arXiv subjects

Ruirui Wu

Publications and source records attributed to Ruirui Wu.

3 recordsLinked to original sources

arXiv2026

Partial data Calder\'{o}n problem for quasilinear conductivities in dimension 2

In this paper, we prove a uniqueness result for the partial data Calder\'{o}n problem with quasilinear conductivity in two dimensions. The proof is based on higher-order linearization and the use of CGO solutions in dimension two that vanish on part of the boundary. Since derivatives of the solutions appear in the integral identity, we need improved remainder estimates, for which we introduce a modification in the choice of the phase, analogous to limiting Carleman weights. We also analyze how combinations of phases produce specific patterns in the products of solutions, which allows us to apply both stationary and nonstationary phase arguments to recover the conductivity.

math.AP↗

arXiv2026

Fractional Vector Calculus and the Fractional Maxwell's Equations

We consider a fractional variant of Maxwell's equations, where the electric and magnetic fields are modeled as two-point fields. To formulate the system, we introduce a fractional curl operator that is compatible with the fractional divergence operator, ensuring the divergence-free condition. A key ingredient is a projection map $\Pi$ that reduces two-point fields to one-point fields. We also define a new fractional Sobolev space whose elements enjoy a fractional Helmholtz decomposition and observe that the projection $\Pi$ is a bijection in this space, which allows us to reformulate the problem entirely in terms of one-point fields. We then prove the well-posedness of the equations in one-point fields in weighted fractional Sobolev spaces, and deduce a corresponding well-posedness result for the two-points fractional Maxwell system. This constitutes a first necessary step towards the resolution of a scattering inverse problem for the fractional Maxwell's equations, which will be the topic of future work.

math.AP↗

arXiv2023

Calder\'{o}n problem for the quasilinear conductivity equation in dimension $2$

In this paper we prove a uniqueness result for the Calder\'{o}n problem for the quasilinear conductivity equation on a bounded domain $\R^2$. The proof of the result is based on the higher order linearization method, which reduces the problem to showing density of products of solutions to the linearized equation and their gradients. In contrast to the higher dimensional case, the proof involves delicate analysis of the correction terms of Bukhgeim type complex geometric solutions (CGOs), which have only limited decay. To prove our results, we construct suitable families of CGOs whose phase functions have and do not have critical points. We also combine stationary phase analysis with $L^p$ estimates for the correction terms of the CGOs.

math.AP↗