digital Object Identifier: doi:10.1007/s00020-005-1410-4
· Zbl 1206.35089
[15] V. D. Didenko and R. Duduchava, “Mellin convolution operators in Bessel potential spaces”, J. Math. Anal. Appl. 443:2 (2016), 707-731. Mathematical Reviews (MathSciNet): MR3514315
Zentralblatt MATH: 1348.47027
Digital Object Identifier: doi:10.1016/j.jmaa.2016.05.043
· Zbl 1348.47027
[16] R. V. Duduchava, “On Noether theorems for singular integral equations”, pp. 19-52 in Proceedings of Symposium on Mechanics and Related Problems of Analysis, vol. 1, 1973. in Russian.
[17] R. Duduchava, “The Green formula and layer potentials”, Integral Equations Operator Theory 41:2 (2001), 127-178. Mathematical Reviews (MathSciNet): MR1847170
Zentralblatt MATH: 1102.31006
Digital Object Identifier: doi:10.1007/BF01295303
· Zbl 1102.31006
[18] R. Duduchava, “Partial differential equations on hypersurfaces”, Mem. Differential Equations Math. Phys. 48 (2009), 19-74. Mathematical Reviews (MathSciNet): MR2603279
Zentralblatt MATH: 1196.35072
· Zbl 1196.35072
[19] R. Duduchava and M. Tsaava, “Mixed boundary value problems for the Laplace-Beltrami equation”, Complex Var. Elliptic Equ. 63:10 (2018), 1468-1496. Mathematical Reviews (MathSciNet): MR3833024
Zentralblatt MATH: 1404.35146
Digital Object Identifier: doi:10.1080/17476933.2017.1385066
· Zbl 1404.35146
[20] R. Duduchava, D. Natroshvili, and E. Shargorodsky, “Basic boundary value problems of thermoelasticity for anisotropic bodies with cuts. I”, Georgian Math. J. 2:2 (1995), 123-140. Mathematical Reviews (MathSciNet): MR1316951
Zentralblatt MATH: 0822.35135
Digital Object Identifier: doi:10.1007/BF02257474
· Zbl 0822.35135
[21] R. Duduchava, D. Natroshvili, and E. Shargorodsky, “Basic boundary value problems of thermoelasticity for anisotropic bodies with cuts. II”, Georgian Math. J. 2:3 (1995), 259-276. Mathematical Reviews (MathSciNet): MR1334881
Zentralblatt MATH: 0822.35136
Digital Object Identifier: doi:10.1007/BF02261700
· Zbl 0822.35136
[22] L. R. Duduchava, D. Mitrea, and M. Mitrea, “Differential operators and boundary value problems on hypersurfaces”, Math. Nachr. 279:9-10 (2006), 996-1023. Mathematical Reviews (MathSciNet): MR2242962
Zentralblatt MATH: 1112.58020
Digital Object Identifier: doi:10.1002/mana.200410407
· Zbl 1112.58020
[23] R. DuDuchava, M. Tsaava, and T. Tsutsunava, “Mixed boundary value problem on hypersurfaces”, Int. J. Differ. Equ. (2014), art.,id.,245350. Mathematical Reviews (MathSciNet): MR3253609
Zentralblatt MATH: 1308.35116
· Zbl 1308.35116
[24] T. Ehrhardt, A. P. Nolasco, and F.-O. Speck, “Boundary integral methods for wedge diffraction problems: the angle \(2\pi/n\), Dirichlet and Neumann conditions”, Oper. Matrices 5:1 (2011), 1-40. Mathematical Reviews (MathSciNet): MR2798793
Zentralblatt MATH: 1216.35013
Digital Object Identifier: doi:10.7153/oam-05-01
· Zbl 1216.35013
[25] T. Ehrhardt, A. P. Nolasco, and F.-O. Speck, “A Riemannn surface approach for diffraction from rational wedges”, Oper. Matrices 8:2 (2014), 301-355. Mathematical Reviews (MathSciNet): MR3224813
Zentralblatt MATH: 1306.35012
Digital Object Identifier: doi:10.7153/oam-08-17
· Zbl 1306.35012
[26] I. Gohberg and N. Krupnik, One-dimensional linear singular integral equations, Operator Theory: Advances and Applications 53-54, Birkhäuser, Basel, 1992. Mathematical Reviews (MathSciNet): MR1138208
