arXiv subjects

Jinping Zhuge

Publications and source records attributed to Jinping Zhuge.

At least 19 recordsLinked to original sources

arXiv2026

Quantitative homogenization of elliptic equations with infinitely many scales

In this paper, we develop a general homogenization theory for elliptic equations with coefficients that oscillate periodically at infinitely many scales $\varepsilon = (\varepsilon_1, \varepsilon_2, \cdots) \in (0,1)^\infty$, with $\varepsilon_1>\varepsilon_2>\cdots$ and $\varepsilon_n \to 0$ as $n \to \infty$. Such problems arise naturally in the study of fractal materials and diffusion in fluids. Under suitable scale-separation assumptions, we prove a qualitative homogenization theorem and obtain optimal $L^2$ convergence rates. We also establish interior and boundary Lipschitz estimates that are uniform in $\varepsilon$.

math.AP↗

arXiv2026

Large-scale harmonic measures and nontangential maximal functions in periodic homogenization

In this paper, we consider the elliptic operators $\mathcal{L}_\varepsilon = -\nabla\cdot (A(X/\varepsilon) \nabla )$ with periodic coefficients in a bounded domain $\Omega$ without any local smoothness assumption on $A = A(Y)$, where $\varepsilon \ll \text{diam}(\Omega)$ is a microscopic scale. Due to the irregularity of the coefficients at $\varepsilon$ scale, we introduce the correct forms of the large-scale nontangential maximal functions for the Dirichlet, Neumann and regularity problems that measure the behaviors of solutions at an $\varepsilon$ distance away from the boundary. The $L^p$ estimates uniform in $\varepsilon$ are established for these nontangential maximal functions for the same and optimal ranges of $p$ as the Laplace operator in the Lipschitz or $C^1$ domains. With some additional regularity assumption on the coefficients, the large-scale estimates combined with the small-scale estimates recover the classical full-scale estimates of the nontangential maximal functions. Our proofs are based on the notion of large-scale $\mathcal{L}_\varepsilon$-harmonic measures, the periodic structure of operators in the transversal direction to the boundaries, and the homogenization tools, including convergence rates and large-scale regularity.

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arXiv2026

Stability of $L^p$ Dirichlet problem under small bi-Lipschitz transformations of domains

We show that small bi-Lipschitz deformations of a Lipschitz domain (with possibly large Lipschitz constant) preserve the solvability of the Dirichlet problem for the Laplacian with boundary data in $L^p$, for the same value of $p>1$. As a consequence, for all $p\in(1,\infty)$, we obtain the solvability of the $L^p$ Dirichlet problem for small Lipschitz perturbations of convex domains, thereby unifying two fundamentally different settings in which such results were previously known: convex and $C^1$ domains. The key ingredient and novelty of our approach is a construction of a change of variables based on a non-constant basis derived from the Green function, which encodes the geometry of the base domain.

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arXiv2025

Quantitative unique continuation for Neumann problem in planar $C^{1,\alpha}$ domains

In this paper, we study the quantitative unique continuation property of the second-order elliptic operators under the vanishing Neumann boundary condition over $C^{1,\alpha}$ or convex domains in two dimensions. We establish the optimal estimates of the number of critical points, doubling index and the total length of level curves. The key idea is to reduce the Neumann problem to the Dirichlet problem, which has been understood better, by a classical duality between an $A$-harmonic function and its stream function.

math.AP↗

arXiv2025

Optimal convergence rates in multiscale elliptic homogenization

This paper is devoted to the quantitative homogenization of multiscale elliptic operator $-\nabla\cdot A_\varepsilon \nabla$, where $A_\varepsilon(x) = A(x/\varepsilon_1, x/\varepsilon_2,\cdots, x/\varepsilon_n)$, $\varepsilon = (\varepsilon_1, \varepsilon_2,\cdots, \varepsilon_n) \in (0,1]^n$ and $\varepsilon_i > \varepsilon_{i+1}$. We assume that $A(y_1,y_2,\cdots, y_n)$ is 1-periodic in each $y_i \in \mathbb{R}^d$ and real analytic. Classically, the method of reiterated homogenization has been applied to study this multiscale elliptic operator, which leads to a convergence rate limited by the ratios $\max \{ \varepsilon_{i+1}/\varepsilon_i: 1\le i\le n-1\}$. In the present paper, under the assumption of real analytic coefficients, we introduce the so-called multiscale correctors and more accurate effective operators, and improve the ratio part of the convergence rate to $\max \{ e^{-c\varepsilon_{i}/\varepsilon_{i+1}}: 1\le i\le n-1 \}$. This convergence rate is optimal in the sense that $c>0$ cannot be replaced by a larger constant. As a byproduct, the uniform Lipschitz estimate is established under a mild double-log scale-separation condition.

math.AP↗

arXiv2024

Wall laws for viscous flows in 3D randomly rough pipes: optimal convergence rates and stochastic integrability

