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Zhongwei Shen

Publications and source records attributed to Zhongwei Shen.

90 records · Page 5Linked to original sources
arXiv2012

Existence, Uniqueness and Anisotropic-Decay-Caused Lifshitz Tails of the Integrated Density of Surface States for Random Surface Models

The current paper is devoted to the study of existence, uniqueness and Lifshitz tails of the integrated density of surface states (IDSS) for Schrödinger operators with alloy type random surface potentials. We prove the existence and uniqueness of the IDSS for negative energies, which is defined as the thermodynamic limit of the normalized eigenvalue counting functions of localized operators on strips with sections being special cuboids. Under the additional assumption that the single-site impurity potential decays anisotropically, we also prove that the IDSS for negative energies exhibits Lifshitz tails near the bottom of the almost sure spectrum in the following three regimes: the quantum regime, the quantum-classical/classical-quantum regime and the classical regime. We point out that the quantum-classical/classical-quantum regime is new for random surface models.

math.SP↗
arXiv2012

Asymptotic Dynamics of a Class of Coupled Oscillators Driven by White Noises

This paper is devoted to the study of the asymptotic dynamics of a class of coupled second order oscillators driven by white noises. It is shown that any system of such coupled oscillators with positive damping and coupling coefficients possesses a global random attractor. Moreover, when the damping and the coupling coefficients are sufficiently large, the global random attractor is a one-dimensional random horizontal curve regardless of the strength of the noises, and the system has a rotation number, which implies that the oscillators in the system tend to oscillate with the same frequency eventually and therefore the so called frequency locking is successful. The results obtained in this paper generalize many existing results on the asymptotic dynamics for a single second order noisy oscillator to systems of coupled second order noisy oscillators. They show that coupled damped second order oscillators with large damping have similar asymptotic dynamics as the limiting coupled first order oscillators as the damping goes to infinite and also that coupled damped second order oscillators have similar asymptotic dynamics as their proper space continuous counterparts, which are of great practical importance.

math.DS↗
arXiv2012

Resolvent Estimates in L^p for the Stokes Operator in Lipschitz Domains

We establish the $L^p$ resolvent estimates for the Stokes operator in Lipschitz domains in $R^d$, $d\ge 3$ for $|\frac{1}{p}-1/2|< \frac{1}{2d} +ε$. The result, in particular, implies that the Stokes operator in a three-dimensional Lipschitz domain generates a bounded analytic semigroup in $L^p$ for $(3/2)-\varep < p< 3+ε$. This gives an affirmative answer to a conjecture of M. Taylor.

math.AP↗
arXiv2012

Homogenization of Green and Neumann Functions

For a family of second-order elliptic operators with rapidly oscillating periodic coefficients, we study the asymptotic behavior of the Green and Neumann functions, using Dirichlet and Neumann correctors. As a result we obtain asymptotic expansions of Poisson kernels and the Dirichlet-to-Neumann maps as well as near optimal convergence rates in $W^{1,p}$ for solutions with Dirichlet or Neumann boundary conditions.

math.AP↗
arXiv2011

Convergence Rates in L^2 for Elliptic Homogenization Problems

We study rates of convergence of solutions in L^2 and H^{1/2} for a family of elliptic systems {L_ε} with rapidly oscillating oscillating coefficients in Lipschitz domains with Dirichlet or Neumann boundary conditions. As a consequence, we obtain convergence rates for Dirichlet, Neumann, and Steklov eigenvalues of {L_ε}. Most of our results, which rely on the recently established uniform estimates for the L^2 Dirichlet and Neumann problems in \cite{12,13}, are new even for smooth domains.

math.AP↗
arXiv2010

Homogenization of Elliptic Systems with Neumann Boundary Conditions

The main purpose of this work is to study uniform regularity estimates for a family of elliptic operators $\{\mathcal{L}_\varepsilon, \varepsilon>0\}$, arising in the theory of homogenization, with rapidly oscillating periodic coefficients. We establish sharp $W^{1,p}$ estimates, Lipschitz estimates, and nontangential maximal function estimates, which are uniform in the parameter $\varepsilon$, on solutions with Neumann boundary conditions in $C^{1,α}$ domains.

math.AP↗
arXiv2010

The Neumann Problem and Helmholtz Decomposition in Convex Domains

We show that the Neumann problem for Laplace's equation in a convex domain $Ω$ with boundary data in $L^p(\partialΩ)$ is uniquely solvable for $1<p<\infty$. As a consequence, we obtain the Helmholtz decomposition of vector fields in $L^p(Ω, \mathbb{R}^d)$.

math.AP↗
arXiv2009

A Bilinear Estimate for Biharmonic Functions in Lipschitz Domains

We show that a bilinear estimate for biharmonic functions in a Lipschitz domain $Ω$is equivalent to the solvability of the Dirichlet problem for the biharmonic equationin $Ω$. As a result, we prove that for any given bounded Lipschitz domain $Ω$ in $\rn{d}$ and $1<q<\infty$, the solvability of the $L^{q}$ Dirichlet problem for $Δ^2 u=0$ in $Ω$ with boundary data in ${\emph{WA}}^{1,q}(\partialΩ)$ is equivalent to that of the $L^p$ regularity problem for $Δ^2 u=0$ in $Ω$ with boundary data in ${\emph{WA}}^{2,p}(\partialΩ)$, where $\frac{1}{p} +\frac{1}{q}=1$. This duality relation, together with known results on the Dirichlet problem, allows us to solve the $L^p$ regularity problemfor $d\ge 4$ and $p$ in certain ranges.

