arXiv subjects

Zihui Zhao

Publications and source records attributed to Zihui Zhao.

28 records · Page 2Linked to original sources

arXiv2021

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

Let $u$ be a harmonic function in a $C^1$-Dini domain $D$ such that $u$ vanishes on a boundary surface ball $\partial D \cap B_{5R}(0)$. We consider an effective version of its singular set (up to boundary) $\mathcal{S}(u):=\{X\in \overline{D}: u(X) = |\nabla u(X)| = 0\} $ and give an estimate of its $(d-2)$-dimensional Minkowski content, which only depends on the upper bound of some modified frequency function of $u$ centered at $0$. Such results are already known in the interior and at the boundary of convex domains, when the standard frequency function is monotone at every point. The novelty of our work on Dini domains is how to compensate for the lack of such monotone quantities at boundary as well as interior points.

math.AP↗

arXiv2020

Two Phase Free Boundary Problem for Poisson Kernels

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, the oscillation of the logarithm of the interior and exterior Poisson kernels yields a great deal of geometric information about the domain. We use techniques from the classical calculus of variations, potential theory, quantitative geometric measure theory to accomplish this. One feature of this work, compared to Bortz-Hofmann PAMS 16 and Kenig-Toro Crelle 06, is that a priori we only require that the domains in question are connected.

math.CA↗

arXiv2020

Uniform rectifiability and elliptic operators satisfying a Carleson measure condition

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 whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying a suitable Carleson measure condition. The result can be viewed as a quantitative analogue of the Wiener criterion adapted to the singular $L^p$ data case. We split our proof on two main steps. In the first one we considered the case in which the desired Carleson measure condition on the coefficients holds with "sufficiently small constant", using a novel application of techniques developed in geometric measure theory. In the second step we establish the final result, that is, the "large constant case". The key elements are a powerful extrapolation argument, which provides a general pathway to self-improve scale-invariant small constant estimates, and a new mechanism to transfer quantitative absolute continuity of elliptic measure between a domain and its subdomains.

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arXiv2020

Uniform rectifiability and elliptic operators satisfying a Carleson measure condition. Part I: The small constant case

The present paper, along with its sequel, 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 whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying a suitable Carleson measure condition. The result can be viewed as a quantitative analogue of the Wiener criterion adapted to the singular $L^p$ data case. This paper addresses the free boundary problem under the assumption of smallness of the Carleson measure of the coefficients. Part II of this work develops an extrapolation argument to bootstrap this result to the general case. The ideas in Part I constitute a novel application of techniques developed in geometric measure theory. They highlight the synergy between several areas. The ideas developed in this paper are well suited to study singularities arising in variational problems in a geometric setting.

math.AP↗

arXiv2020

Uniform rectifiability and elliptic operators satisfying a Carleson measure condition. Part II: The large constant case

The present paper, along with its companion [Hofmann, Martell, Mayboroda, Toro, Zhao, arXiv:1710.06157], 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 whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying a suitable Carleson measure condition. The result can be viewed as a quantitative analogue of the Wiener criterion adapted to the singular $L^p$ data case. The first step in this direction was taken in our previous paper [Hofmann, Martell, Mayboroda, Toro, Zhao, arXiv:1710.06157], where we considered the case in which the desired Carleson measure condition on the coefficients holds with sufficiently small constant. In this paper we establish the final, general result, that is, the "large constant case". The key elements of our approach are a powerful extrapolation argument, which provides a general pathway to self-improve scale-invariant small constant estimates, as well as a new mechanism to transfer quantitative absolute continuity of elliptic measure between a domain and its subdomains.

math.AP↗

arXiv2019

Dirichlet energy-minimizers with analytic boundary

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 minimizing currents with real analytic boundaries and our main result is that their singular set is discrete in 2 dimensions. This confirms (and provides a first step to) a conjecture by B. White \cite{White97} that area minimizing $2$-dimensional currents with real analytic boundaries have a finite number of singularities. We also show that, in any dimension, Dirichlet energy-minimizers with a $C^1$ boundary interface are Hölder continuous at the interface.

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arXiv2018

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

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 class of partial differential equations, analogous to the class of elliptic PDEs in the classical context, is given by linear degenerate equations with the degeneracy suitably depending on the distance to the boundary. The present paper continues this line of research and focuses on the criteria of quantitative absolute continuity of the newly defined harmonic measure with respect to the Hausdorff measure, $ω\in A_\infty(σ)$, in terms of solvability of boundary value problems. The authors establish, in particular, square function estimates and solvability of the Dirichlet problem in BMO for domains with lower-dimensional boundaries under the underlying assumption $ω\in A_\infty(σ)$. More generally, it is proved that in all domains with Ahlfors regular boundaries the BMO solvability of the Dirichlet problem is necessary and sufficient for the absolute continuity of the harmonic measure.

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arXiv2018

Dirichlet problem in domains with lower dimensional boundaries

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 boundaries, e.g., in $Ω:= \mathbb R^n \setminus \mathbb R^d$ with $d 1$ provided that the coefficients satisfy the small Carleson norm condition. Even in the context of the classical case $d=n-1$, (the analogues of) our results are new. The conditions on the coefficients are more relaxed than the previously known ones (most notably, we do not impose any restrictions whatsoever on the first $n-1$ rows of the matrix of coefficients) and the results are more general. We establish local rather than global estimates between the square function and the non-tangential maximal function and, perhaps even more importantly, we establish new Moser-type estimates at the boundary and improve the interior ones.

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arXiv2017

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

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 of the elliptic measure associated to any such operator implies that $Ω$ is a set of locally finite perimeter whose boundary, $\partialΩ$, is rectifiable. As a corollary we show that for this type of operators, absolute continuity of the surface measure with respect to the elliptic measure is enough to guarantee rectifiability of the boundary. In the case that the coefficients are continuous we obtain additional information about $Ω$.

math.AP↗

arXiv2017

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

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 boundary, the BMO solvability of such problems is equivalent to a quantitative absolute continuity of the elliptic measure with respect to the surface measure, i.e. $ω_L\in A_{\infty}(σ)$. This generalizes a previous result on Lipschitz domains by Dindos, Kenig and Pipher.

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