arXiv subjects

Jiuyi Zhu

Publications and source records attributed to Jiuyi Zhu.

34 records · Page 2Linked to original sources

arXiv2020

Nodal sets of Robin and Neumann eigenfunctions

We investigate the measure of nodal sets for Robin and Neumann eigenfunctions in the domain and on the boundary of the domain. A polynomial upper bound for the interior nodal sets is obtained for Robin eigenfunctions in the smooth domain. For the analytic domain, the sharp upper bounds of the interior nodal sets was shown for Robin eigenfunctions. More importantly, we obtain the sharp upper bounds for the boundary nodal sets of Neumann eigenfunctions with new quantitative global Carleman estimates. Furthermore, the sharp doubling inequality and vanishing order of Robin eigenfunctions on the boundary of the domain are obtained.

math.AP↗

arXiv2019

Propagation of smallness in elliptic periodic homogenization

The paper is mainly concerned with an approximate three-ball inequality for solutions in elliptic periodic homogenization. We consider a family of second order operators $\mathcal{L}_ε$ in divergence form with rapidly oscillating and periodic coefficients. It is the first time such an approximate three-ball inequality for homogenization theory is obtained. It implies an approximate quantitative propagation of smallness. The proof relies on a representation of the solution by the Poisson kernel and the Lagrange interpolation technique. Another full propagation of smallness result is also shown.

math.AP↗

arXiv2018

Doubling inequality and nodal sets for solutions of bi-Laplace equations

We investigate the doubling inequality and nodal sets for the solutions of bi-Laplace equations. A polynomial upper bound for the nodal sets of solutions and their gradient is obtained based on the recent development of nodal sets for Laplace eigenfunctions by Logunov. In addition, we derive an implicit upper bound for the nodal sets of solutions. We show two types of doubling inequalities for the solutions of bi-Laplace equations. As a consequence, the rate of vanishing is given for the solutions.

math.AP↗

arXiv2018

Quantitative unique continuation of solutions to higher order elliptic equations with singular coefficients

We investigate the quantitative unique continuation of solutions to higher order elliptic equations with singular coefficients. Quantitative unique continuation described by the vanishing order is a quantitative form of strong unique continuation property. We characterize the vanishing order of solutions for higher order elliptic equations in terms of the norms of coefficient functions in their respective Lebesgue spaces. New versions of quantitative Carleman estimates are established.

math.AP↗

arXiv2017

Quantitative uniqueness of solutions to parabolic equations

We investigate the quantitative uniqueness of solutions to parabolic equations with lower order terms on compact smooth manifolds. Quantitative uniqueness is a quantitative form of strong unique continuation property. We characterize quantitative uniqueness by the rate of vanishing. We can obtain the vanishing order of solutions by $C^{1, 1}$ norm of the potential functions, as well as the $L^\infty$ norm of the coefficient functions. Some quantitative Carleman estimates and three cylinder inequalities are established.

math.AP↗

arXiv2017

Quantitative uniqueness of solutions to second order elliptic equations with singular lower order terms

In this article, we study the quantitative uniqueness of solutions to second order elliptic equations with singular lower order terms. We quantify the strong unique continuation property by estimating the maximal vanishing order of solutions. That is, when $u$ is a non-trivial solution to $\triangle u + W \cdot \nabla u + V u = 0$ in some open, connected subset of $\mathbb R^n$, where $n \geq 3$, we characterize the vanishing order of solutions in terms of the norms of $V$ and $W$ in their respective Lebesgue spaces. Using these maximal order of vanishing estimates, we also establish quantitative unique continuation at infinity results for solutions to $\triangle u + W \cdot \nabla u + V u = 0$ in $\mathbb R^n$. The main tools in our work are new versions of $L^p\to L^q$ Carleman estimates for a range of $p$- and $q$-values.

math.AP↗

arXiv2017

Quantitative uniqueness of solutions to second order elliptic equations with singular potentials in two dimensions

In this article, we study the vanishing order of solutions to second order elliptic equations with singular lower order terms in the plane. In particular, we derive lower bounds for solutions on arbitrarily small balls in terms of the Lebesgue norms of the lower order terms for all admissible exponents. Then we show that a scaling argument allows us to pass from these vanishing order estimates to estimates for the rate of decay of solutions at infinity. Our proofs rely on a new $L^p - L^q$ Carleman estimate for the Laplacian in $\mathbb{R}^2$.

math.AP↗

arXiv2015

Interior nodal sets of Steklov eigenfunctions on surfaces

We investigate the interior nodal sets $\mathcal{N}_λ$ of Steklov eigenfunctions on connected and compact surfaces with boundary. The optimal vanishing order of Steklov eigenfunctions is shown be $Cλ$. The singular sets $\mathcal{S}_λ$ are finite points on the nodal sets. We are able to prove that the Hausdorff measure $H^0(\mathcal{S}_λ)\leq Cλ^2$. Furthermore, we obtain an upper bound for the measure of interior nodal sets $H^1(\mathcal{N}_λ)\leq Cλ^{\frac{3}{2}}$. Here those positive constants $C$ depend only on the surfaces.

