arXiv subjects

Yuchao Yi

Publications and source records attributed to Yuchao Yi.

7 recordsLinked to original sources

arXiv2026

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

Let $W^T$ map Dirichlet data $f$ to time $T$ state $u^f(T, \cdot)$, we study the $K$-normal operator $C^T_K:= (W^T)^*KW^T$ where $(W^T)^*$ is the standard $L^2(dx)$ 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.

math.AP↗

arXiv2026

An inverse problem for compressible Euler's equations

We consider an inverse problem for the compressible Euler's equations in polytropic fluid. We show that by taking active measurements near a particle trajectory one can determine the background flow in a set where pressure waves can propagate from and return to the particle trajectory, under the additional assumption that the flow has nonzero vorticity.

math.AP↗

arXiv2025

Riemannian and Lorentzian Calderón problem under Magnetic Perturbation

We study both the Riemannian and Lorentzian Calderón problem when a family of Dirichlet-to-Neumann maps are given for an open set of magnetic/electromagnetic potentials. For the Riemannian version, by allowing small perturbations of the magnetic potential, we use the Runge Approximation Theorem to show that the metric can be uniquely determined. There is no gauge equivalence in this case. For the Lorentzian version, we use microlocal analysis to construct the trajectory of null-geodesics via generic perturbations of the electromagnetic potential, hence the conformal class of the metric can be constructed. Moreover, we also show, in the Lorentzian case, the same result can be obtained using generic perturbations of the metric itself.

math.AP↗

arXiv2025

Time Separation and Scattering Rigidity for Analytic Lorentzian Manifolds

In this work, we prove the following three rigidity results: (i) in a real-analytic globally hyperbolic spacetime $(M,g)$ without boundary, the time separation function restricted to a thin exterior layer of a unknown compact subset $K \subset M$ determines $K$ up to an analytic isometry, assuming no lightlike cut points in $K$; (ii) in a real-analytic globally hyperbolic spacetime $(M,g)$ with timelike boundary, the boundary time separation function determines $M$ up to an analytic isometry, assuming no lightlike cut points near $M$ and lightlike geodesics are non-trapping; (iii) in a real-analytic Lorentzian manifold $(M,g)$ with timelike boundary, the interior and complete scattering relations near the light cone, each determines $M$ up to an analytic isometry, assuming that lightlike geodesics are non-trapping. We emphasize in all of these three cases we do not assume the convexity of the boundary of the subset or the manifold. Moreover, in (iii) we do not assume causality of the Lorentzian manifold, and allow the existence of cut points. Along the way, we also prove some boundary determination results, the connections between the interior and complete scattering relations, and the connections between the lens data and the scattering relation, for Riemannian manifolds and Lorentzian manifolds with boundaries.

math.DG↗

arXiv2025

Bounded Time Inverse Scattering for Semilinear Dirac Equation

In this paper, we present the first uniqueness result on the bounded time inverse scattering problem for a semilinear Dirac equation with smooth nonlinearity $F(x, z)$ where $(x, z)\in \mathbb{R}^3\times \mathbb{C}^4$ and $x$ is the spatial variable. We show that the solution map, which sends initial data at time 0 to the solution at time $T$, uniquely determines $F(x, z)$ on $x \in \mathbb{R}^3$ and $|z| \leq M$, where $M$ is a constant depend on the solution map, under the assumption that $\partial_z F(x, 0)$ and $\partial^2_z F(x, 0)$ are known. In the proof, we construct a sequence of collisions approaching the initial timeline to simulate a boundary collision. This technique enables us to overcome the difficulties of this hyperbolic system without assumptions on the nonlinearity structure.

math.AP↗

arXiv2025

The Dirichlet-to-Neumann map for Lorentzian Calderón problems with data on disjoint sets

We consider the restricted Dirichlet-to-Neumann map $Λ^{U,V}_{g,A,q}$ for the wave equation with magnetic potential $A$ and scalar potential $q$, on an admissible Lorentzian manifold $(M, g)$ of dimension $n \geq 3$ with boundary. Here $U$ and $V$ are disjoint open subsets of $\partial M$, where we impose the Dirichlet data on $U$ and measure the Neumann-type data on $V$. We use the gliding rays and microlocal analysis to show that, without any a priori information, one can reconstruct the conformal class of the boundary metric $g|_{T\partial M \times T\partial M}$ and the magnetic potential $A|_{T\partial M}$ at recoverable boundary points from $Λ^{U,V}_{g,A,q}$. In particular, the conformal factor and the jet of the metric at those points are determined up to gauge transformations. Moreover, if the metric and the time orientation are known on $U$ (or $V$), then the metric on a larger portion of $V$ (or $U$) can be reconstructed, up to gauge.

math.AP↗

arXiv2024

Wulff inequality for minimal submanifolds in Euclidean space

In this paper, we prove a Wulff inequality for $n$-dimensional minimal submanifolds with boundary in $\mathbb{R}^{n+m}$, where we associate a nonnegative anisotropic weight $Φ: S^{n+m-1}\to \mathbb{R}^{+}$ to the boundary of minimal submanifolds. The Wulff inequality constant depends only on $m$ and $n$, and is independent of the weights. The inequality is sharp if $m=1, 2$ and $Φ$ is the support function of ellipsoids or certain type of centrally symmetric long convex bodies.

math.DG↗