Finite-dimensional approximations to the Poincaré–Steklov operator for general elliptic boundary value problems in domains with cylindrical and periodic exits to infinity

Nazarov, S.

doi:10.1090/mosc/290

Finite-dimensional approximations to the Poincaré–Steklov operator for general elliptic boundary value problems in domains with cylindrical and periodic exits to infinity
HTML articles powered by AMS MathViewer

by S. A. Nazarov
Translated by: V. E. Nazaikinskii

Trans. Moscow Math. Soc. 2019, 1-51

DOI: https://doi.org/10.1090/mosc/290

Published electronically: April 1, 2020

PDF | Request permission

Abstract:

We study formally self-adjoint boundary value problems for elliptic systems of differential equations in domains with periodic (in particular, cylindrical) exits to infinity. Statements of problems in a truncated (finite) domain which provide approximate solutions of the original problem are presented. The integro-differential conditions on the artificially formed end face are interpreted as a finite-dimensional approximation to the Steklov–Poincaré operator, which is widely used when dealing with the Helmholtz equation in cylindrical waveguides. Asymptotically sharp approximation error estimates are obtained for the solutions of the problem with the compactly supported right-hand side in an infinite domain as well as for the eigenvalues in the discrete spectrum (if any). The construction of a finite-dimensional integro-differential operator is based on natural orthogonality and normalization conditions for oscillating and exponential Floquet waves in a periodic quasicylindrical end.

