arXiv subjects

Peihao Zhao

Publications and source records attributed to Peihao Zhao.

8 recordsLinked to original sources

arXiv2020

Quantitative estimates for homogenization of nonlinear elliptic operators in perforated domains

This paper was devoted to study the quantitative homogenization problems for nonlinear elliptic operators in perforated domains. We obtained a sharp error estimate $O(\varepsilon)$ when the problem was anchored in the reference domain $\varepsilonω$. If concerning a bounded perforated domain, one will see a bad influence from the boundary layers, which leads to the loss of the convergence rate by $O(\varepsilon^{1/2})$. Equipped with the error estimates, we developed both interior and boundary Lipschitz estimates at large-scales. As an application, we received the so-called quenched Calderón-Zygumund estimates by Shen's real arguments. To overcome some difficulties, we improved the extension theory from (\cite[Theorem 4.3]{OSY}) to $L^p$-versions with $\frac{2d}{d+1}-ε<p<\frac{2d}{d-1}+ε$ and $0<ε\ll1$. Appealing to this, we established Poincaré-Sobolev inequalities of local type on perforated domains. Some of results in the present literature are new even for related linear elliptic models.

math.AP↗

arXiv2020

suboptimal error estimates for homogenization of linear elasticity systems on perforated domains

In the present work, we established almost-sharp error estimates for linear elasticity systems in periodically perforated domains. The first result was $L^{\frac{2d}{d-1-τ}}$-error estimates $O\big(\varepsilon^{1-\fracτ{2}}\big)$ with $0<τ<1$ for a bounded smooth domain. It followed from weighted Hardy-Sobolev's inequalities and a suboptimal error estimate for the square function of the first-order approximating corrector (which was earliest investigated by C. Kenig, F. Lin, Z. Shen \cite{KLS} under additional regularity assumption on coefficients). The new approach relied on the weighted quenched Calderón-Zygmund estimate (initially appeared in A. Gloria, S. Neukamm, F. Otto's work \cite{Gloria_Neukamm_Otto_2015} for a quantitative stochastic homogenization theory). The second effort was $L^2$-error estimates $O\big(\varepsilon^{\frac{5}{6}}\ln^{\frac{2}{3}}(1/\varepsilon)\big)$ for a Lipschitz domain, followed from a new duality scheme coupled with interpolation inequalities. Also, we developed a new weighted extension theorem for perforated domains, and a real method imposed by Z. Shen \cite{S3} played a fundamental role in the whole project.

math.AP↗

arXiv2020

Local Morrey estimate in Musielak-Orlicz-Sobolev space

Under appropriate assumptions on the $N(Ω)$-fucntion, Morrey estimate is presented in the Musielak-Orlicz-Sobolev space.The assumptions include a new increasing condition on the $x$-derivative of the Young complementary function of the $N(Ω)$-fucntion. The conclusion applies to several important nonlinear examples frequently appeared in mathematical literature.

math.FA↗

arXiv2018

Semiclassical states of a linearly coupled critical fractional Schrödinger system

This paper focuses on the linearly coupled critical fractional Schrödinger system \begin{equation*} \begin{cases} ε^{2s}(-\triangle)^s u +a(x)u=u^p+λv\quad &\text{in}\ \mathbb{R}^N,\\ ε^{2s}(-\triangle)^s v +b(x)v=v^{2_s^*-1}+λu\quad &\text{in}\ \mathbb{R}^N, \end{cases} \end{equation*} where $N>2s,$ $s\in(0,1),$ $p\in(1,2_s^*),$ $ε$ and $λ$ are positive parameters, $a,b\in C{(\mathbb{R}^N)}$ are positive potentials, and $(-\triangle)^s$ is the fractional Laplacian operator. Under certain assumptions on $a$ and $λ,$ we obtain the existence, decay estimates and concentration property of positive vector ground states for small $ε.$ Furthermore, under an additional assumption on potentials $a$ and $b$, we consider the multiplicity of positive vector solutions for small $ε$, which turn out to have similar decay estimate and concentration property to those of the ground state for small $ε$.

math.AP↗

arXiv2018

Convergence rates on periodic homogenization of p-Laplace type equations

In this paper, we find some error estimates for periodic homogenization of p-Laplace type equations under the same structure assumption on homogenized equations. The main idea is that by adjusting the size of the difference quotient of the correctors to make the convergence rate visible. In order to reach our goal, the corresponding flux corrector with some properties are developed. Meanwhile, the shift-arguments is in fact applied down to $\varepsilon$ scale, which leads to a new weighted type inequality for smoothing operator with the weight satisfying Harnack's inequality in small scales. As a result, it is possible to develop some large-scale estimates. We finally mention that our approach brought in a systematic error (this phenomenon will disappear in linear and non-degenerated cases), which was fortunately a quantity $o(\varepsilon)$ here.

math.AP↗

arXiv2018

Quantitative Estimates on Periodic Homogenization of Nonlinear Elliptic Operators

In this paper, we are interested in the periodic homogenization of quasilinear elliptic equations. We obtain error estimates $O(\varepsilon^{1/2})$ for a $C^{1,1}$ domain, and $O(\varepsilon^σ)$ for a Lipschitz domain, in which $σ\in(0,1/2)$ is close to zero. Based upon the convergence rates, an interior Lipschitz estimate, as well as a boundary Hölder estimate can be developed at large scales without any smoothness assumption, and these will implies reverse Hölder estimates established for a $C^1$ domain. By a real method developed by Z.Shen \cite{S3}, we consequently derive a global $W^{1,p}$ estimate for $2\leq p<\infty$. This work may be regarded as an extension of \cite{MAFHL,S5} to a nonlinear operator, and our results may be extended to the related Neumann boundary problems without any real difficulty.

math.AP↗

arXiv2018

The Methods of Layer Potentials for General Elliptic Homogenization Problems in Lipschitz Domains

In terms of layer potential methods, this paper is devoted to study the $L^2$ boundary value problems for nonhomogeneous elliptic operators with rapidly oscillating coefficients in a periodic setting. Under a low regularity assumption on the coefficients, we establish the solvability for Dirichlet, regular and Neumann problems in a bounded Lipschitz domain, as well as, the uniform nontangential maximal function estimates and square function estimates. The main difficulty is reflected in two aspects: (i) we can not treat the lower order terms as a compact perturbation to the leading term due to the low regularity assumption; (ii) the nonhomogeneous systems do not possess a scaling-invariant property in general. Although this work may be regarded as a follow-up to C. Kenig and Z. Shen's in \cite{SZW24}, we make an effort to find a clear way of how to handle the nonhomogeneous operators by using the known results of the homogenous ones. Also, we mention that the periodicity condition plays a key role in the scaling-invariant estimates.

math.AP↗

arXiv2017

Hölder continuity for nonlinear elliptic problem in Musielak-Orlicz-Sobolev space

Under appropriate assumptions on the $N(Ω)$-fucntion, the De Giorgi process is presented in the framework of Musielak-Orlicz-Sobolev space to prove the Hölder continuity of fully nonlinear elliptic problems. As the applications, the Hölder continuity of the minimizers for a class of the energy functionals in Musielak-Orlicz-Sobolev spaces is proved; and furthermore, the Hölder continuity of the weak solutions for a class of fully nonlinear elliptic equations is provided.

math.AP↗