math.AP↗
arXiv2009

Layer Potential Methods for Elliptic Homogenization Problems

In this paper we use the method of layer potentials to study $L^2$ boundary value problems in a bounded Lipschitz domain $Ω$ for a family of second order elliptic systems with rapidly oscillating periodic coefficients, arising in the theory of homogenization. Let $\mathcal{L}_\varepsilon=-\text{div}\big(A(\varepsilon^{-1}X)\nabla \big)$. Under the assumption that $A(X)$ is elliptic, symmetric, periodic and Hölder continuous, we establish the solvability of the $L^2$ Dirichlet, regularity, and Neumann problems for $\mathcal{L}_\varepsilon (u_\varepsilon)=0$ in $Ω$ with optimal estimates uniform in $\varepsilon>0$.

math.AP↗
arXiv2009

Homogenization of Elliptic Boundary Value Problems in Lipschitz Domains

In this paper we study the $L^p$ boundary value problems for $\mathcal{L}(u)=0$ in $\mathbb{R}^{d+1}_+$, where $\mathcal{L}=-\text{div}(A\nabla)$ is a second order elliptic operator with real and symmetric coefficients. Assume that $A$ is {\it periodic} in $x_{d+1}$ and satisfies some minimal smoothness condition in the $x_{d+1}$ variable, we show that the $L^p$ Neumann and regularity problems are uniquely solvable for $1<p<2+δ$. We also present a new proof of Dahlberg's theorem on the $L^p$ Dirichlet problem for $2-δ<p< \infty$ (Dahlberg's original unpublished proof is given in the Appendix). As the periodic and smoothness conditions are imposed only on the $x_{d+1}$ variable, these results extend directly from $\mathbb{R}^{d+1}_+$ to regions above Lipschitz graphs. Consequently, by localization techniques, we obtain uniform $L^p$ estimates for the Dirichlet, Neumann and regularity problems on bounded Lipschitz domains for a family of second order elliptic operators arising in the theory of homogenization. The results on the Neumann and regularity problems are new even for smooth domains.

math.AP↗
arXiv2009

The $L^p$ Regularity Problem on Lipschitz Domains

This paper contains two results on the $L^p$ regularity problem on Lipschitz domains. For second order elliptic systems and $1 2$, the solvability of the $L^p$ regularity problem is equivalent to a weak reverse Hölder condition with exponent $p$.

math.AP↗
arXiv2006

A relationship between the Dirichlet and Regularity Problems for elliptic equations

We study the relationship between the solvability of the $L^p$ Dirichlet problem $(D)_p$ and that of the $L^q$ regularity problem $(R)_q$ for second order elliptic equations with bounded measurable coefficients. It is known that the solvability of $(R)_p$ implies that of $(D)_{p^\prime}$. In this note we show that if $(D)_{p^\prime}$ is solvable, then either $(R)_p$ is solvable or $(R)_q$ is not solvable for any $1<q<\infty$.

math.AP↗
arXiv2006

The L^p Boundary Value Problems on Lipschitz Domains

We develop a new approach to the invertibility of the layer potentials on $L^p$ associated with elliptic equations and systems in Lipschitz domains. As a consequence, for $n\ge 4$ and $(2(n-1)/(n+1))-ε<p<2$, we obtain the solvability of the L^p Neumann type boundary value problems for second order elliptic systems. The analogous results for the biharmonic equation are also established.

math.AP↗
arXiv2005

On Estimates of Biharmonic Functions on Lipschitz and Convex Domains

Using Maz'ya type integral identities with power weights, we obtain new boundary estimates for biharmonic functions on Lipschitz and convex domains in $R^n$. For $n\ge 8$, combined with a result in \cite{S2}, these estimates lead to the solvability of the $L^p$ Dirichlet problem for the biharmonic equation on Lipschitz domains for a new range of $p$. In the case of convex domains, the estimates allow us to show that the $L^p$ Dirichlet problem is uniquely solvable for any $2-\e<p<\infty$ and $n\ge 4$.

math.AP↗
arXiv2005

Necessary and Sufficient Conditions for the Solvability of the $L^p$ Dirichlet Problem On Lipschitz Domains

We study the homogeneous elliptic systems of order $2\ell$ with real constant coefficients on Lipschitz domains in $R^n$, $n\ge 4$. For any fixed $p>2$, we show that a reverse Hölder condition with exponent $p$ is necessary and sufficient for the solvability of the Dirichlet problem with boundary data in $L^p$. We also obtain a simple sufficient condition. As a consequence, we establish the solvability of the $L^p$ Dirichlet problem for $n\ge 4$ and $2-\e< p<\frac{2(n-1)}{n-3} +\e$. The range of $p$ is known to be sharp if $\ell\ge 2$ and $4\le n\le 2\ell +1$. For the polyharmonic equation, the sharp range of $p$ is also found in the case $n=6$, 7 if $\ell=2$, and $n=2\ell+2$ if $\ell\ge 3$.

math.AP↗
arXiv2004

The $L^p$ Dirichlet Problem for Elliptic Systems on Lipschitz Domains

We develop a new approach to the $L^p$ Dirichlet problem via $L^2$ estimates and reverse Holder inequalities. We apply this approach to second order elliptic systems and the polyharmonic equation on a bounded Lipschitz domain $Ω$ in $R^n$. For $n\ge 4$ and $2-ε<p<2(n-1)/(n-3)+ε$, we establish the solvability of the Dirichlet problem with boundary value data in $L^p(\partialΩ)$. In the case of the polyharmonic equation $Δ^{\ell}u=0$ with $\ell\ge 2$, the range of $p$ is sharp if $4\le n \le 2\ell +1$.

math.AP↗