On the limiting amplitude principle for the wave equation with variable coefficients

References

• Bristeau, M.-O., Glowinski, R., Périaux, J. (1998). Controllability methods for the computation of time-periodic solutions; application to scattering. J. Comput. Phys. 147(2):265–292. DOI: 10.1006/jcph.1998.6044.
   Web of Science ®Google Scholar
• Glowinski, R., Rossi, T. (2006). A mixed formulation and exact controllability approach for the computation of the periodic solutions of the scalar wave equation. I. Controllability problem formulation and related iterative solution. Comptes Rendus Mathématique. Académie des Sciences. Paris 343(7):493–498. DOI: 10.1016/j.crma.2006.08.002.
   Google Scholar
• Heikkola, E., Mönkölä, S., Pennanen, A., Rossi, T. (2007). Controllability method for the Helmholtz equation with higher-order discretizations. J. Comput. Phys. 225(2):1553–1576. DOI: 10.1016/j.jcp.2007.02.003.
   Web of Science ®Google Scholar
• Grote, M. J., Tang, J. H. (2019). On controllability methods for the Helmholtz equation. J. Comput. Appl. Math. 358:306–326. DOI: 10.1016/j.cam.2019.03.016.
   Web of Science ®Google Scholar
• Grote, M. J., Nataf, F., Tang, J. H., Tournier, P.-H. (2020). Parallel controllability methods for the Helmholtz equation. Comput. Methods Appl. Mech. Eng. 362:112846. DOI: 10.1016/j.cma.2020.112846.
   Web of Science ®Google Scholar
• Appelö, D., Garcia, F., Runborg, O. (2020). Waveholtz: Iterative solution of the Helmholtz equation via the wave equation. SIAM J. Sci. Comput. 42(4):A1950–A1983. DOI: 10.1137/19M1299062.
   Web of Science ®Google Scholar
• Stolk, C. C. (2021). A time-domain preconditioner for the Helmholtz equation. SIAM J. Sci. Comput. 43(6):A3469–A3502. DOI: 10.1137/20M1359997.
   Google Scholar
• Arnold, A., Geevers, S., Perugia, I., Ponomarev, D. (2022). An adaptive finite element method for high-frequency scattering problems with smoothly varying coefficients. Comput. Math. Appl. 109:1–14. DOI: 10.1016/j.camwa.2022.01.010.
   Web of Science ®Google Scholar
• Tikhonov, A., Samarsky, A. (1948). On the radiation principle. ZHETF 18(2):97–108.
   Google Scholar
• Tikhonov, A., Samarsky, A. (1963). Equations of Mathematical Physics. New York: Pergamon Press.
   Google Scholar
• Ladyzhenskaya, O. (1957). On the principle of limit amplitude. Uspekhi Mat. Nauk 12(3):161–164.
   Google Scholar
• Odeh, F. (1961). Principles of limiting absorption and limiting amplitude in scattering theory. I. Schrödinger’s equation. J. Math. Phys. 2(6):794–800. DOI: 10.1063/1.1724225.
   Web of Science ®Google Scholar
• Odeh, F. (1961). Principles of limiting absorption and limiting amplitude in scattering theory. II. The Wave equation in an inhomogeneous medium. J. Math. Phys. 2(6):800–802. DOI: 10.1063/1.1724226.
   Web of Science ®Google Scholar
• Ramm, A. (1968). On the limiting amplitude principle. Diff. Uravn. 4(4):714–720.
   Google Scholar
• Eidus, D. (1969). The principle of limit amplitude. Russian Math. Surveys 24:97–167. DOI: 10.1070/RM1969v024n03ABEH001348.
   Google Scholar
• Vainberg, B. (1966). Principles of radiation, limit absorption and limit amplitude in the general theory of partial differential equations. Russian Math. Surv. 21(3):115–193. DOI: 10.1070/RM1966v021n03ABEH004157.
   Google Scholar
• Iwasaki,N. (1968). On the principle of limiting amplitude. Publ. RIMS Kyoto Univ. Ser. A 3:373–392. DOI: 10.2977/prims/1195195457.
   Google Scholar
• Ramm, A. (1978). Necessary and sufficient conditions of validity of the limiting amplitude principle. Izv. Vysch. Uch. Zaved. 5(1):96–102.
   Google Scholar
• Tamura, H. (1989). Resolvent estimates at low frequencies and limiting amplitude principle for acoustic propagators. J. Math. Soc. Japan 41(4):549–575. DOI: 10.2969/jmsj/04140549.