math.AP↗

arXiv2015

A lower bound for the nodal sets of Steklov eigenfunctions

We consider the lower bound of nodal sets of Steklov eigenfunctions on smooth Riemannian manifolds with boundary--the eigenfunctions of the Dirichlet-to-Neumann map. Let $N_λ$ be its nodal set. Assume that zero is a regular value of Steklov eigenfunctions. We show that $$H^{n-1}(N_λ)\geq Cλ^{\frac{3-n}{2}}$$ for some positive constant $C$ depending only on the manifold.

math.AP↗

arXiv2015

Lower bounds for interior nodal sets of Steklov eigenfunctions

We study the interior nodal sets, $Z_λ$ of Steklov eigenfunctions in an $n$-dimensional relatively compact manifolds $M$ with boundary and show that one has the lower bounds $|Z_λ|\ge cλ^{\frac{2-n}2}$ for the size of its $(n-1)$-dimensional Hausdorff measure. The proof is based on a Dong-type identity and estimates for the gradient of Steklov eigenfunctions, similar to those in \cite{SZ1} and \cite{SZ2}, respectively.

math.AP↗

arXiv2014

Quantitative uniqueness of elliptic equations

Based on a variant of frequency function, we improve the vanishing order of solutions for Schrödinger equations which describes quantitative behavior of strong uniqueness continuation property. For the first time, we investigate the quantitative uniqueness of higher order elliptic equations and show the vanishing order of solutions. Furthermore, strong unique continuation is established for higher order elliptic equations using this variant of frequency function.

math.AP↗

arXiv2014

Doubling property and vanishing order of Steklov eigenfunctions

The paper is concerned with the doubling estimates and vanishing order of the Steklov eigenfunction on the boundary of a smooth boundary domain $\mathbb R^n$. The eigenfunction is given by a Dirichlet-to-Neumann map. We improve the doubling property shown by Lin and Bellova \cite{BL}. Furthermore, we show that the vanishing order of Steklov eigenfunction is everywhere less than $Cλ$ where $λ$ is the Steklov eigenvalue and $C$ depends only on $Ω$.

math.AP↗

arXiv2014

Indefinite fractional elliptic problem and Liouville theorems

In this paper, we consider the indefinite fractional elliptic problem. A corresponding Liouville-type theorem for the indefinite fractional elliptic equations is established. Furthermore, we obtain a priori bound for solutions in a bounded domain by blowing-up and re-scaling. We also classify the solutions of some degenerate elliptic equation originated from fractional Laplacian.

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arXiv2013

The Maximum Principles and Symmetry results for Viscosity Solutions of Fully Nonlinear Equations

This paper is concerned about maximum principles and radial symmetry for viscosity solutions of fully nonlinear partial differential equations. We obtain the radial symmetry and monotonicity properties for nonnegative viscosity solutions of begin{equation}F (D^2 u)+u^p=0 \quad \quad \mbox{in}\ \mathbb R^n \label{abs}\end{equation} under the asymptotic decay rate $u=o(|x|^{-\frac{2}{p-1}})$ at infinity, where $p>1$ (Theorem 1, Corollary 1). As a consequence of our symmetry results, we obtain the nonexistence of any nontrivial and nonnegative solution when $F$ is the Pucci extremal operators (Corollary 2). Our symmetry and monotonicity results also apply to Hamilton-Jacobi-Bellman or Isaccs equations. A new maximum principle for viscosity solutions to fully nonlinear elliptic equations is established (Theorem 2). As a result, different forms of maximum principles on bounded and unbounded domains are obtained. Radial symmetry, monotonicity and the corresponding maximum principle for fully nonlinear elliptic equations in a punctured ball are shown (Theorem 3). We also investigate the radial symmetry for viscosity solutions of fully nonlinear parabolic partial differential equations (Theorem 4).

math.AP↗

arXiv2012

Liouville-type theorems for fully nonlinear elliptic equation in half spaces

In \cite{LWZ}, we establish Liouville-type theorems and decay estimates for solutions of a class of high order elliptic equations and systems without the boundedness assumptions on the solutions. In this paper, we continue our work in \cite{LWZ} to investigate the role of boundedness assumption in proving Liouville-type theorems for fully nonlinear equations. We remove the boundedness assumption of solutions which was required in the proof of Liouville-type theorems for fully nonlinear elliptic equations or systems in half spaces. We also prove the Liouville-type theorems for supersolutions of a system of fully nonlinear equations with Pucci extremal operators in half spaces.

math.AP↗

arXiv2011

An overdetermined problem in Riesz-potential and fractional Laplacian

The main purpose of this paper is to address two open questions raised by W. Reichel on characterizations of balls in terms of the Riesz potential and fractional Laplacian. For a bounded $C^1$ domain $Ω\subset \mathbb R^N$, we consider the Riesz-potential $$u(x)=\int_Ω\frac{1}{|x-y|^{N-α}} \,dy$$ for $2\leq α\not =N$. We show that $u=$ constant on the boundary of $Ω$ if and only if $Ω$ is a ball. In the case of $ α=N$, the similar characterization is established for the logarithmic potential. We also prove that such a characterization holds for the logarithmic Riesz potential. This provides a characterization for the overdetermined problem of the fractional Laplacian. These results answer two open questions of W. Reichel to some extent.

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