TY - JOUR
T1 - Further results on generalized Holmgren's principle to the Lamé operator and applications
AU - Diao, Huaian
AU - Liu, Hongyu
AU - Wang, Li
PY - 2022/2/5
Y1 - 2022/2/5
N2 - In our earlier paper [14], it is proved that a homogeneous rigid, traction or impedance condition on one or two intersecting line segments together with a certain zero point-value condition implies that the solution to the Lamé system must be identically zero, which is referred to as the generalized Holmgren principle (GHP). The GHP enables us to solve a longstanding inverse scattering problem of determining a polygonal elastic obstacle of general impedance type by at most a few far-field measurements. In this paper, we include all the possible physical boundary conditions from linear elasticity into the GHP study with the soft-clamped, simply-supported as well as the associated impedance-type conditions. We derive a comprehensive and complete characterization of the GHP associated with all of the aforementioned physical conditions. As significant applications, we establish novel unique identifiability results by a few scattering measurements not only for the inverse elastic obstacle problem but also for the inverse elastic diffraction grating problem within polygonal geometry in the most general physical scenario. We follow the general strategy from [14] in establishing the results. However, we develop technically new ingredients to tackle the more general and challenging physical and mathematical setups. It is particularly worth noting that in [14], the impedance parameters were assumed to be constant whereas in this work they can be variable functions.
AB - In our earlier paper [14], it is proved that a homogeneous rigid, traction or impedance condition on one or two intersecting line segments together with a certain zero point-value condition implies that the solution to the Lamé system must be identically zero, which is referred to as the generalized Holmgren principle (GHP). The GHP enables us to solve a longstanding inverse scattering problem of determining a polygonal elastic obstacle of general impedance type by at most a few far-field measurements. In this paper, we include all the possible physical boundary conditions from linear elasticity into the GHP study with the soft-clamped, simply-supported as well as the associated impedance-type conditions. We derive a comprehensive and complete characterization of the GHP associated with all of the aforementioned physical conditions. As significant applications, we establish novel unique identifiability results by a few scattering measurements not only for the inverse elastic obstacle problem but also for the inverse elastic diffraction grating problem within polygonal geometry in the most general physical scenario. We follow the general strategy from [14] in establishing the results. However, we develop technically new ingredients to tackle the more general and challenging physical and mathematical setups. It is particularly worth noting that in [14], the impedance parameters were assumed to be constant whereas in this work they can be variable functions.
KW - Generalized Holmgren's principle
KW - Inverse elastic problems
KW - Lamé system
KW - Polygonal scatterers
KW - Single measurement
KW - Unique identifiability
UR - http://www.scopus.com/inward/record.url?scp=85120649908&partnerID=8YFLogxK
UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-85120649908&origin=recordpage
U2 - 10.1016/j.jde.2021.11.039
DO - 10.1016/j.jde.2021.11.039
M3 - 21_Publication in refereed journal
VL - 309
SP - 841
EP - 882
JO - Journal of Differential Equations
JF - Journal of Differential Equations
SN - 0022-0396
ER -