arXiv subjects

Ziyao Zhao

Publications and source records attributed to Ziyao Zhao.

7 recordsLinked to original sources

arXiv2026

An inverse free boundary problem

We study inverse problems for the elliptic and parabolic obstacle problems from boundary measurements. For the classical elliptic obstacle problem with strictly superharmonic obstacle function, we show that the Dirichlet-to-Neumann map admits a one-sided linearization at every boundary datum lying strictly above the obstacle. The linearized map is the Dirichlet-to-Neumann map for a rough Dirichlet problem on the a priori unknown non-contact set. We show that these linearized Cauchy data uniquely determine the non-contact set up to set of Sobolev $2$-capacity zero and consequently determine the obstacle. Our result applies to the inverse problem for a parabolic obstacle problem where both the coefficient and the obstacle function are time-independent by reducing to the elliptic inverse problem.

math.AP↗

arXiv2026

An inverse problem for inhomogeneous Signorini obstacle

This paper investigates the inverse problem of determining a general Signorini obstacle using boundary measurements. We demonstrate that both the shape of the obstacle and the obstacle function can be uniquely determined from solution measurements taken on an arbitrary open subset of the boundary. This result applies to both the scalar and elasticity versions of the Signorini problem.

math.AP↗

arXiv2026

Inverse problem for the geometric Navier-Stokes equations

We consider the inverse problem of determining a compact Riemannian manifold with boundary from fixed time observations of the solution, restricted to a small subset in space, for the Navier-Stokes system with a local source on the manifold. Our approach is based on a reduction to an inverse problem for an auxiliary hyperbolic Stokes system, via linearization and spectral techniques. We solve the resulting inverse problem by a new generalization of the Boundary Control method.

math.AP↗

arXiv2025

A Hyperbolic Inverse Problem for lower order terms on a closed manifold with disjoint data

We study the unique recovery of time-independent lower order terms appearing in the symmetric first order perturbation of the Riemannian wave equation by sending and measuring waves in disjoint open sets of \textit{a priori} known closed Riemannian manifold. In particular, we show that if the set where we capture the waves satisfies a geometric control condition as well as a certain local symmetry condition for the distance functions, then the aforementioned measurement is sufficient to recover the lower order terms up to the natural gauge. For instance, our result holds if the complement of the receiver set is contained in a simple Riemannian manifold.

math.AP↗

arXiv2025

Unique continuation for the wave equation: the stability landscape

We consider a unique continuation problem for the wave equation given data in a volumetric subset of the space time domain. In the absence of data on the lateral boundary of the space-time cylinder we prove that the solution can be continued with Hölder stability into a certain proper subset of the space-time domain. Additionally, we show that unique continuation of the solution to the entire space-time cylinder with Lipschitz stability is possible given the knowledge of a suitable finite dimensional space in which the trace of the solution on the lateral boundary is contained. These results allow us to design a finite element method that provably converges to the exact solution at a rate that mirrors the stability properties of the continuous problem.

math.NA↗

arXiv2025

An inverse Signorini obstacle problem

We study the inverse problem of determining a Signorini obstacle from boundary measurements for the isotropic elasticity system. We prove that the obstacle can be uniquely determined by a single measurement of displacement and normal stress for the Signorini problem on an open subset of the boundary up to a natural obstruction. In addition to considering the Signorini problem, we develop techniques that can be used to study inverse problems for general differential inequalities.

math.AP↗

arXiv2025

Computational unique continuation with finite dimensional Neumann trace

We consider finite element approximations of unique continuation problems subject to elliptic equations in the case where the normal derivative of the exact solution is known to reside in some finite dimensional space. To give quantitative error estimates we prove Lipschitz stability of the unique continuation problem in the global H1-norm. This stability is then leveraged to derive optimal a posteriori and a priori error estimates for a primal-dual stabilised finite method.

math.NA↗