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Xiujin Chen

Publications and source records attributed to Xiujin Chen.

3 recordsLinked to original sources
arXiv2026

Singular sets of homogeneous Dirichlet heat flows at large time

We study homogeneous Dirichlet heat flows for real symmetric, uniformly elliptic divergence-form operators. We establish quantitative stability of singular sets near homogeneous harmonic polynomials and eigenfunctions. Based on these, in the smooth coefficient case, we prove the spatial singular set at sufficiently large time is codimension-two and has a uniform measure upper bound independent of time. However, this codimension-two property fails in the Lipschitz coefficient case. On a three-dimensional ball, we construct a fixed self-adjoint operator and a heat flow consisting of two eigenmodes whose singular sets contain two-dimensional disks at a sequence of times tending to infinity.

math.AP↗
arXiv2026

Singular Sets and Doubling Properties in Parabolic Homogenization

We study the singular sets of real-valued solutions to periodic parabolic equations with rapidly oscillating coefficients. Under a nondegeneracy condition on the corrector matrix, we obtain a local parabolic $n$-dimensional Hausdorff measure bound and a codimension-two Minkowski estimate. The Minkowski estimate holds at every scale, including scales below the oscillation scale $\varepsilon$, with a constant independent of $\varepsilon$. The bounds depend on a coarse-scale ratio of space-time $L^2$ mass to terminal-slice $L^2$ mass. Independently of corrector nondegeneracy, we prove that this ratio controls doubling at all smaller scales uniformly in $\varepsilon$. The singular-set argument combines summable errors in truncated Gaussian approximations, comparison of caloric polynomials at different time centers, and a stopping cover with microscopic singular-set estimates.

math.AP↗