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Hongjie Dong

Publications and source records attributed to Hongjie Dong.

At least 55 records · Page 3Linked to original sources
arXiv2024

Optimal higher derivative estimates for solutions of the Lamé system with closely spaced hard inclusions

We investigate higher derivative estimates for the Lamé system with hard inclusions embedded in a bounded domain in $\mathbb{R}^{d}$. As the distance $\varepsilon$ between two closely spaced hard inclusions approaches zero, the stress in the narrow regions between the inclusions increases significantly. This stress is captured by the gradient of the solution. The key contribution of this paper is a detailed characterization of this singularity, achieved by deriving higher derivative estimates for solutions to the Lamé system with partially infinite coefficients. These upper bounds are shown to be sharp in two and three dimensions when the domain exhibits certain symmetries. To the best of our knowledge, this is the first work to precisely quantify the singular behavior of higher derivatives in the Lamé system with hard inclusions.

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arXiv2024

Gradient estimates for the conductivity problem with imperfect bonding interfaces

We study the field concentration phenomenon between two closely spaced perfect conductors with imperfect bonding interfaces of low conductivity type. The boundary condition on these interfaces is given by a Robin-type boundary condition. We discover a \textit{new} dichotomy for the field concentration depending on the bonding parameter $γ$. Specifically, we show that the gradient of solution is uniformly bounded independent of $\varepsilon$ (the distance between two inclusions) when $γ$ is sufficiently small. However, the gradient may blow up when $γ$ is large. Moreover, we identify the threshold of $γ$ and the optimal blow-up rates under certain symmetry assumptions. The proof relies on a crucial anisotropic gradient estimate in the thin neck between two inclusions. We develop a general framework for establishing such estimate, which is applicable to a wide range of elliptic equations and boundary conditions.

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arXiv2024

Schauder type estimates for degenerate or singular elliptic equations with DMO coefficients

In this paper, we study degenerate or singular elliptic equations in divergence form $$-\text{div}(x_n^αA\nabla u)=\text{div}(x_n^α\mathbf{g})\quad\text{in }B_1\cap\{x_n>0\}.$$ When $α>-1$, we establish boundary Schauder type estimates under the conormal boundary condition on the flat boundary, provided that the coefficients satisfy Dini mean oscillation (DMO) type conditions. Additionally, as an application, we derive higher-order boundary Harnack principles for uniformly elliptic equations in divergence form with DMO coefficients.

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arXiv2024

Elliptic and parabolic equations with rough boundary data in Sobolev spaces with degenerate weights

We investigate the inhomogeneous boundary value problem for elliptic and parabolic equations in divergence form in the half space $\{x_d > 0\}$, where the coefficients are measurable, singular or degenerate, and depend only on $x_d$. The boundary data are considered in Besov spaces of distributions with negative orders of differentiability in the range $(-1,0]$. The solution spaces are weighted Sobolev spaces with power weights that decay rapidly near the boundary, and are outside the Muckenhoupt $A_p$ class. Sobolev spaces with such weights contain functions that are very singular near the boundary and do not possess a trace on the boundary. Consequently, solutions may not exist for arbitrarily prescribed boundary data and right-hand sides of the equations. We establish a natural structural condition on the right-hand sides of the equations under which the boundary value problem is well-posed.

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arXiv2024

Global BMO-Sobolev Estimates for Second-Order Linear Elliptic Equations on Lipschitz Domains

Let $n \ge 2$ and $Ω\subset \mathbb{R}^n$ be a bounded Lipschitz domain. In this article, we establish first-order global regularity estimates in the scale of BMO spaces on $Ω$ for weak solutions to the second-order elliptic equation $\mathrm{div}(A \nabla u) = \mathrm{div} , \boldsymbol{f}$ in $Ω$. This is achieved under minimal regularity assumptions on $Ω$ and the coefficient matrix $A$, utilizing the pointwise multiplier characterization of the BMO space on $Ω$. As an application, we also obtain global estimates of $\nabla u$ in the Lebesgue space $L^1(Ω)$ when $\boldsymbol{f}$ belongs to the Hardy space on $Ω$.