References

Jindřich Nečas, Les méthodes directes en théorie des équations elliptiques, Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague, 1967 (French). MR 0227584
S. A. Nazarov, Self-adjoint elliptic boundary-value problems. The polynomial property and formally positive operators, J. Math. Sci. (New York) 92 (1998), no. 6, 4338–4353. Some questions of mathematical physics and function theory. MR 1668418, DOI 10.1007/BF02433440
S. A. Nazarov, Polynomial property of selfadjoint elliptic boundary value problems, and the algebraic description of their attributes, Uspekhi Mat. Nauk 54 (1999), no. 5(329), 77–142 (Russian); English transl., Russian Math. Surveys 54 (1999), no. 5, 947–1014. MR 1741662, DOI 10.1070/rm1999v054n05ABEH000204
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. MR 162050, DOI 10.1002/cpa.3160170104
Pavel Exner and Hynek Kovařík, Quantum waveguides, Theoretical and Mathematical Physics, Springer, Cham, 2015. MR 3362506, DOI 10.1007/978-3-319-18576-7
R. Mittra and S. W. Lee, Analytical techniques in the theory of guided waves, Macmillan, New York, 1971.
S. G. Lekhnitskiĭ, Teoriya uprugosti anizotropnogo tela, Izdat. “Nauka”, Moscow, 1977 (Russian). Second edition, revised and augmented. MR 0502604
V. Z. Parton and B. A. Kudryavtsev, Electromagnetoelasticity: Piezoelectrics and Electrically Conductive Solids, Nauka, Moscow, 1988; English transl., Gordon and Breach, New York, 1988.
S. A. Nazarov, Uniform estimates of remainders in asymptotic expansions of solutions to the problem on eigenoscillations of a piezoelectric plate, J. Math. Sci. (N.Y.) 114 (2003), no. 5, 1657–1725. Function theory and applications. MR 1981301, DOI 10.1023/A:1022364812273
S. G. Mikhlin, Variatsionnye metody v matematicheskoĭ fizike, Izdat. “Nauka”, Moscow, 1970 (Russian). Second edition, revised and augmented. MR 0353111
Sergey A. Nazarov and Boris A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, De Gruyter Expositions in Mathematics, vol. 13, Walter de Gruyter & Co., Berlin, 1994. MR 1283387, DOI 10.1515/9783110848915.525
W. G. Mazja, S. A. Nasarow, and B. A. Plamenewski, Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten. II, Mathematische Lehrbücher und Monographien, II. Abteilung: Mathematische Monographien [Mathematical Textbooks and Monographs, Part II: Mathematical Monographs], vol. 83, Akademie-Verlag, Berlin, 1991 (German). Nichtlokale Störungen. [Nonlocal perturbations]. MR 1186211
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, 239 (Russian). MR 607578
S. A. Nazarov, The Navier-Stokes problem in thin or long tubes with periodically varying cross-sections, ZAMM Z. Angew. Math. Mech. 80 (2000), no. 9, 591–612. MR 1778562, DOI 10.1002/1521-4001(200009)80:9<591::AID-ZAMM591>3.0.CO;2-Q
S. A. Nazarov, Finite-dimensional approximations of the Steklov-Poincaré operator for the Helmholtz equation in periodic waveguides, J. Math. Sci. (N.Y.) 232 (2018), no. 4, Problems in mathematical analysis. No. 93 (Russian), 461–502. MR 3830394, DOI 10.1007/s10958-018-3890-1
V. Baronian, A. S. Bonnet-Ben Dhia, and E. Lunéville, Transparent boundary conditions for the harmonic diffraction problem in an elastic waveguide, J. Comput. Appl. Math. 234 (2010), no. 6, 1945–1952. MR 2644191, DOI 10.1016/j.cam.2009.08.045
Vahan Baronian, Anne-Sophie Bonnet-Ben Dhia, Sonia Fliss, and Antoine Tonnoir, Iterative methods for scattering problems in isotropic or anisotropic elastic waveguides, Wave Motion 64 (2016), 13–33. MR 3494271, DOI 10.1016/j.wavemoti.2016.02.005
V. I. Lebedev and V. I. Agoshkov, Operatory Puankare-Steklova i ikh prilozheniya v analize, Akad. Nauk SSSR, Vychisl. Tsentr, Moscow, 1983 (Russian). MR 827980
S. A. Nazarov, A criterion for the existence of decaying solutions in the problem of a resonator with a cylindrical waveguide, Funktsional. Anal. i Prilozhen. 40 (2006), no. 2, 20–32, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 40 (2006), no. 2, 97–107. MR 2256860, DOI 10.1007/s10688-006-0016-1
Calvin Hayden Wilcox, Scattering theory for diffraction gratings, Applied Mathematical Sciences, vol. 46, Springer-Verlag, New York, 1984. MR 725334, DOI 10.1007/978-1-4612-1130-3