This paper is concerned with effective approximations and wall laws of viscous laminar flows in 3D pipes with randomly rough boundaries. The random roughness is characterized by the boundary oscillation scale $\varepsilon \ll 1 $ and a probability space with ergodicity quantified by functional inequalities. The results in this paper generalize the previous work for 2D channel flows with random Lipschitz boundaries to 3D pipe flows with random boundaries of John type. Moreover, we establish the optimal convergence rates and substantially improve the stochastic integrability obtained in the previous studies. Additionally, we prove a refined version of the Poiseuille's law in 3D pipes with random boundaries, which seems unaddressed in the literature. Our systematic approach combines several classical and recent ideas (particularly from homogenization theory), including the Saint-Venant's principle for pipe flows, the large-scale regularity theory over rough boundaries, and applications of functional inequalities.

math.AP↗

arXiv2024

Observability inequalities for heat equations with potentials

This paper is mainly concerned with the observability inequalities for heat equations with time-dependent Lipschtiz potentials. The observability inequality for heat equations asserts that the total energy of a solution is bounded above by the energy localized in a subdomain with an observability constant. For a bounded measurable potential $V = V(x,t)$, the factor in the observability constant arising from the Carleman estimate is best known to be $\exp(C\|V\|_{\infty}^{2/3})$ (even for time-independent potentials). In this paper, we show that, for Lipschtiz potentials, this factor can be replaced by $\exp(C(\|\nabla V\|_{\infty}^{1/2} +\|\partial_tV\|_{\infty}^{1/3} ))$, which improves the previous bound $\exp(C\|V\|_{\infty}^{2/3})$ in some typical scenarios. As a consequence, with such a Lipschitz potential, we obtain a quantitative regular control in a null controllability problem. In addition, for the one-dimensional heat equation with some time-independent bounded measurable potential $V = V(x)$, we obtain the optimal observability constant.

math.OC↗

arXiv2024

Convergence rates of eigenvalue problems in perforated domains: the case of small volume

This paper is concerned with the Dirichlet eigenvalue problem for Laplace operator in a bounded domain with periodic perforation in the case of small volume. We obtain the optimal quantitative error estimates independent of the spectral gaps for an asymptotic expansion, with two leading terms, of Dirichlet eigenvalues. We also establish the convergence rates for the corresponding eigenfunctions. Our approach uses a known reduction to a degenerate elliptic eigenvalue problem for which a quantitative analysis is carried out.

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arXiv2024

Uniform Calder\'{o}n-Zygmund estimates in multiscale elliptic homogenization

This paper is concerned with the elliptic equation $-\text{div} (A_\varepsilon \nabla u_\varepsilon) = \text{div} f$ in a bounded $C^1$ domain, where $A_\varepsilon$ takes a form of $A_\varepsilon(x) = A(x/\varepsilon_1, x/\varepsilon_2,\cdots, x/\varepsilon_n)$, with $A(y_1,y_2,\cdots,y_n)$ being 1-periodic in each $y_i$. We prove the uniform Calder\'{o}n-Zygmund estimate, namely, the uniform $L^p$ boundedness of the linear map $f\mapsto \nabla u_\varepsilon$ for any $p\in (1,\infty)$ with a constant independent of small parameters $(\varepsilon_1,\varepsilon_2,\cdots, \varepsilon_n) \in (0,1]^n$. Our result includes the uniform Calder\'{o}n-Zygmund estimate in quasiperiodic elliptic homogenization (even without the Diophantine condition), which was previously unknown. The proof novelly combines the Dirichlet's theorem on the simultaneous Diophantine approximation from number theory, a technique of reperiodization, reiterated periodic homogenization and a large-scale real-variable argument. Using the idea of reperiodization, we also obtain some large-scale or mesoscopic-scale Lipschitz estimates.

math.AP↗

arXiv2024

The sharp estimate of nodal sets for Dirichlet Laplace eigenfunctions in polytopes

Let $P$ be a bounded $n$-dimensional Lipschitz polytope, and let $φ_λ$ be a Dirichlet Laplace eigenfunction in $P$ corresponding to the eigenvalue $λ$. We show that the $(n-1)$-dimensional Hausdorff measure of the nodal set of $φ_λ$ does not exceed $C(P)\sqrtλ$. Our result extends the previous ones in quaisconvex domains (including $C^1$ and convex domains) to general polytopes that are not necessarily quasiconvex.

math.AP↗

arXiv2023

Uniform regularity for degenerate elliptic equations in perforated domains

This paper is concerned with a class of degenerate elliptic equations with rapidly oscillating coefficients in periodically perforated domains, which arises in the study of spectrum problems for uniformly elliptic equations in perforated domains. We establish a quantitative convergence rate and obtain the uniform weighted Lipschitz and $W^{1,p}$ estimates.

math.AP↗

arXiv2023

Nodal sets of Dirichlet eigenfunctions in quasiconvex Lipschitz domains

We introduce the class of quasiconvex Lipschitz domains, which covers both $C^1$ and convex domains, to the study of boundary unique continuation for elliptic operators. In particular, we prove the upper bound of the size of nodal sets for Dirichlet eigenfunctions of general elliptic equations in bounded quasiconvex Lipschitz domains. Our result is new even for Laplace operator in convex domains.