Zentralblatt MATH: 0743.45004
· Zbl 0743.45004
[27] G. C. Hsiao and W. L. Wendland, Boundary integral equations, Applied Mathematical Sciences 164, Springer, Berlin, 2008. Mathematical Reviews (MathSciNet): MR2441884
Zentralblatt MATH: 1157.65066
· Zbl 1157.65066
[28] D. Kapanadze and B.-W. Schulze, Crack theory and edge singularities, Mathematics and its Applications 561, Kluwer Academic Publishers Group, Dordrecht, 2003. Mathematical Reviews (MathSciNet): MR2023308
Zentralblatt MATH: 1053.58010
· Zbl 1053.58010
[29] A. I. Komech, N. J. Mauser, and A. E. Merzon, “On Sommerfeld representation and uniqueness in scattering by wedges”, Math. Methods Appl. Sci. 28:2 (2005), 147-183. Mathematical Reviews (MathSciNet): MR2110262
Zentralblatt MATH: 1072.35180
Digital Object Identifier: doi:10.1002/mma.553
· Zbl 1072.35180
[30] V. A. Kozlov, V. G. Mazya, and J. Rossmann, Spectral problems associated with corner singularities of solutions to elliptic equations, Mathematical Surveys and Monographs 85, American Mathematical Society, Providence, 2001. Mathematical Reviews (MathSciNet): MR1788991
Zentralblatt MATH: 0965.35003
· Zbl 0965.35003
[31] P. D. Lax and A. N. Milgram, “Parabolic equations”, pp. 167-190 in Contributions to the theory of partial differential equations, Annals of Mathematics Studies 33, Princeton University Press, Princeton, 1954. Mathematical Reviews (MathSciNet): MR0067317
Zentralblatt MATH: 0058.08703
· Zbl 0058.08703
[32] J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, vol. 1, Springer, Heidelberg, 1972. Mathematical Reviews (MathSciNet): MR0350178
Zentralblatt MATH: 0223.35039
· Zbl 0223.35039
[33] G. D. Malyuzhinets, “The excitation, reflection and radiation of surface waves in a wedge-like region with given face impedances”, Dokl. Akad. Nauk SSSR 121 (1958), 436-439. Mathematical Reviews (MathSciNet): MR0099201
[34] E. Meister, “Some multiple-part Wiener-Hopf problems in mathematical physics”, pp. 359-407 in Mathematical models and methods in mechanics, edited by W. a. a. Fiszdon and K. Wilmański, Banach Center Publ. 15, PWN, Warsaw, 1985. Mathematical Reviews (MathSciNet): MR874849
Zentralblatt MATH: 0549.35110
· Zbl 0549.35110
[35] E. Meister, “Some solved and unsolved canonical problems of diffraction theory”, pp. 320-336 in Differential equations and mathematical physics, edited by I. W. Knowles and Y. Saitō, Lecture Notes in Math. 1285, Springer, Berlin, 1987. Mathematical Reviews (MathSciNet): MR921283
Zentralblatt MATH: 0651.35019
· Zbl 0651.35019
[36] E. Meister and F.-O. Speck, “Some multidimensional Wiener-Hopf equations with applications”, pp. 217-262 in Trends in applications of pure mathematics to mechanics, vol. 2, Monographs Stud. Math. 5, Pitman, London, 1979. Mathematical Reviews (MathSciNet): MR566531
Zentralblatt MATH: 0415.45001
· Zbl 0415.45001
[37] E. Meister and F.-O. Speck, “Modern Wiener-Hopf methods in diffraction theory”, pp. 130-171 in Ordinary and partial differential equations, vol. 2, edited by B. D. Sleeman and R. J. Jarvis, Pitman Res. Notes Math. Ser. 216, Longman Sci. Tech., Harlow, 1989. Mathematical Reviews (MathSciNet): MR1031728
Zentralblatt MATH: 0689.35024
· Zbl 0689.35024
[38] E. Meister, F. Penzel, F.-O. Speck, and F. S. Teixeira, “Some interior and exterior boundary-value problems for the Helmholtz equation in a quadrant”, Proc. Roy. Soc. Edinburgh Sect. A 123:2 (1993), 275-294. Mathematical Reviews (MathSciNet): MR1215413
