Elastic scattering by unbounded rough surfaces: solvability in weighted Sobolev spaces

References

• Abubakar I. Scattering of plane elastic waves at rough surface I. Proc. Cambridge Philos. Soc. 1962;58:136–157.
   Web of Science ®Google Scholar
• DeSanto JA. Scattering by rough surfaces. In: Pike R, Sabatier P, editors. Scattering and inverse scattering in pure and applied science. New York: Academic Press; 2002. p. 15–36.
   Google Scholar
• Elfouhaily TM, Guerin CA. A critical survey of approximate scattering wave theories from random rough surfaces. Waves Random Media. 2004;14:R1–R40.
   Web of Science ®Google Scholar
• Ogilvy JA. Theory of wave scattering from random rough surfaces. Bristol: Adam Hilger; 1991.
   Google Scholar
• Voronovich AG. Wave scattering from rough surfaces. Berlin: Springer; 1994.
   Google Scholar
• Warnick KF, Chew WC. Numerical simulation methods for rough surface scattering. Waves Random Media. 2001;11:R1–R30.
   Web of Science ®Google Scholar
• Fokkema JT. Reflection and transmission of elastic waves by the spatially periodic interface between two solids (theory of the integral-equation method). Wave Motion. 1980;2:375–393.
   Web of Science ®Google Scholar
• Fokkema JT, van den Berg PM. Elastodynamic diffraction by a periodic rough surface (stress-free boundary). J. Acoust. Soc. Amer. 1977;62:1095–1101.
   Web of Science ®Google Scholar
• Sherwood JWC. Elastic wave propagation in a semi-infinite solid medium. Proc. Phys. Soc. 1958;71:207–219.
   Google Scholar
• Chandler-Wilde SN, Ross C. Scattering by rough surfaces: the Dirichlet problem for the Helmholtz equation in a non-locally perturbed half-plane. Math. Meth. Appl. Sci. 1996;19:959–976.
   Web of Science ®Google Scholar
• Chandler-Wilde SN, Zhang B. A uniqueness result for scattering by infinite rough surfaces. SIAM J. Appl. Math. 1998;58:1774–1790.
   Web of Science ®Google Scholar
• Chandler-Wilde SN, Heinemeyer E, Potthast R. Acoustic scattering by mildly rough unbounded surfaces in three dimensions. SIAM J. Appl. Math. 2006;66:1002–1026.
   Web of Science ®Google Scholar
• Thomas M. Analysis of rough surface scattering problems [PhD thesis]. University of Reading, Reading: UK; 2006.
   Google Scholar
• Zhang B, Chandler-Wilde SN. Acoustic scattering by an inhomogeneous layer on a rigid plate. SIAM J. Appl. Math. 1998;58:1931–1950.
   Web of Science ®Google Scholar
• Chandler-Wilde SN, Zhang B. Scattering of electromagnetic waves by rough interfaces and inhomogeneous layers. SIAM J. Math. Anal. 1999;30:559–583.
   Web of Science ®Google Scholar
• Natroshvili D, Arens T, Chandler-Wilde SN. Uniqueness, existence, and integral equation formulations for interface scattering problems. Memoirs Differ. Equ. Math. Phys. 2003;30:105–146.
   Google Scholar
• Chandler-Wilde SN, Monk P. Existence, uniqueness and variational methods for scattering by unbounded rough surfaces. SIAM J. Math. Anal. 2005;37:598–618.
   Web of Science ®Google Scholar
• Chandler-Wilde SN, Monk P, Thomas M. The mathematics of scattering by unbounded, rough, inhomogeneous layers. J. Comp. Appl. Math. 2007;204:549–559.
   Web of Science ®Google Scholar
• Lechleiter A, Ritterbusch S. A variational method for wave scattering from penetrable rough layers. IMA J. Appl. Math. 2009;26:1–26.
   Google Scholar
• Arens T. Uniqueness for elastic wave scattering by rough surfaces. SIAM J. Math. Anal. 2001;33:461–471.
   Web of Science ®Google Scholar
• Arens T. Existence of solution in elastic wave scattering by unbounded rough surfaces. Math. Meth. Appl. Sci. 2002;25:507–528.
   Web of Science ®Google Scholar
• Elschner J, Hu G. Elastic scattering by unbounded rough surfaces. SIAM J. Math. Anal. 2012;44:4101–4127.
   Web of Science ®Google Scholar
• Duran M, Muga I, Nedelec JC. The outgoing time-harmonic elastic waves in a half-plane with free boundary. SIAM J. Appl. Math. 2011;71:443–464.
   Web of Science ®Google Scholar
• Duran M, Muga I, Nedelec JC. The Helmholtz equation in a locally perturbed half-space with non-absorbing boundary. Arch. Ration. Mech. Anal. 2009;191:143–172.
   Web of Science ®Google Scholar
• Chandler-Wilde SN, Elschner J. Variational approach in weighted Sobolev spaces to scattering by unbounded rough surfaces. SIAM J. Math. Anal. 2010;42:2554–2580.
   Web of Science ®Google Scholar
• Elschner J, Yamamoto M. An inverse problem in periodic diffractive optics: reconstruction of Lipschitz grating profiles. Appl. Anal. 2002;81:1307–1328.
   Google Scholar
• Arens T. The scattering of plane elastic waves by a one-dimensional periodic surface. Math. Meth. Appl. Sci. 1999;22:55–72.
   Web of Science ®Google Scholar
• Elschner J, Hu G. Variational approach to scattering of plane elastic waves by diffraction gratings. Math. Meth. Appl. Sci. 2010;33:1924–1941.
   Web of Science ®Google Scholar
• Elschner J, Hu G. Scattering of plane elastic waves by three-dimensional diffraction gratings. Math. Models Methods Appl. Sci. 2012;22:1150019.
   Web of Science ®Google Scholar
• Parenti C. Operatori pseudo-differenziali in Rn e applicazioni. Ann. Mat. Pura Appl. 1972;93:359–389.
   Google Scholar
• Volevich LR, Mehtiev AG. Multipliers of Fourier integrals in weighted classes and imbedding theorems. Izv. Vyssh. Uchebn. Zaved. Mat. 1967;65:27–38. Russian.
   Google Scholar
• Charalambopoulos A, Gintides D, Kiriaki K. Radiation conditions for rough surfaces in linear elasticity. Q. J. Mech. Appl. Math. 2002;55:421–441.
   Web of Science ®Google Scholar
• Hsiao GC, Wendland WL. Boundary integral equations. Berlin: Springer; 2008.
   Google Scholar
• Nitsche JA. On Korn’s second inequality. RAIRO Anal. Numer. 1981;15:237–248.
   Google Scholar
• Dalibard A-L, Gérard-Varet D. Effective boundary condition at a rough surface starting from a slip condition. J. Differ. Equ. 2011;251: 3450–3487.
   Web of Science ®Google Scholar
• Kress R. Inverse elastic scattering from a crack. Inverse Prob. 1996;12:667–684.
   Web of Science ®Google Scholar
• Eskin GI. Boundary value problems for elliptic pseudodifferential equations. Vol. 52, Translations of Mathematical Monographs. Providence (RI): American Mathematical Society; 1981.
   Google Scholar
• Chandler-Wilde SN, Heinemeyer E, Potthast R. A well-posed integral equation formulation for three dimensional rough surface scattering. Proc. R. Soc. A. 2006;462:3683–3705.
   Google Scholar