Zihui Zhao

activity

20162026

A_{\infty}

condition of the elliptic measure in uniform domains

15 citations · 58 across the 12 of their papers we have counts for

collaborators

14 papers

math.AP2026

Generic boundary behavior of Dirichlet eigenfunctions

Alec Payne, Zihui Zhao

We extend Uhlenbeck's classical results on the generic properties of Laplacian eigenfunctions. In particular, we find a residual set of $C^{k}$ domains in $R^{n}$ , $k > n$ , su…

math.AP2026

Singularity formation and self-shrinkers for the parabolic Bernoulli problem

Hunter Liu, Sebastian Munoz, Zihui Zhao

We study singularity formation for classical solutions of the parabolic Bernoulli free boundary problem and the associated self-similar profiles. Inspired by Ricci and mean curvatu…

math.CV2022

Examples of non-Dini domains with large singular sets

Carlos Kenig, Zihui Zhao

Let $u$ be a non-trivial harmonic function in a domain $D \subset R^{d}$ which vanishes on an open set of the boundary. In a recent paper, we showed that if $D$ is a $C^{1}$ -Di…

math.AP2021

Expansion of harmonic functions near the boundary of Dini domains

Carlos Kenig, Zihui Zhao

Let $u$ be a harmonic function in a $C^{1}$ -Dini domain, such that $u$ vanishes on an open set of the boundary. We show that near every point in the open set, $u$ can be written uniq…

math.AP2021

Elliptic measures for Dahlberg-Kenig-Pipher operators: Asymptotically optimal estimates

Simon Bortz, Tatiana Toro, Zihui Zhao

Questions concerning quantitative and asymptotic properties of the elliptic measure corresponding to a uniformly elliptic divergence form operator have been the focus of recent stu…

math.AP2021★ 10 cited

Boundary unique continuation on $C^{1}$ -Dini domains and the size of the singular set

Carlos Kenig, Zihui Zhao

Let $u$ be a harmonic function in a $C^{1}$ -Dini domain $D$ such that $u$ vanishes on a boundary surface ball $\partial D \cap B_{5 R} (0)$ . We consider an effective version of its sin…

math.AP2020

Optimal Poisson kernel regularity for elliptic operators with Hölder-continuous coefficients in vanishing chord-arc domains

Simon Bortz, Tatiana Toro, Zihui Zhao

We show that if $Ω$ is a vanishing chord-arc domain and $L$ is a divergence-form elliptic operator with Hölder-continuous coefficient matrix, then $lo g k_{L} \in V M O$ , where $k_{L}$ i…

math.AP2020★ 5 cited

Uniform rectifiability and elliptic operators satisfying a Carleson measure condition

Steve Hofmann, José María Martell, Svitlana Mayboroda +2

The present paper establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of w…

math.CA2019★ 1 cited

Two Phase Free Boundary Problem for Poisson Kernels

Simon Bortz, Max Engelstein, Max Goering +2

We provide a potential theoretic characterization of vanishing chord-arc domains under minimal assumptions. In particular we show that, if a domain has Ahlfors regular boundary, th…

math.AP2019★ 1 cited

Dirichlet energy-minimizers with analytic boundary

Camillo De Lellis, Zihui Zhao

In this paper, we consider multi-valued graphs with a prescribed real analytic interface that minimize the Dirichlet energy. Such objects arise as a linearized model of area minimi…

math.AP2018★ 10 cited

Dirichlet problem in domains with lower dimensional boundaries

Joseph Feneuil, Svitlana Mayboroda, Zihui Zhao

The present paper pioneers the study of the Dirichlet problem with $L^{q}$ boundary data for second order operators with complex coefficients in domains with lower dimensional bounda…

math.AP2018★ 9 cited

Square function estimates, BMO Dirichlet problem, and absolute continuity of harmonic measure on lower-dimensional sets

Svitlana Mayboroda, Zihui Zhao

In the recent work [DFM1, DFM2] G. David, J. Feneuil, and the first author have launched a program devoted to an analogue of harmonic measure for lower-dimensional sets. A relevant…

math.AP2017★ 7 cited

Boundary rectifiability and elliptic operators with $W^{1, 1}$ coefficients

Tatiana Toro, Zihui Zhao

We consider second order divergence form elliptic operators with $W^{1, 1}$ coefficients, in a uniform domain $Ω$ with Ahlfors regular boundary. We show that the $A_{\infty}$ property…

math.CA2016★ 15 cited

BMO solvability and the $A_{\infty}$ condition of the elliptic measure in uniform domains

Zihui Zhao

We consider the Dirichlet boundary value problem for divergence form elliptic operators with bounded measurable coefficients. We prove that for uniform domains with Ahlfors regular…

Generic boundary behavior of Dirichlet eigenfunctions

Singularity formation and self-shrinkers for the parabolic Bernoulli problem

Examples of non-Dini domains with large singular sets

Expansion of harmonic functions near the boundary of Dini domains

Elliptic measures for Dahlberg-Kenig-Pipher operators: Asymptotically optimal estimates

Boundary unique continuation on C1-Dini domains and the size of the singular set

Optimal Poisson kernel regularity for elliptic operators with Hölder-continuous coefficients in vanishing chord-arc domains

Uniform rectifiability and elliptic operators satisfying a Carleson measure condition

Two Phase Free Boundary Problem for Poisson Kernels

Dirichlet energy-minimizers with analytic boundary

Dirichlet problem in domains with lower dimensional boundaries

Square function estimates, BMO Dirichlet problem, and absolute continuity of harmonic measure on lower-dimensional sets

Boundary rectifiability and elliptic operators with W1,1 coefficients

BMO solvability and the A∞​ condition of the elliptic measure in uniform domains

Boundary unique continuation on $C^{1}$ -Dini domains and the size of the singular set

Boundary rectifiability and elliptic operators with $W^{1, 1}$ coefficients

BMO solvability and the $A_{\infty}$ condition of the elliptic measure in uniform domains