The low frequency limit of the exterior Dirichlet problem for the reduced wave equation

References

• MacCamy , R.C. 1965 . Low frequency acoustic oscillations . Quart. Appl. Math , 23 : 247 – 256 .
   Google Scholar
• Eidus , D.M. 1969 . The principle of limit amplitude . Russ. Math. Surv , 24 : 97 – 167 .
   Google Scholar
• Giesecke , B. 1964 . Zum Dirichletschen Prinzip fur selbstadjungierte elliptische DifFerentialope-ratoren . Math.Z. , 86 : 54 – 62 .
   Google Scholar
• Gilbarg , D. and Trudinger , N.S. 1977 . Elliptic partial differential equations of second order , Berlin : Springer-Verlag .
   Google Scholar
• Jager , W. 1967 . Zur Theorie der Schwingungsgleichung mit variablen Koeffizienten in Aufiengebieten . Math.Z , 102 : 62 – 88 .
   Google Scholar
• Kress , R. 1979 . On the limiting behavior of solutions to the boundary integral equations associated with time harmonic wave equations for small frequencies . Math. Meth. in the Appl. Sci , 1 : 89 – 100 .
   Google Scholar
• Kress , R. 1987 . On the Low Wave Number Asymptotics for the Two-dimensional Exterior Di-richlet Problem for the Reduced Wave Equation . Math. Meth. in the Appl. Sci , 9 : 335 – 341 .
   Web of Science ®Google Scholar
• Lax , P. and Phillips , R. 1967 . Scattering theory Academic Press New York
   Google Scholar
• Leis , R. 1965 . Zur Dirichletschen Aufienraumaufgabe der Helmholtzschen Schwingungsgleichung . Math.Z , 90 : 205 – 211 .
   Google Scholar
• Leis , R. 1986 . Initial boundary value problems in mathematical physics , Stuttgart : Teubner &Wiley .
   Google Scholar
• Littman , W. , Stampacchia , G. and Weinberger , H.F. 1963 . Regular points for elliptic equations with discontinuous coefficients . Ann. Sc. Norm. Sup. Pisa , XVII : 43 – 77 .
   Google Scholar
• Magnus , W. , Oberhettinger , F. and Soni , R.P. 1966 . Formulas and Theorems for the Special Functions of Mathematical Physics , Berlin : Springer-Verlag .
   Google Scholar
• Meyers , N.G. and Serrin , J. 1960 . The exterior Dirichlet problem for second order elliptic partial differential equations . J. of Math, and Mech , 9 : 513 – 538 .
   Web of Science ®Google Scholar
• Morgenrother , N. and Werner , P. 1989 . On the Principles of Limiting Absorption and Limit Amplitude for a Class of Locally Perturbed Waveguides. Part 2: Time-dependent Theory . Math. Meth. in the Appl. Sci , 11 : 1 – 25 .
   Web of Science ®Google Scholar
• Ramm , A.G. 1986 . Scattering by obstacles Reidel , Dordecht .
   Google Scholar
• Ramm , A.G. 1986 . Behavior of the solutions to exterior boundary value problems at low fre-quencies . J. Math. Anal. Appl , 117 : 561 – 569 .
   Google Scholar
• Ramm , A.G. , Weaver , O.L. , Week , N. and Witsch , K.J. 1986 . Dissipative Maxwell's Equations at Low Frequencies . Math. Meth. in the . Appl. sci , 117
   Google Scholar
• Shibata , Y. 1983 . On the global existence theorem of classical solutions of second order fully nonlinear hyperbolic equations with first order dissipation in the exterior domain . Tsukuba J. Math , 7 : 1 – 68 .
   Google Scholar
• H. Chr , Stuwe and Werner , P. 1989 . Remarks on the low frequency asymptotics for the reduced wave equation and the Schrodinger equation in low-dimensional spaces . Asymptotic Analysis , 2 : 179 – 202 .
   Google Scholar
• Tsutsumi , Y. 1984 . Local energy decay of solutions to the free Schrodinger equation in exterior domains . J. Fac. Sci. Univ. Tokyo, sect. IA, Math , 31 : 97 – 108 .
   Google Scholar
• Vainberg , B.R. 1968 . On the analytic properties of the resolvent for a certain class of operator pencils . Math. USSR Sb , 6 : 241 – 272 .
   Google Scholar
• Vainberg , B.R. 1973 . On exterior elliptic problems polynomially depending on a spectral parameter, and the asymptotic behavior for large time of solutions of nonstationary problems . Math. USSR Sb. , 21 : 221 – 239 .
   Google Scholar
• Week , N. 1969 . Klassische Losungen sind auch schwache Losungen . Archiv der Mathematik , 20 : 628 – 637 .
   Google Scholar
• Werner , P. 1962 . Randwertprobleme der mathematischen Akustik . Arch. Rat. Mech. Anal , 10 : 29 – 62 .
   Web of Science ®Google Scholar
• Werner , P. 1985 . Zur Asymptotik der Wellengleichung und der Warmeleitungsgleichung in zwei-dimensionalen Auftenraumen . Math. Meth. in the Appl. Sci , 7 : 170 – 201 .
   Google Scholar
• Werner , P. 1986 . Low frequency asymptotics for the reduced wave equation in two-dimensional exterior spaces . Math. Meth. in the Appl. Sci , 8 : 134 – 156 .
   Google Scholar
• Wilcox , C.H. 1975 . Scattering theory for the d’Alembert equation in exterior domains Lecture Notes in Mathematics , Berlin : Springer-Verlag .
   Google Scholar