Zentralblatt MATH: 0794.35029
Digital Object Identifier: doi:10.1017/S0308210500025671
· Zbl 0794.35029
[39] A. Moura Santos, F.-O. Speck, and F. S. Teixeira, “Minimal normalization of Wiener-Hopf operators in spaces of Bessel potentials”, J. Math. Anal. Appl. 225:2 (1998), 501-531. Mathematical Reviews (MathSciNet): MR1644280
Zentralblatt MATH: 0915.47020
Digital Object Identifier: doi:10.1006/jmaa.1998.6041
· Zbl 0915.47020
[40] B. Noble, Methods based on the Wiener-Hopf technique for the solution of partial differential equations, International Series of Monographs on Pure and Applied Mathematics 7, Pergamon, London, 1958. Mathematical Reviews (MathSciNet): MR0102719
Zentralblatt MATH: 0082.32101
· Zbl 0082.32101
[41] A. F. dos Santos and F. S. Teixeira, “The Sommerfeld problem revisited: solution spaces and the edge conditions”, J. Math. Anal. Appl. 143:2 (1989), 341-357. Mathematical Reviews (MathSciNet): MR1022540
Zentralblatt MATH: 0713.35023
Digital Object Identifier: doi:10.1016/0022-247X(89)90045-0
· Zbl 0713.35023
[42] A. M. Santos, F. O. Speck, and F. S. Teixeira, “Compatibility conditions in some diffraction problems”, pp. 25-38 in Direct and inverse electromagnetic scattering, edited by A. H. Serbest and S. R. Cloude, Pitman Res. Notes Math. Ser. 361, Longman, Harlow, 1996. Mathematical Reviews (MathSciNet): MR1463158
Zentralblatt MATH: 0869.35102
· Zbl 0869.35102
[43] F.-O. Speck, “Mixed boundary value problems of the type of Sommerfeld”s half-plane problem”, Proc. Roy. Soc. Edinburgh Sect. A 104:3-4 (1986), 261-277. Mathematical Reviews (MathSciNet): MR877905
Zentralblatt MATH: 0626.35020
Digital Object Identifier: doi:10.1017/S0308210500019211
· Zbl 0626.35020
[44] F.-O. Speck, “On the reduction of linear systems related to boundary value problems”, pp. 391-406 in Operator theory, pseudo-differential equations, and mathematical physics, edited by Y. I. Karlovich et al., Oper. Theory Adv. Appl. 228, Birkhäuser, Basel, 2013. Mathematical Reviews (MathSciNet): MR3025506
Zentralblatt MATH: 1276.47026
· Zbl 1276.47026
[45] F.-O. Speck, R. A. Hurd, and E. Meister, “Sommerfeld diffraction problems with third kind boundary conditions”, SIAM J. Math. Anal. 20:3 (1989), 589-607. Mathematical Reviews (MathSciNet): MR990866
Zentralblatt MATH: 0677.47021
Digital Object Identifier: doi:10.1137/0520042
· Zbl 0677.47021
[46] H. Triebel, Interpolation theory, function spaces, differential operators, 2nd ed., Johann Ambrosius Barth, Heidelberg, 1995. Mathematical Reviews (MathSciNet): MR1328645
Zentralblatt MATH: 0830.46028
· Zbl 0830.46028
[47] P. Y. Ufimtsev, Theory of edge diffraction in electromagnetics, Tech Science Press, Encino, 2003. Zentralblatt MATH: 1036.78001
· Zbl 1036.78001
[48] V. B. Vasiliev, Wave factorization of elliptic symbols: theory and applications, Kluwer Academic Publishers, Dordrecht, 2000. Mathematical Reviews (MathSciNet): MR1795504
[49] W. L. Wendland, E. Stephan, and G. C. Hsiao, “On the integral equation method for the plane mixed boundary value problem of the Laplacian”, Math. Methods Appl. Sci. 1:3 (1979), 265-321. Mathematical Reviews (MathSciNet): MR548943
Digital Object Identifier: doi:10.1002/mma.1670010302
· Zbl 0461.65082
[50] P. · Zbl 0969.35042
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