math.AP↗

arXiv2023

Spectral inequality for Schrödinger equations with power growth potentials

We prove a spectral inequality for Schrödinger equations with power growth potentials, which particularly confirms a conjecture in \cite{DSV}. This spectral inequality depends on the decaying density of the sensor sets, and the growth rate of potentials. The proof relies on three-ball inequalities derived from modified versions of quantitative global and local Carleman estimates that take advantage of the gradient information of the potentials.

math.AP↗

arXiv2022

Higher-order boundary layers and regularity for Stokes systems over rough boundaries

In this paper, we study the large-scale boundary regularity for the Stokes system in periodically oscillating John domains. Our main contribution is the construction of boundary layer correctors of arbitrary order. This is a significant generalization of the known results restricted to the first and second orders. As an application, we prove the large-scale regularity estimate, as well as a Liouville theorem, of arbitrary order for the Stokes system. Our results are also related to higher-order boundary layer tails and wall laws in viscous fluids over rough boundaries.

math.AP↗

arXiv2021

Compactness and stable regularity in multiscale homogenization

In this paper we develop some new techniques to study the multiscale elliptic equations in the form of $-\text{div} \big(A_\varepsilon \nabla u_{\varepsilon} \big) = 0$, where $A_\varepsilon(x) = A(x, x/\varepsilon_1,\cdots, x/\varepsilon_n)$ is an $n$-scale oscillating periodic coefficient matrix, and $(\varepsilon_i)_{1\le i\le n}$ are scale parameters. We show that the $C^α$-Hölder continuity with any $α\in (0,1)$ for the weak solutions is stable, namely, the constant in the estimate is uniform for arbitrary $(\varepsilon_1, \varepsilon_2, \cdots, \varepsilon_n) \in (0,1]^n$ and particularly is independent of the ratios between $\varepsilon_i$'s. The proof uses an upgraded method of compactness, involving a scale-reduction theorem by $H$-convergence. The Lipschitz estimate for arbitrary $(\varepsilon_i)_{1\le i\le n}$ still remains open. However, for special laminate structures, i.e., $A_\varepsilon(x) = A(x,x_1/\varepsilon_1, \cdots, x_d/\varepsilon_n)$, we show that the Lipschitz estimate is stable for arbitrary $(\varepsilon_1, \varepsilon_2, \cdots, \varepsilon_n) \in (0,1]^n$. This is proved by a technique of reperiodization.

math.AP↗

arXiv2021

Large-scale regularity for the stationary Navier-Stokes equations over non-Lipschitz boundaries

In this paper we address the large-scale regularity theory for the stationary Navier-Stokes equations in highly oscillating bumpy John domains. These domains are very rough, possibly with fractals or cusps, at the microscopic scale, but are amenable to the mathematical analysis of the Navier-Stokes equations. We prove: (i) a large-scale Calderón-Zygmund estimate, (ii) a large-scale Lipschitz estimate, (iii) large-scale higher-order regularity estimates, namely, $C^{1,γ}$ and $C^{2,γ}$ estimates. These nice regularity results are inherited only at mesoscopic scales, and clearly fail in general at the microscopic scales. We emphasize that the large-scale $C^{1,γ}$ regularity is obtained by using first-order boundary layers constructed via a new argument. The large-scale $C^{2,γ}$ regularity relies on the construction of second-order boundary layers, which allows for certain boundary data with linear growth at spatial infinity. To the best of our knowledge, our work is the first to carry out such an analysis. In the wake of many works in quantitative homogenization, our results strongly advocate in favor of considering the boundary regularity of the solutions to fluid equations as a multiscale problem, with improved regularity at or above a certain scale.

math.AP↗

arXiv2021

Doubling inequalities and nodal sets in periodic elliptic homogenization

We prove explicit doubling inequalities and obtain uniform upper bounds (under $(d-1)$-dimensional Hausdorff measure) of nodal sets of weak solutions for a family of linear elliptic equations with rapidly oscillating periodic coefficients. The doubling inequalities, explicitly depending on the doubling index, are proved at different scales by a combination of convergence rates, a three-ball inequality from certain "analyticity", and a monotonicity formula of a frequency function. The upper bounds of nodal sets are shown by using the doubling inequalities, approximations by harmonic functions and an iteration argument.

math.AP↗

arXiv2021

Regularity theory of elliptic systems in $\varepsilon$-scale flat domains

We consider the linear elliptic systems or equations in divergence form with periodically oscillating coefficients. We prove the large-scale boundary Lipschitz estimate for the weak solutions in domains satisfying the so-called $\varepsilon$-scale flatness condition, which could be arbitrarily rough below $\varepsilon$-scale. This particularly generalizes Kenig and Prange's work in [32] and [33] by a quantitative approach.

math.AP↗