×

Finite-dimensional approximations to the Poincaré-Steklov operator for general elliptic boundary value problems in domains with cylindrical and periodic exits to infinity. (English. Russian original) Zbl 1445.35157

Trans. Mosc. Math. Soc. 2019, 1-51 (2019); translation from Tr. Mosk. Mat. O.-va 80, No. 1, 1-62 (2019).
The paper is focused on the boundary value problem \[\left\{\begin{array}{l} \mathcal{L}(x,\nabla_x)u(x)-\lambda \mathcal{R}(x)u(x)=f(x),\,\,\,x\in\Xi,\\ \mathcal{B}(x,\nabla_x)u(x)=0,\,\,\,x\in\partial \Xi, \end{array}\right.\tag{1}\] where the waveguide \(\Xi\) is a domain with smooth \((d-1)\)-dimensional boundary \(\partial \Xi\), given by \(\Xi=\Theta\cup \Pi^+\), with the resonator \(\Theta\) – an open set in the half-space \(\mathbb{R}_-^d=\{x:\,\,z<0\}\), and the periodic “sleeve” \(\Pi^+=\{x\in \Pi:\,\,z\ge 0\}\). Here \(\Pi\) is a domain of the Euclidian space \(\mathbb{R}^d\), \(d\ge 2\), given by \(\Pi=\{x=(y,z):\,\,(y,z\pm 1)\in \Pi\}\), (\(z=x_d\)), with a bounded periodicity cell \(\overline\omega=\{x=(x_1,\ldots,x_{d-1},x_d)\in \Pi:\,\,|z|=|x_d|<1/2\}\). In addition, \(\Xi\) has a periodic exit to infinity, \(\lambda\) is a nonnegative spectral parameter, \(f=(f_1,\ldots,f_n)^T\) is a given vector function, and \(u=(u_1,\ldots,u_n)^T\) is the unknown vector function. \(\mathcal{L}\) is a self-adjoint second-order differential operator in the domain \(\Xi\) defined by \(\mathcal{L}(x,\nabla_x)=\overline{D(-\nabla_x)}^T\mathcal{A}(x)D(\nabla_x)\), with \(\mathcal{A}\) a \(N\times N\) Hermitian and positive definite matrix function on \(\overline{\Xi}\), the matrix operator \(\mathcal{B}\) is defined by \(\mathcal{B}u=\mathcal{P}\mathcal{N}u-(\mathbb{I}_n-\mathcal{P})u\), with the operator of Neumann boundary conditions on \(\partial \Xi\) given by \(\mathcal{N}(x,\nabla_x)=\overline{D(\nu(x))}^T\mathcal{A}(x)D(\nabla_x)\), where \(\nabla_x=\text{grad}\), and \(\nu=(\nu_1,\ldots,\nu_d)^T\) is the unit exterior normal vector on \(\partial \Xi\). \(\mathcal{R}\) is a Hermitian and positive definite smooth \(n\times n\) matrix function, and \(\mathcal{P}\) is a smooth \(n\times n\) matrix function which is an orthogonal projection in the space \(\mathbb{C}^n\). The author studies the Steklov-Poincaré operator in domains with periodic exits to infinity, which arises when problem \((1)\) in an infinite waveguide is restricted to a bounded subdomain (for example, the resonator \(\Theta\)), by introducing some artificial boundary conditions. Also finite-dimensional approximation of the Steklov-Poincaré operator is discussed, and asymptotically sharp error estimates for the solutions of the problem and for the eigenvalues in the discrete spectrum are obtained. The case of waveguides with several exits to infinity, and other available generalizations are finally presented.

