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Jongkeun Choi

Publications and source records attributed to Jongkeun Choi.

At least 19 recordsLinked to original sources
arXiv2026

Failure of zero extension in parabolic Sobolev spaces

We show that spatial zero extension across the boundary may fail in parabolic Sobolev spaces $\mathring{\mathcal{H}}^1_p((0,T) \times Ω)$, which can also be characterized as $$ L_p(0,T;\mathring{W}^1_p(Ω))\cap W^1_p(0,T; W^{-1}_{p}(Ω)). $$ More precisely, for any $p\in [1, \infty)$, we construct a function $u\in \mathring{\mathcal{H}}^1_p((0,T)\times \mathbb{R}^d_+)$ whose zero extension does not belong to $\mathcal{H}^1_p((0,T)\times \mathbb{R}^d)$. The obstruction occurs even for a flat boundary and is caused by a self-similar boundary layer concentrated at the initial-boundary corner, which produces a boundary supported normal flux defect after zero extension. We also discuss the suitability of various Sobolev-type spaces as solution spaces for parabolic equations in divergence form.

math.AP↗
arXiv2025

Refined regularity at critical points for linear elliptic equations

We investigate the regularity of solutions to linear elliptic equations in both divergence and non-divergence forms, particularly when the principal coefficients have Dini mean oscillation. We show that if a solution $u$ to a divergence-form equation satisfies $Du(x^o)=0$ at a point, then the second derivative $D^2u(x^o)$ exists and satisfies sharp continuity estimates. As a consequence, we obtain ``$C^{2,α}$ regularity'' at critical points when the coefficients of $L$ are $C^α$. This result refines a theorem of Teixeira (Math. Ann. 358 (2014), no. 1--2, 241--256) in the linear setting, where both linear and nonlinear equations were considered. We also establish an analogous result for equations in non-divergence form.

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arXiv2025

Regularity of elliptic equations in double divergence form and applications to Green's function estimates

We investigate the regularity of elliptic equations in double divergence form, where the leading coefficients satisfying the Dini mean oscillation condition. We prove that the solutions are differentiable on the zero level set and derive a pointwise bound for the derivative, which substantially improve a recent result by Leitão, Pimentel, and Santos (Anal. PDE 13(4):1129--1144, 2020). As an application, we establish global pointwise estimates for the Green's function of second-order uniformly elliptic operators in non-divergence form, considering Dini mean oscillation coefficients in bounded $C^{1,α}$ domains. This result extends a recent work by Chen and Wang (Electron. J. Probab. 28(36):54 pp, 2023).

math.AP↗
arXiv2021

Optimal regularity of mixed Dirichlet-conormal boundary value problems for parabolic operators

We obtain the regularity of solutions in Sobolev spaces for the mixed Dirichlet-conormal problem for parabolic operators in cylindrical domains with time-dependent separations, which is the first of its kind. Assuming the boundary of the domain to be Reifenberg-flat and the separation to be locally sufficiently close to a Lipschitz function of $m$ variables, where $m=0,\ldots,d-2$, with respect to the Hausdorff distance, we prove the unique solvability for $p\in (2(m+2/(m+3),2(m+2)/(m+1)))$. In the case when $m=0$, the range $p\in(4/3,4)$ is optimal in view of the known results for Laplace equations.

math.AP↗
arXiv2021

Gradient estimates for Stokes and Navier-Stokes systems with piecewise DMO coefficients

We study stationary Stokes systems in divergence form with piecewise Dini mean oscillation coefficients and data in a bounded domain containing a finite number of subdomains with $C^{1,\rm{Dini}}$ boundaries. We prove that if $(u, p)$ is a weak solution of the system, then $(Du, p)$ is bounded and piecewise continuous. The corresponding results for stationary Navier-Stokes systems are also established, from which the Lipschitz regularity of the stationary $H^1$-weak solution in dimensions $d=2,3,4$ is obtained.

math.AP↗
arXiv2019

Optimal regularity for a Dirichlet-conormal problem in Reifenberg flat domain

We study the divergence form second-order elliptic equations with mixed Dirichlet-conormal boundary conditions. The unique $W^{1,p}$ solvability is obtained with $p$ being in the optimal range $(4/3,4)$. The leading coefficients are assumed to have small mean oscillations and the boundary of domain is Reifenberg flat. We also assume that the two boundary conditions are separated by some Reifenberg flat set of co-dimension $2$ on the boundary.

math.AP↗
arXiv2019

Green functions for pressure of Stokes systems

We study Green functions for the pressure of stationary Stokes systems in a (possibly unbounded) domain $Ω\subset \mathbb{R}^d$, where $d\ge 2$. We construct the Green function when coefficients are merely measurable in one direction and have Dini mean oscillation in the other directions, and $Ω$ is such that the divergence equation is solvable there. We also establish global pointwise bounds for the Green function and its derivatives when coefficients have Dini mean oscillation and $Ω$ has a $C^{1,\rm{Dini}}$ boundary. Green functions for the flow velocity of Stokes systems are also considered.

math.AP↗
arXiv2019

Weighted $L_{p,q}$-estimates for higher order elliptic and parabolic systems with $\mathrm{BMO}_x$ coefficients on Reifenberg flat domains

We prove weighted $L_{p,q}$-estimates for divergence type higher order elliptic and parabolic systems with irregular coefficients on Reifenberg flat domains. In particular, in the parabolic case the coefficients do not have any regularity assumptions in the time variable. As functions of the spatial variables, the leading coefficients are permitted to have small mean oscillations. The weights are in the class of Muckenhoupt weights $A_p$. We also prove the solvability of the systems in weighted Sobolev spaces.

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arXiv2018

Estimates for Green functions of Stokes systems in two dimensional domains

We prove the existence and pointwise bounds of the Green functions for stationary Stokes systems with measurable coefficients in two dimensional domains. We also establish pointwise bounds of the derivatives of the Green functions under a regularity assumption on the $L_1$-mean oscillations of the coefficients.

math.AP↗
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.

math.AP↗
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.

math.AP↗
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 $(Du, p)$.

math.AP↗
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.

math.AP↗
arXiv2017

The Green function for the Stokes system with measurable coefficients

We study the Green function for the stationary Stokes system with bounded measurable coefficients in a bounded Lipschitz domain $Ω\subset \mathbb{R}^n$, $n\ge 3$. We construct the Green function in $Ω$ under the condition $(\bf{A1})$ that weak solutions of the system enjoy interior Hölder continuity. We also prove that $(\bf{A1})$ holds, for example, when the coefficients are $\mathrm{VMO}$. Moreover, we obtain the global pointwise estimate for the Green function under the additional assumption $(\bf{A2})$ that weak solutions of Dirichlet problems are locally bounded up to the boundary of the domain. By proving a priori $L^q$-estimates for Stokes systems with $\mathrm{BMO}$ coefficients on a Reifenberg domain, we verify that $(\bf{A2})$ is satisfied when the coefficients are $\mathrm{VMO}$ and $Ω$ is a bounded $C^1$ domain.

math.AP↗
arXiv2017

Fundamental solutions for stationary Stokes systems with measurable coefficients

We establish the existence and the pointwise bound of the fundamental solution for the stationary Stokes system with measurable coefficients in the whole space $\mathbb{R}^d$, $d \ge 3$, under the assumption that weak solutions of the system are locally Hölder continuous. We also discuss the existence and the pointwise bound of the Green function for the Stokes system with measurable coefficients on $Ω$, where $Ω$ is an unbounded domain such that the divergence equation is solvable. Such a domain includes, for example, half space and an exterior domain.

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