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arXiv2024

Log-type ultra-analyticity of elliptic equations with gradient terms

It is well known that every solution of an elliptic equation is analytic if its coefficients are analytic. However, less is known about the ultra-analyticity of such solutions. This work addresses the problem of elliptic equations with lower-order terms, where the coefficients are entire functions of exponential type. We prove that every solution satisfies a quantitative logarithmic ultra-analytic bound and demonstrate that this bound is sharp. The results suggest that the ultra-analyticity of solutions to elliptic equations cannot be expected to achieve the same level of ultra-analyticity as the coefficients.

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arXiv2024

Higher order parabolic systems with piecewise DMO and Hölder continuous coefficients

In this paper, we are concerned with divergence form, higher-order parabolic systems in a cylindrical domain with a finite number of subdomains. We establish $L_\infty$ and Schauder estimates of solutions when the leading coefficients and the non-homogeneous term exhibit piecewise Dini mean oscillation and piecewise Hölder continuity, respectively. To the best of our knowledge, our results are new for higher-order elliptic and parabolic systems.

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arXiv2024

The Local-well-posedness of the relativistic Vlasov-Maxwell-Landau system with the specular reflection boundary condition

We prove the local-in-time well-posedness of the relativistic Vlasov-Maxwell-Landau system in a bounded domain $Ω$ with the specular reflection condition. Our result covers the case when $Ω$ is a non-convex domain, e.g., solid torus. To the best of our knowledge, this is the first local well-posedness result for a nonlinear kinetic model with a self-consistent magnetic effect in a three-dimensional $\textbf{bounded}$ domain.

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arXiv2024

Global well-posedness for supercritical SQG with perturbations of radially symmetric data

We study the global well-posedness of the supercritical dissipative surface quasi-geostrophic (SQG) equation, a key model in geophysical fluid dynamics. While local well-posedness is known, achieving global well-posedness for large initial data remains open. Motivated by enhanced decay in radial solutions, we aim to establish global well-posedness for small perturbations of potentially large radial data. Our main result shows that for small perturbations of radial data, the SQG equation admits a unique global solution.

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arXiv2023

Asymptotics of the solution to the perfect conductivity problem with $p$-Laplacian

We study the perfect conductivity problem with closely spaced perfect conductors embedded in a homogeneous matrix where the current-electric field relation is the power law $J=σ|E|^{p-2}E$. The gradient of solutions may be arbitrarily large as $\varepsilon$, the distance between inclusions, approaches to 0. To characterize this singular behavior of the gradient in the narrow region between two inclusions, we capture the leading order term of the gradient. This is the first gradient asymptotics result on the nonlinear perfect conductivity problem.

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arXiv2023

On higher regularity of Stokes systems with piecewise Hölder continuous coefficients

In this paper, we consider higher regularity of a weak solution $({\bf u},p)$ to stationary Stokes systems with variable coefficients. Under the assumptions that coefficients and data are piecewise $C^{s,δ}$ in a bounded domain consisting of a finite number of subdomains with interfacial boundaries in $C^{s+1,μ}$, where $s$ is a positive integer, $δ\in (0,1)$, and $μ\in (0,1]$, we show that $D{\bf u}$ and $p$ are piecewise $C^{s,δ_μ}$, where $δ_μ=\min\big\{\frac{1}{2},μ,δ\big\}$. Our result is new even in the 2D case with piecewise constant coefficients.

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arXiv2023

Nondivergence form degenerate linear parabolic equations on the upper half space

We study a class of nondivergence form second-order degenerate linear parabolic equations in $(-\infty, T) \times {\mathbb R}^d_+$ with the homogeneous Dirichlet boundary condition on $(-\infty, T) \times \partial {\mathbb R}^d_+$, where ${\mathbb R}^d_+ = \{x =(x_1,x_2,\ldots, x_d) \in {\mathbb R}^d\,:\, x_d>0\}$ and $T\in {(-\infty, \infty]}$ is given. The coefficient matrices of the equations are the product of $μ(x_d)$ and bounded positive definite matrices, where $μ(x_d)$ behaves like $x_d^α$ for some given $α\in (0,2)$, which are degenerate on the boundary $\{x_d=0\}$ of the domain. The divergence form equations in this setting were studied in [14]. Under a partially weighted VMO assumption on the coefficients, we obtain the wellposedness and regularity of solutions in weighted Sobolev spaces. Our research program is motivated by the regularity theory of solutions to degenerate viscous Hamilton-Jacobi equations.