MSC:

35J57 Boundary value problems for second-order elliptic systems
47F05 General theory of partial differential operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Nec J. Necas, Les m\'ethodes directes en th\'eorie des \'equations elliptiques, Masson-Academia, Paris-Prague, 1967. · Zbl 1225.35003
[2] na217 S. A. Nazarov, Self-adjoint elliptic boundary-value problems. The polynomial property and formally positive operators, Probl. Mat. Anal. 16 (1997), 167-192; English transl., J. Math. Sci. New York 92 (1998), no. 6, 4338-4353. · Zbl 0945.35029
[3] na262 S. A. Nazarov, The polynomial property of self-adjoint elliptic boundary-value problems and an algebraic description of their attributes, Usp. Mat. Nauk 54 (1999), no. 5, 77-142; English transl., Russ. Math. Surv. 54 (1999), no. 5, 947-1014. · Zbl 0970.35026
[4] ADN2 S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions: II, Comm. Pure Appl. Math. 17 (1964), 35-92. · Zbl 0123.28706
[5] ExKo P. Exner and H. Kovar\^k, Quantum waveguides, Springer, Cham, 2015. · Zbl 1314.81001
[6] Mitra R. Mittra and S. W. Lee, Analytical techniques in the theory of guided waves, Macmillan, New York, 1971. · Zbl 0227.35002
[7] Lekh S. G. Lekhnitskii, Theory of elasticity of an anisotropic body, Nauka, Moscow, 1977; English transl., Mir, Moscow, 1981. · Zbl 0467.73012
[8] PaKu V. Z. Parton and B. A. Kudryavtsev, Electromagnetoelasticity: Piezoelectrics and Electrically Conductive Solids, Nauka, Moscow, 1988; English transl., Gordon and Breach, New York, 1988.
[9] na317 S. A. Nazarov, Uniform estimates of remainders in asymptotic expansions of solutions to the problem on eigenoscillations of a piezoelectric plate, Probl. Mat. Anal. 25 (2003), 99-188; English transl., J. Math. Sci. New York 114 (2003), no. 5, 1657-1725. · Zbl 1054.35108
[10] Mikh S. G. Mikhlin, Variational Methods in Mathematical Physics, Nauka, Moscow, 1970; English transl., Pergamon Press, Oxford, 1964. · Zbl 0119.19002
[11] NaPl S. A. Nazarov and B. A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, De Gruyter, Berlin, 1994. · Zbl 0806.35001
[12] MaNaPl V. Maz’ya, S. Nazarov, and B. Plamenevskij, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, vols. 1, 2, Akademie-Verlag, Berlin, 1991; English transl., Birkh\"auser, Basel, 2000.
[13] na28 S. A. Nazarov, On the asymptotics with respect to a parameter of the solution of a boundary value problem with periodic coefficients in a cylinder, Differ. Uravn. Primen. 30 (1981), 27-45. (Russian) \pagebreak · Zbl 0522.35009
[14] na269 S. A. Nazarov, The Navier-Stokes problem in thin or long tubes with periodically varying cross-section, ZAMM 80 (2000), no. 9, 591-612. · Zbl 0991.76013
[15] NaScal S. A. Nazarov, Finite-dimensional approximations of the Steklov-Poincar\'e operator for the Helmholtz equation in periodic waveguides, Probl. Mat. Anal. 93 (2018), 53-87; English transl., J. Math. Sci. New York 232 (2018), no. 4, 461-502. · Zbl 1401.35039
[16] BBD1 V. Baronian, A.-S. Bonnet-Ben Dhia, and E. Lun\'eville, Transparent boundary conditions for the harmonic diffraction problem in an elastic waveguide, J. Comput. Appl. Math. 234 (2010), no. 6, 1945-1952. · Zbl 1405.35121
