Stability for the acoustic scattering problem for sound-hard scatterers

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November 2013, 7(4): 1307-1329. doi: 10.3934/ipi.2013.7.1307

Stability for the acoustic scattering problem for sound-hard scatterers

Giorgio Menegatti ^1, and Luca Rondi ^2,

1.	Dipartimento di Matematica e Geoscienze, Università degli Studi di Trieste, via Valerio 12/1, 34127 Trieste, Italy
2.	Dipartimento di Matematica e Informatica, Università degli Studi di Trieste, via Valerio, 12/1, 34127 Trieste

Received January 2013 Revised April 2013 Published November 2013

Abstract
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We study the stability for the direct acoustic scattering problem with sound-hard scatterers with minimal regularity assumptions on the scatterers. The main tool we use for this purpose is the convergence in the sense of Mosco.
We obtain uniform decay estimates for scattered fields and we investigate how a sound-hard screen may be approximated by thin sound-hard obstacles.

Keywords: scattering, Helmholtz equation, Neumann problem, decay estimates, Mosco convergence..

Mathematics Subject Classification: Primary: 35P25; Secondary: 49J4.

Citation: Giorgio Menegatti, Luca Rondi. Stability for the acoustic scattering problem for sound-hard scatterers. Inverse Problems and Imaging, 2013, 7 (4) : 1307-1329. doi: 10.3934/ipi.2013.7.1307

References:

[1]	R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
[2]	D. Bucur and N. Varchon, Stability of the Neumann problem for variations of boundary, C. R. Acad. Sci. Paris Sér. I Math., 331 (2000), 371-374. doi: 10.1016/S0764-4442(00)01668-2.
[3]	D. Bucur and N. Varchon, Boundary variation for a Neumann problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 29 (2000), 807-821.
[4]	A. Chambolle and F. Doveri, Continuity of Neumann linear elliptic problems on varying two-dimensional bounded open sets, Comm. Partial Differential Equations, 22 (1997), 811-840. doi: 10.1080/03605309708821285.
[5]	D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, Berlin Heidelberg New York, 1998.
[6]	G. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhäuser, Boston Basel Berlin, 1993. doi: 10.1007/978-1-4612-0327-8.
[7]	G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures: existence and approximation results, Arch. Ration. Mech. Anal., 162 (2002), 101-135. doi: 10.1007/s002050100187.
[8]	A. Giacomini, A stability result for Neumann problems in dimension $N\geq 3$, J. Convex Anal., 11 (2004), 41-58.
[9]	R. Kress, On the low wave number asymptotics for the two-dimensional exterior Dirichlet problem for the reduced wave equation, Math. Meth. Appl. Sci., 9 (1987), 335-341. doi: 10.1002/mma.1670090126.
[10]	J. Li, H. Liu, L. Rondi and G. Uhlmann, Regularized transformation-optics cloaking for the Helmholtz equation: from partial cloak to full cloak, preprint, arXiv:1301.7013 (2013).
[11]	F. Murat, The Neumann sieve, in Nonlinear Variational Problems, Pitman, Boston, 1985, 24-32.
[12]	L. Rondi, Unique determination of non-smooth sound-soft scatterers by finite\-ly many far-field measurements, Indiana Univ. Math. J., 52 (2003), 1631-1662. doi: 10.1512/iumj.2003.52.2394.
[13]	L. Rondi, Unique continuation from Cauchy data in unknown non-smooth domains, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 5 (2006), 189-218.
[14]	C. H. Wilcox, Scattering Theory for the d'Alembert Equation in Exterior Domains, Springer-Verlag, Berlin New York, 1975.

show all references

References:

[1]	R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
[2]	D. Bucur and N. Varchon, Stability of the Neumann problem for variations of boundary, C. R. Acad. Sci. Paris Sér. I Math., 331 (2000), 371-374. doi: 10.1016/S0764-4442(00)01668-2.
[3]	D. Bucur and N. Varchon, Boundary variation for a Neumann problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 29 (2000), 807-821.
[4]	A. Chambolle and F. Doveri, Continuity of Neumann linear elliptic problems on varying two-dimensional bounded open sets, Comm. Partial Differential Equations, 22 (1997), 811-840. doi: 10.1080/03605309708821285.
[5]	D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, Berlin Heidelberg New York, 1998.
[6]	G. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhäuser, Boston Basel Berlin, 1993. doi: 10.1007/978-1-4612-0327-8.
[7]	G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures: existence and approximation results, Arch. Ration. Mech. Anal., 162 (2002), 101-135. doi: 10.1007/s002050100187.
[8]	A. Giacomini, A stability result for Neumann problems in dimension $N\geq 3$, J. Convex Anal., 11 (2004), 41-58.
[9]	R. Kress, On the low wave number asymptotics for the two-dimensional exterior Dirichlet problem for the reduced wave equation, Math. Meth. Appl. Sci., 9 (1987), 335-341. doi: 10.1002/mma.1670090126.
[10]	J. Li, H. Liu, L. Rondi and G. Uhlmann, Regularized transformation-optics cloaking for the Helmholtz equation: from partial cloak to full cloak, preprint, arXiv:1301.7013 (2013).
[11]	F. Murat, The Neumann sieve, in Nonlinear Variational Problems, Pitman, Boston, 1985, 24-32.
[12]	L. Rondi, Unique determination of non-smooth sound-soft scatterers by finite\-ly many far-field measurements, Indiana Univ. Math. J., 52 (2003), 1631-1662. doi: 10.1512/iumj.2003.52.2394.
[13]	L. Rondi, Unique continuation from Cauchy data in unknown non-smooth domains, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 5 (2006), 189-218.
[14]	C. H. Wilcox, Scattering Theory for the d'Alembert Equation in Exterior Domains, Springer-Verlag, Berlin New York, 1975.

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