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Hongjie Dong

Publications and source records attributed to Hongjie Dong.

At least 127 records · Page 7Linked to original sources
arXiv2018

Higher-order parabolic equations with VMO assumptions and general boundary conditions with variable leading coefficients

We prove weighted mixed $L_{p}(L_{q})$-estimates, with $p,q\in(1,\infty)$, for higher-order elliptic and parabolic equations on the half space $\mathbb{R}^{d+1}_{+}$ and on domains with general boundary conditions which satisfy the Lopatinskii--Shapiro condition. We assume that the elliptic operators $A$ have leading coefficients which are in the class of vanishing mean oscillations both in the time and the space variables, and that the boundary conditions have variable leading coefficients. The proofs are based on and generalize the estimates recently obtained by the authors in [DG17]. [DG17] H.Dong and C.Gallarati. Higher order elliptic and parabolic equations with VMO assumptions and general boundary conditions. Accepted for publication in Journal of Functional Analysis. See: arXiv:1702.03254.

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arXiv2018

Weighted mixed-norm $L_p$-estimates for elliptic and parabolic equations in non-divergence form with singular degenerate coefficients

In this paper, we study both elliptic and parabolic equations in non-divergence form with singular degenerate coefficients. Weighted and mixed-norm $L_p$-estimates and solvability are established under some suitable partially weighted BMO regularity conditions on the coefficients. When the coefficients are constants, the operators are reduced to extensional operators which arise in the study of fractional heat equations and fractional Laplace equations. Our results are new even in this setting and in the unmixed case. For the proof, we establish both interior and boundary Lipschitz estimates for solutions and for higher order derivatives of solutions to homogeneous equations. We then employ the perturbation method by using the Fefferman-Stein sharp function theorem, the Hardy-Littlewood maximum function theorem, as well as a weighted Hardy's inequality.

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arXiv2018

Regularity theory for parabolic equations with singular degenerate coefficients

In this paper, we study parabolic equations in divergence form with coefficients that are singular degenerate as some Muckenhoupt weight functions in one spatial variable. Under certain conditions, weighted reverse Hölder's inequalities are established. Lipschitz estimates for weak solutions are proved for homogeneous equations with singular degenerate coefficients depending only on one spatial variable. These estimates are then used to establish interior, boundary, and global weighted estimates of Calderón-Zygmund type for weak solutions, assuming that the coefficients are partially VMO (vanishing mean oscillations) with respect to the considered weights. The solvability in weighted Sobolev spaces is also achieved. Our results are new even for elliptic equations, and non-trivially extend known results for uniformly elliptic and parabolic equations. The results are also useful in the study of fractional elliptic and parabolic equations with measurable coefficients.

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arXiv2018

Gradient estimates for Stokes systems with Dini mean oscillation coefficients

We study the stationary Stokes system in divergence form. The coefficients are assumed to be merely measurable in one direction and have Dini mean oscillations in the other directions. We prove that if $(u,p)$ is a weak solution of the system, then $(Du,p)$ is bounded and its certain linear combinations are continuous. We also prove a weak type-$(1,1)$ estimate for $(Du,p)$ under a stronger assumption on the $L^1$-mean oscillation of the coefficients. The corresponding results up to the boundary on a half ball are also established. These results are new even for elliptic equations and systems.

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arXiv2018

Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces

We study regularity criteria for the $d$-dimensional incompressible Navier-Stokes equations. We prove if $u\in L_{\infty}^tL_d^x((0,T)\times \mathbb{R}^d_+)$ is a Leray-Hopf weak solution vanishing on the boundary and the pressure $p$ satisfies a local condition $\|p\|_{L_{2-1/d}(Q(z_0,1)\cap (0,T)\times \mathbb{R}^d_+)}\leq K$ for some constant $K>0$ uniformly in $z_0$, then $u$ is regular up to the boundary in $(0,T)\times \mathbb{R}^d_+$. Furthermore, when $T=\infty$, $u$ tends to zero as $t\rightarrow \infty$. We also study the local problem in half unit cylinder $Q^+$ and prove that if $u\in L^t_{\infty}L^x_d(Q^+)$ and $ p\in L_{2-1/d}(Q^+)$, then $u$ is Hölder continuous in the closure of the set $Q^+(1/4)$. This generalizes a result by Escauriaza, Seregin, and Šverák to higher dimensions and domains with boundary.

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arXiv2018

On the $W^2_p$ Estimate for Oblique Derivative Problem in Lipschitz Domains

The aim of this paper is to establish $W^2_p$ estimate for non-divergence form second-order elliptic equations with the oblique derivative boundary condition in domains with small Lipschitz constants. Our result generalizes those in [14, 15], which work for $C^{1,α}$ domains with $α> 1-1/p$. As an application, we also obtain a solvability result. An extension to fully nonlinear elliptic equations with the oblique derivative boundary condition is also discussed.

