arXiv subjects

Hongjie Dong

Publications and source records attributed to Hongjie Dong.

At least 19 recordsLinked to original sources

arXiv2026

Regular boundary points and the Dirichlet problem for elliptic equations in double divergence form

We study the Dirichlet problem for second-order elliptic operators in double divergence form, which arise as formal adjoints of non-divergence form operators and include the stationary Fokker-Planck-Kolmogorov equation. Assuming that the leading coefficients have Dini mean oscillation and that the lower-order coefficients satisfy natural integrability conditions, we construct the Perron solution in arbitrary bounded domains. We prove that a boundary point is regular with respect to the operator if and only if it satisfies the classical Wiener criterion for the Laplacian. In particular, the Dirichlet problem is uniquely solvable in every bounded domain that is regular for the Laplacian.

math.AP↗

arXiv2026

Weighted mixed-norm estimates for fractional parabolic equations with space-time nonlocal operators

We establish weighted mixed-norm estimates for fractional parabolic equations \begin{equation*} \partial_t^αu=Lu-λu+f \text{ in } (0,T)\times\mathbb{R}^d, \end{equation*} with nonlocal operators in both time and space. Here, $\partial_t^α$ is the Caputo derivative of order $α\in(0,1)$, and $L$ is a spatially nonlocal operator of order $σ\in(0,2)$ whose kernel is merely measurable in time. We also obtain the corresponding estimates in the odd mixed-norm spaces, where the inner integration is taken in time and the outer one in space. The estimates are robust in the limit $α\to1$ and $σ\to2$. We also establish unique solvability when either $T<\infty$ or $λ>0$. The proof is based on a direction-by-direction extension argument.

math.AP↗

arXiv2026

Several counterexamples to kinetic Schauder estimates

In this paper, we construct counterexamples to the Schauder estimate for kinetic Fokker--Planck equations, disproving \cite[Conjecture~1.3]{HW}. The conjectured estimate would control second velocity derivatives using only velocity Hölder regularity of the coefficients and forcing, with no spatial regularity. We also give a zero-forcing variant, where the same {conditions are imposed on} a bounded zeroth-order coefficient. Finally, in dimensions $d\geq 2$, we give endpoint examples showing that bounded velocity-independent forcing may produce bounded distributional stationary solutions whose second velocity derivatives are not locally bounded.

math.AP↗

arXiv2026

Sharp Logarithmic Ultra-analyticity for Fractional and Nonlocal Elliptic Equations

It is well known that solutions of elliptic equations inherit analyticity from analytic coefficients, while much less is understood about the inheritance of ultra-analytic regularity, especially for nonlocal equations. This paper develops a systematic Fourier-analytic framework to study fractional and more general nonlocal pure-potential equations whose potentials satisfy ultra-analytic derivative bounds. We prove sharp quantitative logarithmic ultra-analytic estimates for normalized solutions, and show that both the logarithmic power and the leading constant involving the fractional exponent are optimal in natural periodic model examples. We also establish a general transfer principle for weighted ultra-analytic scales, which reveals why standard scales are not preserved, and singles out a natural family of invariant ultra-analytic spaces.

math.AP↗

arXiv2026

Singular-degenerate parabolic systems with the conormal boundary condition on the upper half space

We prove the well-posedness and regularity of solutions in mixed-norm weighted Sobolev spaces for a class of second-order parabolic and elliptic systems in divergence form in the half-space $\mathbb{R}^d_+ = \{x_d > 0\}$ subject to the conormal boundary condition. Our work extends results previously available for scalar equations to the case of systems of equations. The leading coefficients are the product of $x_d^α$ and bounded non-degenerate matrices, where $α\in (-1,\infty)$. The leading coefficients are assumed to be merely measurable in the $x_d$ variable, and to have small mean oscillations in small cylinders with respect to the other variables. If the parameter $α>0$, the lower-order coefficients are allowed to blow-up near the boundary. Our results readily generalize to infinite-dimensional equations in general real and complex Hilbert spaces.

math.AP↗

arXiv2026

Global well-posedness of the one-phase Muskat problem with surface tension

In this paper, we establish the global well-posedness of the one-phase Muskat problem with surface tension for small initial data. This problem describes the motion of the interface separating a wet region from a dry region within a porous medium, a process governed by Darcy's law. Although physically essential, the inclusion of surface tension introduces an additional challenge. We prove that if the initial free boundary is sufficiently small in $H^s$, $s>d/2+1$, then the problem admits a unique global strong solution. Moreover, the solution converges to zero in Lipschitz norm as $t\rightarrow\infty$. To the best of our knowledge, this work constitutes the first global well-posedness result for the one-phase Muskat problem with surface tension.

