Leakage detection, collision relation, and self-adjoint cancellation in multi-velocity systems

2026/06/23 by Yuchao Yi
#math.AP

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Abstract

Let W^T map Dirichlet data f to time T state u^f(T, ⋅), we study the K-normal operator C^T_K:= (W^T)^*KW^T where (W^T)^* is the standard L²(dx) adjoint. We prove that it is a locally finite sum of Fourier Integral Operators (FIOs) away from the glancing directions with canonical relation involving collision data at time T between pairs of velocities. However, if the operator is M-self-adjoint, then the canonical relation for C^T_M loses the collision data. Thus, when system satisfies certain coupling requirement, we demonstrate off-polarization leakage detection and introduce a new collision rigidity problem. Combine these two parts, we prove an inverse problem of velocity recovery from C^T_I for multi-velocity wave models and variable coefficient isotropic elasticity system, where I is the identity matrix.

Citations

• Determining anisotropic real‐analytic conductivities by boundary measurements
• Singularities of boundary value problems. I
• Reflection of singularities of solutions to systems of differential equations

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