   Web of Science ®Google Scholar
• Morawetz, C. (1962). The limiting amplitude principle. Commun. Pure Appl. Math. 15(3):349–361. DOI: 10.1002/cpa.3160150303.
   Web of Science ®Google Scholar
• Cassier, M., Hazard, C., Joly, P. (2017). Spectral theory for Maxwell’s equations at the interface of a metamaterial. Part I: generalized Fourier transform. Commun. Partial Differ. Equ. 42(11):1707–1748. DOI: 10.1080/03605302.2017.1390675.
   Web of Science ®Google Scholar
• Cassier, M., Hazard, C., Joly, P. (2022). Spectral theory for Maxwell’s equations at the interface of a metamaterial. Part II: Limiting absorption, limiting amplitude principles and interface resonance. Commun. Partial Differ. Equ. 47(6):1217–1295. DOI: 10.1080/03605302.2022.2051188.
   Web of Science ®Google Scholar
• Carvalho, C., Ciarlet Jr., P., and Scheid, C. (2022). Limiting amplitude principle and resonances in plasmonic structures with corners: numerical investigation. Comput. Methods Appl. Mech. Eng. 388:114207. DOI: 10.1016/j.cma.2021.114207.
   Web of Science ®Google Scholar
• Beals, M. (1996). Global time decay of the amplitude of a reflected wave. In: Partial Differential Equations and Mathematical Physics, Progress in Nonlinear Differential Equations and Their Applications, Vol. 21. Boston: Springer, pp. 25–44.
   Google Scholar
• Arnold, A., Geevers, S., Perugia, I., Ponomarev, D. (2022). On the exponential time-decay for the one-dimensional wave equation with variable coefficients. Commun. Pure Appl. Anal. 21(10):3389–3405. DOI: 10.3934/cpaa.2022105.
   Web of Science ®Google Scholar
• Engquist, B., Zhao, H. (2018). Approximate separability of the Green’s function of the Helmholtz equation in the high frequency limit. Commun. Pure Appl. Math. 71(11):2220–2274. DOI: 10.1002/cpa.21755.
   Web of Science ®Google Scholar
• Bouclet, J.-M., Burq, N. (2021). Sharp resolvent and time-decay estimates for dispersive equations on asymptotically Euclidean backgrounds. Duke Math. J. 170(11):2575–2629. DOI: 10.1215/00127094-2020-0080.
   Web of Science ®Google Scholar
• Brezis, H., Mironescu, P. (2018). Gagliardo-Nirenberg inequalities and non-inequalities: the full story. Annales de l’Institut Henri Poincaré C, Analyse non linéaire 35(5):1355–1376. DOI: 10.1016/j.anihpc.2017.11.007.
   Web of Science ®Google Scholar
• Chazarain, J., Piriou, A. (2011). Introduction to the Theory of Linear Partial Differential Equations. Amsterdam: Elsevier.
   Google Scholar
• Eidus, D. (1965). On the principle of limiting absorption [translation from Russian]. Amer. Math. Soc. Transl. 47(2):157–191.
   Google Scholar
• Engquist, B., Majda, A. (1977). Absorbing boundary conditions for numerical simulation of waves. Proc. Natl. Acad. Sci. 74(5):1765–1766. DOI: 10.1073/pnas.74.5.1765.
   PubMed Web of Science ®Google Scholar
• Evans, L. (2010). Partial Differential Equations, 2nd ed. Providence, RI: AMS.
   Google Scholar
• Gilbarg, D., Trudinger, N. S. (1977). Elliptic Partial Differential Equations of Second Order. Berlin: Springer.
   Google Scholar
• Graham, I. G., Pembery, O. R., Spence, E. A. (2019). The Helmholtz equation in heterogeneous media: a priori bounds, well-posedness, and resonances. J. Differ. Equ. 266(6):2869–2923. DOI: 10.1016/j.jde.2018.08.048.
   Web of Science ®Google Scholar
• Grote, M. J., Keller, J. B. (1995). Exact nonreflecting boundary conditions for the time dependent wave equation. SIAM J. Appl. Math. 55(2):280–297. DOI: 10.1137/S0036139993269266.
   Web of Science ®Google Scholar
• Olver, F. W. J., Lozier, D. W., Boisvert, R. F., Clark, C. W. (2010). NIST Handbook of Mathematical Functions. Cambridge: Cambridge University Press.
   Google Scholar
• Shapiro, J. (2018). Local energy decay for Lipschitz wavespeeds. Commun. Partial Differ. Equ. 43(5):839–858. DOI: 10.1080/03605302.2018.1475491.
   Web of Science ®Google Scholar