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Lihe Wang

Publications and source records attributed to Lihe Wang.

13 recordsLinked to original sources
arXiv2025

Boundary pointwise regularity for the Poisson problem on uniform domain

In this paper, we study the boundary pointwise regularity for the Poisson problem on domains with rough boundaries, specifically uniform domains. In general, it is not straightforward to define weak solutions for non-zero boundary data on such domains. To address this, we introduce a novel definition of weak solutions tailored to the setting of uniform domains. Remarkably, this definition allows for the analysis of the regularity of weak solutions. In particular, by establishing an energy inequality, we prove the boundary pointwise $C^\alpha$ regularity by using compactness methods under the admissible condition. Furthermore, by exploiting the the linear structure of solutions with respective to the harmonic functions, we establish boundary pointwise $C^{1,\alpha}$ and $C^{2,\alpha}$ regularities when the boundary data and the domain boundary are pointwise $C^{1,\alpha}$ and $C^{2,\alpha }$, respectively.

math.AP↗
arXiv2024

Singular metrics with nonnegative scalar curvature and RCD

We show that a uniformly Euclidean metric with isolated singularity on $M^n = T^n \# M_0$, where $4\leq n\leq 7$ or $n\geq 4$, $M_0$ spin, and nonnegative scalar curvature on the smooth part is Ricci flat and extends smoothly over the singularity. This confirms Schoen's Conjecture in these cases. The key to the proof is to show that the space has nonnegative synthetic Ricci curvature, i.e., an $RCD(0, n)$ space. Our result also holds when the singular set consists of a finite union of submanifolds (of possibly different dimensions) intersecting transversally under additional assumption on the co-dimension and the location of the singular set.

math.DG↗
arXiv2024

Interior Second Order H\"{o}lder Regularity for Stokes systems

Global second order H\"{o}lder regularity for Stokes systems can be obtained by global Schauder estimates, which are actually a priori estimates and were established by Solonnikov [20] and [23] with appropriate compatible conditions. This paper will investigate the corresponding interior regularity which unfortunately may fail in general from Serrin's counterexample (cf. [19]). However, we discover interior $C^{2,\alpha}$ regularity for velocity and interior $C^{1,\alpha}$ regularity for pressure in spatial variables, and furthermore, for curl of velocity, we find its gradient belongs to $C^{\alpha, \frac{\alpha}2}$, that is, possesses H\"{o}lder continuity in both space and time directions. The interesting phenomenon here is that no continuity in time variable is assumed for both the coefficients and the righthand side terms. The estimates for velocity and its curl are achieved pointwisely and the results are sharp indicated by a counterexample.

math.AP↗
arXiv2023

$C^{0}$-regularity for solutions of elliptic equations with distributional coefficients

In this paper, the continuity of solutions for elliptic equations in divergence form with distributional coefficients is considered. Inspired by the discussion on necessary and sufficient conditions for the form boundedness of elliptic operators by Maz'ya and Verbitsky (Acta Math., 188, 263-302, 2002 and Comm. Pure Appl. Math., 59, 1286-1329, 2006), we propose two kinds of sufficient conditions, which are some Dini decay conditions and some integrable conditions named Kato class or $K^{1}$ class, to show that the weak solution of the Schrödinger type elliptic equation with distributional coefficients is continuous and give an almost optimal priori estimate. These estimates can clearly show that how the coefficients and nonhomogeneous terms influence the regularity of solutions. The $\ln$-Lipschitz regularity and Hölder regularity are also obtained as corollaries which cover the classical De Giorgi's Hölder estimates.

math.AP↗
arXiv2023

Partial Regularity of Navier-Stokes Equations

We prove, with a more geometric approach, that the solutions to the Navier-Stokes equations are regular up to a set of Hausdorff dimension 1. The main tool for the proof is a new compactness lemma and the monotonicity property of harmonic functions.

math.AP↗
arXiv2022

Interior $L_{p}$ regularity for Stokes systems

A new iteration method is represented to study the interior $L_{p}$ regularity for Stokes systems both in divergence form and in non-divergence form. By the iteration, we improve the integrability of derivatives of solutions for Stokes systems step by step; after infinitely many steps, $L_{p}$ regularity is achieved; and in each step, the maximal function method is used where solutions and their derivatives are involved simultaneously in each scale. The H\"{o}lder continuity of the coefficients in spatial variables is assumed to compensate the different scalings between the solutions and their derivatives.