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. MR 954784, DOI 10.1137/0725044
I. M. Gel′fand, Expansion in characteristic functions of an equation with periodic coefficients, Doklady Akad. Nauk SSSR (N.S.) 73 (1950), 1117–1120 (Russian). MR 0039154
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, 239 (Russian). MR 607578
P. A. Kuchment, Floquet theory for partial differential equations, Uspekhi Mat. Nauk 37 (1982), no. 4(226), 3–52, 240 (Russian). MR 667973
M. M. Skriganov, Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators, Trudy Mat. Inst. Steklov. 171 (1985), 122 (Russian). MR 798454
Peter Kuchment, Floquet theory for partial differential equations, Operator Theory: Advances and Applications, vol. 60, Birkhäuser Verlag, Basel, 1993. MR 1232660, DOI 10.1007/978-3-0348-8573-7
I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. MR 0246142
Alexander V. Sobolev and Jonathan Walthoe, Absolute continuity in periodic waveguides, Proc. London Math. Soc. (3) 85 (2002), no. 3, 717–741. MR 1936818, DOI 10.1112/S0024611502013631
T. A. Suslina and R. G. Shterenberg, Absolute continuity of the spectrum of the Schrödinger operator with the potential concentrated on a periodic system of hypersurfaces, Algebra i Analiz 13 (2001), no. 5, 197–240 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 13 (2002), no. 5, 859–891. MR 1882869
I. Kachkovskiĭ and N. Filonov, Absolute continuity of the spectrum of a periodic Schrödinger operator in a multidimensional cylinder, Algebra i Analiz 21 (2009), no. 1, 133–152 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 21 (2010), no. 1, 95–109. MR 2553054, DOI 10.1090/S1061-0022-09-01087-5
Keith Miller, Nonunique continuation for uniformly parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients, Arch. Rational Mech. Anal. 54 (1974), 105–117. MR 342822, DOI 10.1007/BF00247634
N. Filonov, Second-order elliptic equation of divergence form having a compactly supported solution, J. Math. Sci. (New York) 106 (2001), no. 3, 3078–3086. Function theory and phase transitions. MR 1906035, DOI 10.1023/A:1011379807662
M. N. Demchenko, On the nonuniqueness of the continuation of the solution of the Maxwell system, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 393 (2011), no. Matematicheskie Voprosy Teorii Rasprostraneniya Voln. 41, 80–100, 261 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 185 (2012), no. 4, 554–566. MR 2870205, DOI 10.1007/s10958-012-0938-5
M. S. Agranovič and M. I. Višik, Elliptic problems with a parameter and parabolic problems of general type, Uspehi Mat. Nauk 19 (1964), no. 3 (117), 53–161 (Russian). MR 0192188
S. A. Nazarov and B. A. Plamenevskiĭ, Radiation conditions for selfadjoint elliptic problems, Dokl. Akad. Nauk SSSR 311 (1990), no. 3, 532–536 (Russian); English transl., Soviet Math. Dokl. 41 (1990), no. 2, 274–277 (1991). MR 1075678
S. A. Nazarov and B. A. Plamenevskiĭ, Radiation principles for selfadjoint elliptic problems, Differential equations. Spectral theory. Wave propagation (Russian), Probl. Mat. Fiz., vol. 13, Leningrad. Univ., Leningrad, 1991, pp. 192–244, 308 (Russian, with Russian summary). MR 1341639
S. A. Nazarov, Umov-Mandel′shtam radiation conditions in elastic periodic waveguides, Mat. Sb. 205 (2014), no. 7, 43–72 (Russian, with Russian summary); English transl., Sb. Math. 205 (2014), no. 7-8, 953–982. MR 3242645, DOI 10.1070/sm2014v205n07abeh004405
Sergey A. Nazarov and Jari Taskinen, Radiation conditions for the linear water-wave problem in periodic channels, Math. Nachr. 290 (2017), no. 11-12, 1753–1778. MR 3683459, DOI 10.1002/mana.201600313
N. A. Umov, Equations of motion of energy in bodies, Ulrich & Schulze Printing House, Odessa, 1874. (Russian)
J. H. Poynting, On the transfer of energy in the electromagnetic field, Phil. Trans. Roy. Soc. London 175 (1884), 343–361.
L. I. Mandel′shtam, Lektsii po optike, teorii otnositel′nosti i kvantovoĭ mekhanike, Izdat. “Nauka”, Moscow, 1972 (Russian). Edited and with a preface by S. M. Rytov. MR 0392467