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arXiv2018

Green functions of conormal derivative problems for stationary Stokes system

We study Green functions for stationary Stokes systems satisfying the conormal derivative boundary condition. We establish existence, uniqueness, and various estimates for the Green function under the assumption that weak solutions of the Stokes system are continuous in the interior of the domain. Also, we establish the global pointwise bound for the Green function under the additional assumption that weak solutions of the conormal derivative problem for the Stokes system are locally bounded up to the boundary. We provide some examples satisfying such continuity and boundedness properties.

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arXiv2018

Gradient estimates for Stokes systems in domains

We study the stationary Stokes system with Dini mean oscillation coefficients in a domain having $C^{1,\rm{Dini}}$ boundary. We prove that if $(u, p)$ is a weak solution of the system with zero Dirichlet boundary condition, then $(Du,p)$ is continuous up to the boundary. We also prove a weak type-$(1,1)$ estimate for $(Du, p)$.

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arXiv2017

Fundamental solutions for second order parabolic systems with drift terms

We construct fundamental solutions of second-order parabolic systems of divergence form with bounded and measurable leading coefficients and divergence free first-order coefficients in the class of $BMO^{-1}_x$, under the assumption that weak solutions of the system satisfy a certain local boundedness estimate. We also establish Gaussian upper bound for such fundamental solutions under the same conditions.

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arXiv2017

On $C^1$, $C^2$, and weak type-$(1,1)$ estimates for linear elliptic operators: Part II

We extend and improve the results in \cite{DK16}: showing that weak solutions to full elliptic equations in divergence form with zero Dirichlet boundary conditions are continuously differentiable up to the boundary when the leading coefficients have Dini mean oscillation and the lower order coefficients verify certain conditions. Similar results are obtained for non-divergence form equations. We extend the weak type-(1, 1) estimates in \cite{DK16} and \cite{Es94} up to the boundary and derive a Harnack inequality for non-negative adjoint solutions to non-divergence form elliptic equations, when the leading coefficients have Dini mean oscillation.

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arXiv2017

Conormal derivative problems for stationary Stokes system in Sobolev spaces

We prove the solvability in Sobolev spaces of the conormal derivative problem for the stationary Stokes system with irregular coefficients on bounded Reifenberg flat domains. The coefficients are assumed to be merely measurable in one direction, which may differ depending on the local coordinate systems, and have small mean oscillations in the other directions. In the course of the proof, we use a local version of the Poincaré inequality on Reifenberg flat domains, the proof of which is of independent interest.

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arXiv2017

On Schauder estimates for a class of nonlocal fully nonlinear parabolic equations

We obtain Schauder estimates for a class of concave fully nonlinear nonlocal parabolic equations of order $σ\in (0,2)$ with rough and non-symmetric kernels. As a application, we prove that the solution to a translation invariant equation with merely bounded data is $C^σ$ in $x$ variable and $Λ^1$ in $t$ variable, where $Λ^1$ is the Zygmund space.

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arXiv2017

Weighted $L_q$-estimates for stationary Stokes system with partially BMO coefficients

We prove the unique solvability of solutions in Sobolev spaces to the stationary Stokes system on a bounded Reifenberg flat domain when the coefficients are partially BMO functions, i.e., locally they are merely measurable in one direction and have small mean oscillations in the other directions. Using this result, we establish the unique solvability in Muckenhoupt type weighted Sobolev spaces for the system with partially BMO coefficients on a Reifenberg flat domain. We also present weighted a priori $L_q$-estimates for the system when the domain is the whole Euclidean space or a half space.

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arXiv2016

On $L_p$-estimates for elliptic and parabolic equations with $A_p$ weights

We prove generalized Fefferman-Stein type theorems on sharp functions with $A_p$ weights in spaces of homogeneous type with either finite or infinite underlying measure. We then apply these results to establish mixed-norm weighted $L_p$-estimates for elliptic and parabolic equations/systems with (partially) BMO coefficients in regular or irregular domains.

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arXiv2016

Dini estimates for nonlocal fully nonlinear elliptic equations

We obtain Dini type estimates for a class of concave fully nonlinear nonlocal elliptic equations of order $σ\in (0,2)$ with rough and non-symmetric kernels. The proof is based on a novel application of Campanato's approach and a refined $C^{σ+α}$ estimate in [8].

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arXiv2016

Partial Schauder estimates for second-order elliptic and parabolic equations: a revisit

Under various conditions, we establish Schauder estimates for both divergence and non-divergence form second-order elliptic and parabolic equations involving Hölder semi-norms not with respect to all, but only with respect to some of the independent variables. A novelty of our results is that the coefficients are allowed to be merely measurable with respect to the other independent variables.

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arXiv2016

On $C^1$, $C^2$, and weak type-$(1,1)$ estimates for linear elliptic operators

We show that any weak solution to elliptic equations in divergence form is continuously differentiable provided that the modulus of continuity of coefficients in the $L^1$-mean sense satisfies the Dini condition. This in particular answers a question recently raised by Yanyan Li and allows us to improve a result of Brezis. We also prove a weak type-$(1,1)$ estimate under a stronger assumption on the modulus of continuity. The corresponding results for non-divergence form equations are also established.

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