math.AP↗
arXiv2022

Boundary Lipschitz regularity of solutions for general semilinear elliptic equations in divergence form

In this paper, we study the nonhomogeneous Dirichlet problem concerning general semilinear elliptic equations in divergence form. We establish that the boundary Lipschitz regularity of solutions under some more weaker conditions on the coefficients, the boundary, the boundary function and the nonhomogeneous term. In particular, we assume that the nonhomogeneous term satisfies Dini continuity condition and Lipschitz Newtonian potential condition, which will be the optimal conditions to obtain the boundary Lipschitz regularity of solutions.

math.AP↗
arXiv2021

Boundary Lipschitz regularity of solutions for semilinear elliptic equations in divergence form

In this paper, we consider the pointwise boundary Lipschitz regularity of solutions for the semilinear elliptic equations in divergence form mainly under some weaker assumptions on nonhomogeneous term and the boundary. If the domain satisfies C^{1,\text{Dini}} condition at a boundary point, and the nonhomogeneous term satisfies Dini continuous condition and Lipschitz Newtonian potential condition, then the solution is Lipschitz continuous at this point. Furthermore, we generalize this result to Reifenberg C^{1,\text{Dini}} domains.

math.AP↗
arXiv2020

Pointwise Regularity for Fully Nonlinear Elliptic Equations in General Forms

In this paper, we develop systematically the pointwise regularity for viscosity solutions of fully nonlinear elliptic equations in general forms. In particular, the equations with quadratic growth (called natural growth) in the gradient are covered. We obtain a series of interior and boundary pointwise $C^{k,\alpha}$ regularity ($k\geq 1$ and $0<\alpha<1$). In addition, we also derive the pointwise $C^k$ regularity ($k\geq 1$) and $C^{k,\mathrm{lnL}}$ regularity ($k\geq 0$), which correspond to the end points $\alpha=0$ and $\alpha=1$ respectively. Some regularity results are new even for the linear equations. Moreover, the minimum requirements are imposed to obtain above regularity and our proofs are simple.

math.AP↗
arXiv2016

Inhomogeneous Hopf-Oleĭnik Lemma and Applications. Part IV: Sharp Krylov Boundary Gradient Type Estimates for Solutions to Fully Nonlinear Differential Inequalities with unbounded coefficients and $C^{1,Dini}$ boundary data

In this paper we provide another application of the Inhomogeneous Hopf-Ole\uınik Lemma (IHOL) proved in \cite{BM-IHOL-PartI} or \cite{Boyan-2}. As a matter of fact, we also provide a new and simpler proof of a slightly weaker version IHOL for the uniformly elliptic fully nonlinear case which is sufficient for most purposes. The paper has essentially two parts. In the first part, we use IHOL for unbounded RHS to develop a Caffarelli's "Lipschitz implies $C^{1,α}$" approach to prove Ladyzhenskaya-Uraltseva boundary gradient type estimates for functions in $S^{*}(γ, f)$ that vanishes on the boundary. Here, unbounded RHS means that $f\in L^{q}$ with $q>n$. This extends the celebrated Krylov's boundary gradient estimate proved in \cite{Krylov}. A Phragmén-Lindelöf classification result for solutions in half spaces is recovered from these estimates. Moreover, a Hölder estimate up to the boundary (in the half-ball) for $u(x)/x_{n}$ is obtained. In the second part, we extend the previous results for functions in $S^{*}(γ, σ, f)$ where $γ,f\in L^{q}$ with $q>n$ that have a $C^{1,Dini}$ boundary data on a $W^{2,q}$ domain. Here, we use an "improvement of flatness" strategy suited to the unbounded coefficients scenario. As a consequence of that, a quantitative version of IHOL under pointwise $C^{1,Dini}$ boundary regularity is obtained.

math.AP↗
arXiv2008

Semicircle Law of Vandermonde Ensemble

In the present paper, we give a simple proof of the level density of fixed trace square ensemble.We derive the integral equation of the level density of fixed trace square ensemble.Then we analyze the asymptotic behavior of the level density.

math-ph↗