I. I. Vorovič and V. A. Babeško, Dinamicheskie smeshannye zadachi teorii uprugosti dlya neklassicheskikh oblasteĭ, “Nauka”, Moscow, 1979 (Russian). MR 566583
V. G. Maz′ja and B. A. Plamenevskiĭ, The coefficients in the asymptotics of solutions of elliptic boundary value problems with conical points, Math. Nachr. 76 (1977), 29–60 (Russian). MR 601608, DOI 10.1002/mana.19770760103
S. A. Nazarov, The constants in the asymptotic expansion of solutions of elliptic boundary value problems with periodic coefficients in a cylinder, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. vyp. 3 (1985), 16–22, 131 (Russian, with English summary). MR 813870
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243
S. A. Nazarov, Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide, Theoret. and Math. Phys. 167 (2011), no. 2, 606–627. Translation of Teoret. Mat. Fiz. 167 (2011), no. 2, 239–263. MR 3166368, DOI 10.1007/s11232-011-0046-6
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. MR 1610688, DOI 10.1093/qjmam/51.1.1
Anders Holst and Dmitri Vassiliev, Edge resonance in an elastic semi-infinite cylinder, Appl. Anal. 74 (2000), no. 3-4, 479–495. MR 1757552, DOI 10.1080/00036810008840829
S. A. Nazarov, Elastic waves trapped by a homogeneous anisotropic semicylinder, Mat. Sb. 204 (2013), no. 11, 99–130 (Russian, with Russian summary); English transl., Sb. Math. 204 (2013), no. 11-12, 1639–1670. MR 3155865, DOI 10.1070/sm2013v204n11abeh004353
I. V. Kamotskiĭ and S. A. Nazarov, Elastic waves localized near periodic families of defects, Dokl. Akad. Nauk 368 (1999), no. 6, 771–773 (Russian); English transl., Dokl. Phys. 44 (1999), no. 10, 715–717. MR 1749046
S. A. Nazarov, Localized elastic fields in periodic waveguides with defects, Prikl. Mekh. Tekhn. Fiz. 52 (2011), no. 2, 183–194 (Russian, with Russian summary); English transl., J. Appl. Mech. Tech. Phys. 52 (2011), no. 2, 311–320. MR 2830642, DOI 10.1134/S0021894411020192
Franz Rellich, Über das asymptotische Verhalten der Lösungen von $\Delta u+\lambda u=0$ in unendlichen Gebieten, Jber. Deutsch. Math.-Verein. 53 (1943), 57–65 (German). MR 17816
V. A. Kondrat′ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obšč. 16 (1967), 209–292 (Russian). MR 0226187
V. A. Kondrat′ev, The smoothness of the solution of the Dirichlet problem for second order elliptic equations in a piecewise smooth domain, Differencial′nye Uravnenija 6 (1970), 1831–1843 (Russian). MR 0282052
V. A. Kondrat′ev, Singularities of the solution of the Dirichlet problem for a second order elliptic equation in the neighborhood of an edge, Differencial′nye Uravnenija 13 (1977), no. 11, 2026–2032, 2109 (Russian). MR 0486987
S. A. Nazarov, Estimates near an edge for the solution of the Neumann problem for an elliptic system, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. vyp. 1 (1988), 37–42, 117 (Russian, with English summary); English transl., Vestnik Leningrad Univ. Math. 21 (1988), no. 1, 52–59. MR 946462
S. A. Nazarov and B. A. Plamenevskiĭ, Asymptotics of the spectrum of the Neumann problem in singularly degenerate thin domains. I, Algebra i Analiz 2 (1990), no. 2, 85–111 (Russian); English transl., Leningrad Math. J. 2 (1991), no. 2, 287–311. MR 1062264
Sergei A. Nazarov and Jari Taskinen, Spectral gaps for periodic piezoelectric waveguides, Z. Angew. Math. Phys. 66 (2015), no. 6, 3017–3047. MR 3428452, DOI 10.1007/s00033-015-0561-7
M. I. Višik and L. A. Lyusternik, Regular degeneration and boundary layer for linear differential equations with small parameter, Uspehi Mat. Nauk (N.S.) 12 (1957), no. 5(77), 3–122 (Russian). MR 0096041
M. Š. Birman and M. Z. Solomjak, Spektral′naya teoriya samosopryazhennykh operatorov v gil′bertovom prostranstve, Leningrad. Univ., Leningrad, 1980 (Russian). MR 609148
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 (Russian, with Russian summary); English transl., Comput. Math. Math. Phys. 53 (2013), no. 6, 702–720. MR 3252907, DOI 10.1134/S0965542513060122
Sergei A. Nazarov and Keijo M. Ruotsalainen, A rigorous interpretation of approximate computations of emebedded eigenfrequencies of water waves, Z. Anal. Anwend. 35 (2016), no. 2, 211–242. MR 3489047, DOI 10.4171/ZAA/1563