math.AP↗

arXiv2026

Nontangential Maximal Function estimates for the elliptic Mixed Boundary Value Problem with variable coefficients

We consider an elliptic operator $L$ with variable, merely bounded, and measurable coefficients on a Lipschitz domain, and study solutions to $Lu=0$ that attain given Neumann and Dirichlet-regularity data on different parts of the boundary. The boundary data lies in $L^p$ or $W^{1,p}$ respectively, and we show nontangential maximal function estimates of the gradient of the solution. This mixed boundary value problem generalizes the pure Dirichlet, regularity, and Neumann problem with rough boundary data in $L^p$, and the already established mixed boundary value problem for the Laplacian.

math.AP↗

arXiv2026

On nondivergence form linear parabolic and elliptic equations with degenerate coefficients

We establish the unique solvability in weighted mixed-norm Sobolev spaces for a class of degenerate parabolic and elliptic equations in the upper half space. The operators are in nondivergence form, with the leading coefficients given by $x_d^2a_{ij}$, where $a_{ij}$ is bounded, uniformly nondegenerate, and measurable in $(t,x_d)$ except $a_{dd}$, which is measurable in $t$ or $x_d$. In the remaining spatial variables, they have weighted small mean oscillations. In addition, we investigate the optimality of the function spaces associated with our results.

math.AP↗

arXiv2026

Recent developments on elliptic equations from composites

When inclusions in a composite are separated by a very small gap, high contrast between the inclusion and matrix properties can induce strong amplification of the underlying field inside the narrow region. Quantifying this field concentration phenomenon is important both for the theory of composite materials and for practical applications. This survey reviews substantial progress over the past three decades. In particular, we survey a set of elliptic equations and systems for which optimal estimates or sharp asymptotic characterizations have been obtained, and we highlight several interesting open questions.

math.AP↗

arXiv2026

Serrin's overdetermined theorem within Lipschitz domains

Let $Ω\subset\mathbb R^n$ be a Lipschitz domain. We prove that, $Ω$ satisfies the following Serrin-type overdetermined system $$u \in W^{1,2}(\mathbb R^n), \quad u=0\ \text{ a.e. in }\mathbb R^n\setminus Ω,\quad Δu=\mathbf{c}\mathscr{H}^{n-1}|_{\partial^*Ω} - \mathbf{1}_Ω\,dx,$$ in the weak sense if and only if $Ω$ is a ball. Here $\mathscr H^{n-1}$ denotes the $(n-1)$-dimensional Hausdorff measure. Moreover, a generalization of our method in the anisotropic setting is discussed. Our approach offers an alternative proof to [15] in the case of Lipschitz domains, introducing a novel viewpoint to settle [18, Question 7.1].

math.AP↗

arXiv2026

Remarks on the convex integration technique applied to singular stochastic partial differential equations

Singular stochastic partial differential equations informally refer to the partial differential equations with rough random force that leads to the products in the nonlinear terms becoming ill-defined. Besides the theories of regularity structures and paracontrolled distributions, the technique of convex integration has emerged as a possible approach to construct a solution to such singular stochastic partial differential equations. We review recent developments in this area, and also demonstrate that an application of the convex integration technique to prove non-uniqueness seems unlikely for a particular singular stochastic partial differential equation, specifically the $Φ^{4}$ model from quantum field theory.

math.PR↗

arXiv2026

Gradient estimates for the $p$-Laplacian perfect conductivity problem with partially flat and $C^{1,γ}$ inclusions

In this paper, we investigate the gradient estimates for solutions to the perfect conductivity problem with two closely spaced perfect conductors embedded in a homogeneous matrix, modeled by $p$-Laplacian elliptic equations. We first prove that the gradient of the solution remains bounded when the conductors possess partially ``flat" boundaries. This contrasts with the case involving strictly convex inclusions, where the gradient can blow up. Second, for conductors with $C^{1,γ}$ boundaries ($γ\in(0,1)$), we establish both upper and lower bounds on the gradient, with optimal blow-up rates. Furthermore, we provide precise asymptotic expansions in some special cases.