[17] BBD2 V. Baronian, A.-S. Bonnet-Ben Dhia, S. Fliss, and A. Tonnoir, Iterative methods for scattering problems in isotropic or anisotropic elastic waveguides, Wave Motion 64 (2016), 13-33. · Zbl 1469.74069
[18] StPo V. I. Lebedev and V. I. Agoshkov, Poincar\'e-Steklov operators and their applications in analysis, Otd. Vychisl. Matem. Akad. Nauk SSSR, Moscow, 1983. (Russian) · Zbl 0547.47029
[19] na365 S. A. Nazarov, A criterion for the existence of decaying solutions in the problem on a resonator with a cylindrical waveguide, Funkts. Anal. Prilozh. 40 (2006), no. 2, 20-32; English transl., Funct. Anal. Appl. 40 (2006), no. 2, 97-107. · Zbl 1162.35342
[20] Wilcox C. H .Wilcox, Scattering theory of diffraction gratings, Springer, New York, 1984. · Zbl 0541.76001
[21] BBDsome M. Lenoir and A. Tounsi, The localized finite element method and its application to the two-dimensional sea-keeping problem, SIAM J. Numer. Anal. 25 (1988), no. 4, 729-752. · Zbl 0656.76008
[22] Gel I. M. Gelfand, Expansion in eigenfunctions of equations with periodic coefficients, Dokl. Akad. Nauk SSSR 73 (1950), no. 6, 1117-1120. · Zbl 0037.34505
[23] na17 S. A. Nazarov, Elliptic boundary value problems with periodic coefficients in a cylinder, Izv. Akad. Nauk SSSR, Ser. Mat. 45 (1981), no. 1, 101-112; English transl., Math. USSR Izv. 18, 89-98 (1982). · Zbl 0483.35032
[24] KuchUMN P. A. Kuchment, Floquet theory for partial differential equations, Usp. Mat. Nauk 37 (1982), no. 4(226), 3-52; English transl., Russ. Math. Surv. 37 (1982), no. 4, 1-60. · Zbl 0519.35003
[25] Skrig M. M. Skriganov, Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators, Trudy Mat. Inst. Steklov. 171 (1985), 3-122; English transl., Proc. Steklov Inst. Math. 171 (1987), 1-121. · Zbl 0615.47004
[26] Kuchbook P. Kuchment, Floquet theory for partial differential equations, Birch\"auser, Basel, 1993. · Zbl 0789.35002
[27] GoKr I. C. Gohberg and M. G. Krein, Introduction to the theory of linear nonselfadjoint operators in Hilbert space, Nauka, Moscow, 1965; English transl., Transl. Math. Monographs, vol. 18, Amer. Math. Soc., Providence, RI, 1969. · Zbl 0138.07803
[28] T1 A. V. Sobolev and J. Walthoe, Absolute continuity in periodic waveguides, Proc. London Math. Soc. (3) 85 (2002), no. 3, 717-741. · Zbl 1247.35077
[29] T2 T. A. Suslina and R. G. Shterenberg, Absolute continuity of the spectrum of the Schr\"odinger operator with the potential concentrated on a periodic system of hypersurfaces, Algebra Anal. 13 (2001), no. 5, 197-240; English transl., St. Petersbg. Math. J. 13 (2002), no. 5, 859-891. · Zbl 1068.35122
[30] T3 I. Kachkovskii and N. Filonov, Absolute continuity of the Schr\"odinger operator spectrum in a multidimensional cylinder, Algebra Anal. 21 (2009), no. 1, 133-152; English transl., St. Petersbg. Math. J. 21 (2010), no. 1, 95-109. · Zbl 1194.35284
[31] T4 K. Miller, Nonunique continuation for uniformly parabolic and elliptic equations in self-adjoint divergence form with H\"older continuous coefficients, Arch. Rat. Mech. Anal. 54 (1974), no. 2, 105-117. · Zbl 0289.35046
