PreprintarXiv licencearXiv
Singular sets of homogeneous Dirichlet heat flows at large time
Abstract
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.
Related
- Singular Aubry--Mather Sets for a Homogeneous Single-Input Control System
- Entropy on regular sets and periodic-orbit growth for singular flows
- Lack of exponential stability for thermoelastic Timoshenko and Bresse systems with full homogeneous Dirichlet boundary conditions
- Fractal dimensions and quasisymmetric packing-minimalities of some homogeneous Moran sets
- Positive normalized solutions for a singular regularized p(x)-Laplacian Dirichlet problem
BibTeX
@misc{chen2026singular,
title = {{Singular sets of homogeneous Dirichlet heat flows at large time}},
author = {Xiujin Chen and Junyuan Wang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05053},
eprint = {2611.05053},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05053},
}
RIS
TY - UNPB
TI - Singular sets of homogeneous Dirichlet heat flows at large time
AU - Xiujin Chen
AU - Junyuan Wang
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05053
UR - https://arxiv.org/abs/2611.05053
AB - 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.
ER -