On generalized Holmgren’s principle to the Lamé operator with applications to inverse elastic problems

Further results on generalized Holmgren's principle to the Lamé operator and applications

“…This paper is continuation of our earlier paper [9] on a novel unique continuation principle for the Lamé operator and its applications to several challenging inverse elastic problems. We begin by briefly introducing the background and motivation of our study and referring to [9] for more related discussions.…”

“…It is noted that u is real-analytic in Ω since L is an elliptic PDO with constant coefficients. The classical Holmgren's uniqueness principle (HUP) states that if any two different conditions from B j (u) = 0, 1 ≤ j ≤ 4, are satisfied on Γ h , then u ≡ 0 in Ω; see [9] for more relevant background discussion on this aspect. In [9], we show that under ultra weaker conditions in two scenarios, the Holmgren principle still holds, which is referred to as the generalized Holmgren principle (GHP).…”

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In our earlier paper [9], 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.

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Numerical Methods for Semilinear Fractional Diffusion Equations with Time Delay

“…The benchmarking numerical examples verify the effectiveness and efficiency of our numerical scheme. For the future study, we plan to apply the newly derived numerical methods to some inverse problems in the fractional setting, say in particular the Schiffer problem, which is a longstanding problem in the inverse scattering theory [4,9,17,18,25,[28][29][30][31][32][33][34][35][36][37][38], but was recently solved in the fractional setting associated with the fractional Helmholtz equation [5].…”

The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation

“…Then based on the decoupling results of the shear and pressure waves, [29], [31] had shown the uniqueness of the third or fourth kind impenetrable polyhedral scatterers by a minimal number of far-field measurements. Moreover, Gintides [17] had proved a local uniqueness result for a rigid scatterer with one incident wave by applying the Faber-Krahn inequality; Diao, Liu and Wang [13] have shown that a generalized impedance obstacle as well as its boundary impedance can be determined by using at most four far-field patterns, where they have generalized the classical Holmgren's uniqueness principle for the Lamé operator in two aspects.…”