[32] T5 N. Filonov, Second-order elliptic equation of divergence form having a compactly supported solution,Probl. Mat. Anal. 22 (2001), 246-257; English transl., J. Math. Sci. New York 106 (2001), no. 3, 3078-3086. · Zbl 0991.35020
[33] T6 M. N. Demchenko, Nonunique continuation for the Maxwell system, Zap. Nauchn. Sem. POMI 393 (2011), 80-100; English transl., J. Math. Sci. (N. Y.) 185 (2012), no. 4, 554-566. · Zbl 1259.78008
[34] AgVi M. S. Agranovich and M. I. Vishik, Elliptic problems with a parameter and parabolic problems of general type, Usp. Mat. Nauk 19 (1964), no. 3(117), 53-161; English transl., Russ. Math. Surv. 19 (1964), no. 3, 53-157. · Zbl 0137.29602
[35] na128 S. A. Nazarov and B. A. Plamenevskij, On radiation conditions for selfadjoint elliptic problems, Dokl. Akad. Nauk SSSR 311 (1990), no. 3, 532-536; English transl., Sov. Math. Dokl. 41 (1990), no. 2, 274-277. · Zbl 0725.35072
[36] na147 S. A. Nazarov and B. A. Plamenevskij, Radiation principles for self-adjoint elliptic problems, Prob. Matem. Fiz. 13 (1991), 192-244. (Russian)
[37] na569 S. A. Nazarov, Umov-Mandelshtam radiation conditions in elastic periodic waveguides, Mat. Sb. 205 (2014), no. 7, 43-72; English transl., Sb. Math. 205 (2014), no. 7, 953-982. · Zbl 1304.35677
[38] na642 S. A. Nazarov and J. Taskinen, Radiation conditions for the linear water-wave problem in periodic channels, Math. Nachr. 290 (2017), no. 11-12, 1753-1778. · Zbl 1379.35068
[39] Umov N. A. Umov, Equations of motion of energy in bodies, Ulrich & Schulze Printing House, Odessa, 1874. (Russian)
[40] Poynt J. H. Poynting, On the transfer of energy in the electromagnetic field, Phil. Trans. Roy. Soc. London 175 (1884), 343-361. · JFM 17.1028.01
[41] Mand L. I. Mandelshtam, Lectures on optics, relativity theory, and quantum mechanics, vol. 2, Izd. Akad. Nauk SSSR, Moscow, 1947. (Russian)
[42] VoBa I. I. Vorovich and V. A. Babeshko, Dynamic mixed problems of elasticity theory for nonclassical domains, Nauka, Moscow, 1979. (Russian) · Zbl 0515.73027
[43] MaPl1 V. G. Maz’ya and B. A. Plamenevskij, On the coefficients in the asymptotics of solutions of elliptic boundary value problems in domains with conical points, Math. Nachr. 76 (1977), 29-60. (Russian) · Zbl 0359.35024
[44] na68 S. A. Nazarov, On the constants in the asymptotics of solutions of elliptic boundary-value problems with periodic coefficients in a cylinder, Vestn. Leningr. Univ. (1985), no. 15, Mat. Mekh. Astron., no. 3, 16-22; English transl., Vestn. Leningr. Univ. Math. 18 (1985), no. 3, 16-22. · Zbl 0605.35006
[45] LiMa J.-L. Lions and E. Magenes, Probl\`emes aux limites non homog\`enes et applications, Dunod, Paris, 1970. · Zbl 0197.06701
[46] na489 S. A. Nazarov, Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide, Teor. Mat. Fiz. 167(2011), no. 2, 239-263; English transl., Theor. Math. Phys. 167 (2011), no. 2, 606-627. · Zbl 1274.81109
[47] VasTimo I. Roitberg, D. Vassiliev, and T. Weidl, Edge resonance in an elastic semi-strip, Quart. J. Mech. Appl. Math. 51 (1998), no. 1, 1-13. · Zbl 0926.74033
[48] VasEl A. Holst and D. Vassiliev, Edge resonance in an elastic semi-infinite cylinder, Appl. Anal. 74 (2000), no. 3-4, 479-495. · Zbl 1162.74374
