arXiv subjects

Yingying Cai

Publications and source records attributed to Yingying Cai.

4 recordsLinked to original sources

arXiv2026

Dimension Drop for Harmonic Measure on Ahlfors Regular Boundaries

We provide quantitative estimates for the dimension drop of harmonic measure. We show that for a domain $\Omega = \mathbb{R}^{n+1} \setminus E$ where $E$ is an $s$-Ahlfors regular compact set satisfying a uniform $L^2$-based non-flatness condition $\beta_2 \ge \delta_0$, the dimension of its harmonic measure is strictly less than $s$ for $s \in (n - c\delta_0^2, n]$. For planar domains, we establish an analogous quantitative threshold $s_0 = 1 - c\delta_0^2$ under Azzam's uniform non-flatness condition $\beta_\infty + \beta_{\operatorname{hole}} \ge \delta_0$.

math.AP↗

arXiv2025

Quantitative unique continuation for Neumann problem in planar $C^{1,α}$ domains

In this paper, we study the quantitative unique continuation property of the second-order elliptic operators under the vanishing Neumann boundary condition over $C^{1,α}$ or convex domains in two dimensions. We establish the optimal estimates of the number of critical points, doubling index and the total length of level curves. The key idea is to reduce the Neumann problem to the Dirichlet problem, which has been understood better, by a classical duality between an $A$-harmonic function and its stream function.

math.AP↗

arXiv2024

Unique continuation at the boundary for divergence form elliptic equations on quasiconvex domains

Let $Ω\subset \mathbb{R}^d$ be a quasiconvex Lipschitz domain and $A(x)$ be a $d \times d$ uniformly elliptic, symmetric matrix with Lipschitz coefficients. Assume a nontrivial $u$ solves $-\nabla \cdot (A(x) \nabla u) = 0$ in $Ω$, and $u$ vanishes on $Σ= \partial Ω\cap B$ for some ball $B$. The main contribution of this paper is to demonstrate the existence of a countable collection of open balls $(B_i)_i$ such that the restriction of $u$ to $B_i \cap Ω$ maintains a consistent sign. Furthermore, for any compact subset $K$ of $Σ$, the set difference $K \setminus \bigcup_i B_i$ is shown to possess a Minkowski dimension that is strictly less than $d - 1 - ε$. As a consequence, we prove Lin's conjecture in quasiconvex domains.

math.AP↗

arXiv2024

The sharp estimate of nodal sets for Dirichlet Laplace eigenfunctions in polytopes

Let $P$ be a bounded $n$-dimensional Lipschitz polytope, and let $φ_λ$ be a Dirichlet Laplace eigenfunction in $P$ corresponding to the eigenvalue $λ$. We show that the $(n-1)$-dimensional Hausdorff measure of the nodal set of $φ_λ$ does not exceed $C(P)\sqrtλ$. Our result extends the previous ones in quaisconvex domains (including $C^1$ and convex domains) to general polytopes that are not necessarily quasiconvex.

math.AP↗