Corners and edges always scatter

Consider time-harmonic acoustic scattering problems governed by the Helmholtz equation in two and three dimensions. We prove that bounded penetrable obstacles with corners or edges scatter every incident wave nontrivially, provided the function of refractive index is real-analytic. Moreover, if such a penetrable obstacle is a convex polyhedron or polygon, then its shape can be uniquely determined by the far-field pattern over all observation directions incited by a single incident wave. Our arguments are elementary and rely on the expansion of solutions to the Helmholtz equation.

[1]  H. Haddar,et al.  Transmission Eigenvalues in Inverse Scattering Theory , 2012 .

[2]  Fioralba Cakoni,et al.  The Existence of an Infinite Discrete Set of Transmission Eigenvalues , 2010, SIAM J. Math. Anal..

[3]  Lucas Chesnel,et al.  Strongly oscillating singularities for the interior transmission eigenvalue problem , 2013 .

[4]  Victor Isakov,et al.  On uniqueness in th invese transmission scattering problem , 1990 .

[5]  A. Nachman,et al.  Reconstructions from boundary measurements , 1988 .

[7]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

[8]  John Sylvester,et al.  Transmission Eigenvalues , 2008, SIAM J. Math. Anal..

[9]  D. Colton,et al.  The interior transmission problem , 2007 .

[10]  Johannes Elschner,et al.  Uniqueness in inverse transmission scattering problems for multilayered obstacles , 2011 .

[11]  O. Yu. Imanuvilov,et al.  Inverse Boundary Value Problem for Schrödinger Equation in Two Dimensions , 2011, SIAM J. Math. Anal..

[12]  John Sylvester,et al.  Corners Always Scatter , 2012, 1211.1848.

[13]  D. Colton,et al.  THE INVERSE SCATTERING PROBLEM FOR TIME-HARMONIC ACOUSTIC WAVES IN AN INHOMOGENEOUS MEDIUM , 1988 .

[14]  Y. Y. Belov,et al.  Inverse Problems for Partial Differential Equations , 2002 .

[15]  N I Grinberg,et al.  The Factorization Method for Inverse Problems , 2007 .

[16]  Rainer Kress,et al.  Uniqueness in inverse obstacle scattering (acoustics) , 1993 .

[17]  A. Bukhgeǐm,et al.  Recovering a potential from Cauchy data in the two-dimensional case , 2008 .

[18]  Jingzhi Li,et al.  Uniqueness in determining refractive indices by formally determined far-field data , 2013, 1308.5955.

[19]  Masahiro Yamamoto,et al.  Uniqueness in Identification of the Support of a Source Term in an Elliptic Equation , 2003, SIAM J. Math. Anal..

[20]  Uniqueness in inverse elastic scattering from unbounded rigid surfaces of rectangular type , 2014 .

[21]  Guo Zhang,et al.  Reconstruction from boundary measurements for less regular conductivities , 2012, 1212.0727.

[22]  John Sylvester,et al.  Discreteness of Transmission Eigenvalues via Upper Triangular Compact Operators , 2011, SIAM J. Math. Anal..

[23]  Victor Isakov,et al.  Inverse obstacle problems , 2009 .

[24]  J. Sylvester,et al.  A global uniqueness theorem for an inverse boundary value problem , 1987 .