Non-radiating elastic sources in inhomogeneous elastic media at corners with applications

Abstract This paper is concerned with non-radiating elastic sources in inhomogeneous elastic media. We demonstrate that the value of non-radiating elastic sources must vanish at convex corners of their support, provided the sources exhibit Hölder continuous regularity near the corner. Additionally, their gradient must satisfy intricate algebraic relationships with the angles defining the underlying corners, assuming the sources have <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> regularity with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> in the neighborhood of the corners. Our analysis employs complex geometrical optics solutions as test functions within a partial differential system to conduct asymptotic analysis near the corners. These characterizations enable us to establish unique identifiability results for determining the position and shape of radiating elastic sources from a single far-field measurement, both locally and globally. The uniqueness of such identification is a longstanding challenge in inverse scattering with a rich history. Specifically, when the support of a radiating elastic source is a convex polygon and the source is Hölder continuous at the corners, we can simultaneously determine the source’s shape and its values at the corners. Furthermore, when the source function exhibits <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> regularity in the neighborhood of a corner, the gradient at that corner can typically be determined. Additionally, when the support includes a convex sectorial corner and the elastic source satisfies certain generic conditions, we demonstrate that such a source must radiate at any frequency.