[49] na554 S. A. Nazarov, Elastic waves trapped by a homogeneous anisotropic semicylinder, Mat. Sb. 204 (2013), no. 11, 99-130; English transl., Sb. Math. 204 (2013), no. 11, 1639-1670. · Zbl 1293.35325
[50] na257 I. V. Kamotskii and S. A. Nazarov, Elastic waves localized near periodic families of defects, Dokl. Ross. Akad. Nauk 368 (1999), no. 6, 771-773; English transl., Dokl. Phys. 44 (1999), no. 10, 715-717. · Zbl 1060.74559
[51] na485 S. A. Nazarov, Localized elastic fields in periodic waveguides with defects, Prikl. Mekh. Tekh. Fiz. 52 (2011), no. 2, 183-194; English transl., J. Appl. Mech. Tech. Phys. 52 (2011), no. 2, 311-320. · Zbl 1272.74342
[52] Rel F. Rellich, \"Uber das asymptotische Verhalten der L\"osungen von \(\Delta u+\lambda u=0\) in unendlichen Gebieten, Jahresber. Deutsch. Math. Verein. 53 (1943), 57-65. · Zbl 0028.16401
[53] Ko V. A. Kondratiev, Boundary problems for elliptic equations in domains with conical or angular points, Tr. Mosk. Mat. Ob. 16 (1967), 279-292; English transl., Trans. Mosc. Math. Soc. 16 (1967), 227-313. · Zbl 0194.13405
[54] Ko1 V. A. Kondratiev, The smoothness of the solution of the Dirichlet problem for second order elliptic equations in a piecewise smooth domain, Differ. Uravn. 6 (1970), no. 10, 1831-1843. (Russian) · Zbl 0209.41104
[55] Ko2 V. A. Kondratiev, Singularities of a solution of Dirichlet’s problem for a second-order elliptic equation in the neighborhood of an edge, Differ. Uravn. 13 (1977), no. 11, 2026-2032; English transl., Differ. Equations 13 (1977), no. 11, 1411-1415. · Zbl 0394.35027
[56] na93 S. A. Nazarov, Estimates near the edge of the solution of the Neumann problem for an elliptic system, Vestn. Leningr. Univ. Ser. I (1988), no. 1, 37-42; English transl., Vestn. Leningr. Univ. Math. 21 (1988), no. 1, 52-59. · Zbl 0684.35021
[57] na129 S. A. Nazarov and B. A. Plamenevskij, Asymptotics of the spectrum of the Neumann problem in singularly perturbed thin domains: I, Algebra Anal. 2 (1990), no. 2, 85-111; English transl., Leningr. Math. J. 2 (1991), no. 2, 287-311. · Zbl 0751.35027
[58] na600 S. A. Nazarov and J. Taskinen, Spectral gaps for periodic piezoelectric waveguides, ZAMP 66 (2015), 3017-3047. · Zbl 1333.35275
[59] ViLu M. I. Vishik and L. A. Lyusternik, Regular degeneration and boundary layer for linear differential equations with small parameter, Usp. Mat. Nauk 12 (1957), no. 5(77), 3-122. · Zbl 0087.29602
[60] BiSo M. Sh. Birman and M. Z. Solomjak, Spectral theory of self-adjoint operators in Hilbert space, Izd. Leningrad. Univ., Leningrad, 1980; English transl., D. Reidel, Dordrecht, 1987. · Zbl 0744.47017
[61] na538 S. A. Nazarov, Scheme for interpretation of approximately computed eigenvalues embedded in a continuous spectrum, Zh. Vychisl. Mat. Mat. Fiz. 53 (2013), no. 6, 878-897; English transl., Comput. Math. Math. Phys. 53 (2013), no. 6, 702-720. · Zbl 1299.76235
[62] NaRu S. A. Nazarov and K. M. Ruotsalainen, A rigorous interpretation of approximate computations of embedded eigenfrequencies of water waves, Z. Anal. Anwend. 35 (2016), no. 2, 211-242. · Zbl 1338.74056
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.