math.AP↗

arXiv2025

$L_p$-estimates for nonlocal equations with general Lévy measures

We consider nonlocal operators of the form \begin{equation*} L_t u(x) = \int_{\mathbb{R}^d} \left( u(x+y)-u(x)-\nabla u(x)\cdot y^{(σ)} \right) ν_t(dy), \end{equation*} where $ν_t$ is a general Lévy measure of order $σ\in(0,2)$. We allow this class of Lévy measures to be very singular and impose no regularity assumptions in the time variable. Continuity of the operators and the unique strong solvability of the corresponding nonlocal parabolic equations in $L_p$ spaces are established. We also demonstrate that, depending on the ranges of $σ$ and $d$, the operator can or cannot be treated in weighted mixed-norm spaces.

math.AP↗

arXiv2025

Higher derivative estimates for Stokes equations with closely spaced rigid inclusions in three dimensions

In this paper, we establish higher-order derivative estimates for the Stokes equations in a three-dimensional domain containing two closely spaced rigid inclusions. We construct a sequence of auxiliary functions via an inductive process to isolate the leading singular terms of higher-order derivatives within the narrow region between the inclusions. For a class of convex inclusions of general shapes, the construction of three-dimensional auxiliary functions -- unlike the two-dimensional case -- relies on the decay properties of solutions to a class of two-dimensional partial differential equations with singular coefficients. Taking advantage of this, we obtain pointwise upper bounds of derivatives up to the seventh order for general inclusions. Under additional symmetry conditions, we derive optimal estimates for derivatives of arbitrary order. Consequently, we obtain precise blow-up rates for the Cauchy stress and its higher-order derivatives in the narrow region between the inclusions.

math.AP↗

arXiv2025

Spatial $C^1$, $C^2$, and Schauder estimates for nonstationary Stokes equations with Dini mean oscillation coefficients

We establish the spatial differentiability of weak solutions to nonstationary Stokes equations in divergence form with variable viscosity coefficients having $L_2$-Dini mean oscillations. As a corollary, we derive local spatial Schauder estimates for such equations if the viscosity coefficient belongs to $C^α_x$. Similar results also hold for strong solutions to nonstationary Stokes equations in nondivergence form.

math.AP↗

arXiv2025

The analysis of resonant frequencies and blow-up estimates of close-to-touching subwavelength resonators in the two-dimensional Helmholtz system

In this paper, we investigate wave scattering by a pair of closely spaced inclusions embedded in a homogeneous medium, characterized by a high contrast physical parameters. The system is modeled by the two-dimensional Helmholtz equation. We show that this configuration exhibits two sub-wavelength resonant modes, whose frequencies display distinct leading-order asymptotic behaviors. These findings differ significantly from those in the three-dimensional Helmholtz setting. Furthermore, we provide a quantitative analysis of the gradient blow-up rates for the wave field localized between the two resonators.

math.AP↗

arXiv2025

$L_p$-estimates of the conormal derivative problem for parabolic equations with time measurable coefficients and $A_p$-weights

This paper investigates weighted mixed-norm estimates for divergence-type parabolic equations on Reifenberg-flat domains with the conormal derivative boundary condition. The leading coefficients are assumed to be merely measurable in the time variable and to have small mean oscillations in the spatial variables. In deriving the boundary estimates, we overcome a regularity issue by employing half-time derivative estimates.

math.AP↗

arXiv2025

Optimal gradient estimates for conductivity problems with imperfect low-conductivity interfaces

This paper studies field concentration between two nearly touching conductors separated by imperfect low-conductivity interfaces, modeled by Robin boundary conditions. It is known that for any sufficiently small interfacial bonding parameter $γ> 0$, the gradient remains uniformly bounded with respect to the separation distance $\varepsilon$. In contrast, for the perfect bonding case ($γ= 0$, corresponding to the perfect conductivity problem), the gradient may blow up as $\varepsilon \to 0$ at a rate depending on the dimension. In this work, we establish optimal pointwise gradient estimates that explicitly depend on both $γ$ and $\varepsilon$ in the regime where these parameters are small. These estimates provide a unified framework that encompasses both the previously known bounded case ($γ> 0$) and the singular blow-up scenario ($γ= 0$), thus furnishing a complete and continuous characterization of the gradient behavior throughout the transition in $γ$. The key technical achievement is the derivation of new regularity results for elliptic equations as $γ\to0$, along with a case dichotomy based on the relative sizes of $γ$ and a distance function $δ(x')$. Our results hold for strictly relatively convex conductors in all dimensions $n \geq 2$.

math.AP↗