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arXiv2023

Optimal estimates for transmission problems including relative conductivities with different signs

We study the gradient and higher order derivative estimates for the transmission problem in the presence of closely located inclusions. We show that in two dimensions, when relative conductivities of circular inclusions have different signs, the gradient and higher order derivatives are bounded independent of $\varepsilon$, the distance between the inclusions. We also show that for general smooth strictly convex inclusions, when one inclusion is an insulator and the other one is a perfect conductor, the derivatives of any order is bounded independent of $\varepsilon$ in any dimensions $n \ge 2$.

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arXiv2023

Sobolev estimates for singular-degenerate quasilinear equations beyond the $A_2$ class

We study a conormal boundary value problem for a class of quasilinear elliptic equations in bounded domain $Ω$ whose coefficients can be degenerate or singular of the type $\text{dist}(x, \partial Ω)^α$, where $\partial Ω$ is the boundary of $Ω$ and $α\in (-1, \infty)$ is a given number. We establish weighted Sobolev type estimates for weak solutions under a smallness assumption on the weighted mean oscillations of the coefficients in small balls. Our approach relies on a perturbative method and several new Lipschitz estimates for weak solutions to a class of singular-degenerate quasilinear equations.

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arXiv2023

The insulated conductivity problem with $p$-Laplacian

We study the insulated conductivity problem with closely spaced insulators embedded in a homogeneous matrix where the current-electric field relation is the power law $J = |E|^{p-2}E$. The gradient of solutions may blow up as $\varepsilon$, the distance between insulators, approaches to 0. In 2D, we prove an upper bound of the gradient to be of order $\varepsilon^{-α}$, where $α= 1/2$ when $p \in(1,3]$ and any $α> 1/(p-1)$ when $p > 3$. We provide examples to show that this exponent is almost optimal. In dimensions $n \ge 3$, we prove an upper bound of order $\varepsilon^{-1/2 + β}$ for some $β> 0$, and show that $β\nearrow 1/2$ as $n \to \infty$.

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arXiv2023

Time fractional parabolic equations with partially SMO coefficients

We present the unique solvability in Sobolev spaces of time fractional parabolic equations in divergence and non-divergence forms. The leading coefficients are merely measurable in $(t,x_1)$ for $a^{ij}$, $1 \leq i,j \leq d$, $(i,j) \neq (1,1)$. The coefficient $a^{11}$ is merely measurable locally either in $t$ or $x_1$. As functions of the remaining variables, the coefficients have small mean oscillations. We consider mixed norm Sobolev spaces with Muckenhoupt weights. Our results generalize previous work on parabolic equations with time fractional derivatives to a much larger class of coefficients and solution spaces.

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arXiv2022

Jacobian determinants for (nonlinear) gradient of planar $\infty$-harmonic functions and applications

In dimension 2, we introduce a distributional Jacobian determinant $\det DV_β(Dv)$ for the nonlinear complex gradient $(x_1,x_2)\mapsto |Dv|^β(v_{x_1},-v_{x_2})$ for any $β>-1$, whenever $v\in W^{1,2 }_{\text{loc}}$ and $β|Dv|^{1+β}\in W^{1,2}_{\text{loc}}$. Then for any planar $\infty$-harmonic function $u$, we show that such distributional Jacobian determinant is a nonnegative Radon measure with some quantitative local lower and upper bounds. We also give the following two applications. (i) Applying this result with $β=0$, we develop an approach to build up a Liouville theorem, which improves that of Savin [33]. Precisely, if $u$ is $\infty$-harmonic functions in whole ${\mathbb R}^2$ with $$ \liminf_{R\to\infty}\inf_{c\in\mathbb R}\frac1 {R^3}\int_{B(0,R)}|u(x)-c|\,dx<\infty,$$ then $u=b+a\cdot x$ for some $b\in{\mathbb R}$ and $a\in{\mathbb R}^2$. (ii) Denoting by $u_p$ the $p$-harmonic function having the same nonconstant boundary condition as $u$, we show that $\det DV_β(Du_p) \to \det DV_β(Du)$ as $p\to\infty$ in the weak-$\star$ sense in the space of Radon measure. Recall that $V_β(Du_p)$ is always quasiregular mappings, but $V_β(Du)$ is not in general.

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