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Doyoon Kim

Publications and source records attributed to Doyoon Kim.

At least 37 records · Page 2Linked to original sources
arXiv2019

$L_q$-estimates for stationary Stokes system with coefficients measurable in one direction

We study the stationary Stokes system with variable coefficients in the whole space, a half space, and on bounded Lipschitz domains. In the whole and half spaces, we obtain a priori $\dot W^1_q$-estimates for any $q\in [2,\infty)$ when the coefficients are merely measurable functions in one fixed direction. For the system on bounded Lipschitz domains with a small Lipschitz constant, we obtain a $W^1_q$-estimate and prove the solvability for any $q\in (1,\infty)$ when the coefficients are merely measurable functions in one direction and have locally small mean oscillations in the orthogonal directions in each small ball, where the direction is allowed to depend on the ball.

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.

math.AP↗
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

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↗
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

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.

math.AP↗
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.

math.AP↗
arXiv2015

Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces

We prove the unique solvability in weighted Sobolev spaces of non-divergence form elliptic and parabolic equations on a half space with the homogeneous Neumann boundary condition. All the leading coefficients are assumed to be only measurable in the time variable and have small mean oscillations in the spatial variables. Our results can be applied to Neumann boundary value problems for {\em stochastic} partial differential equations with BMO$_x$ coefficients.

math.AP↗
arXiv2014

On the impossibility of $W_p^2$ estimates for elliptic equations with piecewise constant coefficients

In this paper, we present counterexamples showing that for any $p\in (1,\infty)$, $p\neq 2$, there is a non-divergence form uniformly elliptic operator with piecewise constant coefficients in $\mathbb{R}^2$ (constant on each quadrant in $\mathbb{R}^2$) for which there is no $W^2_p$ estimate. The corresponding examples in the divergence case are also discussed. One implication of these examples is that the ranges of $p$ are sharp in the recent results obtained in [4,5] for non-divergence type elliptic and parabolic equations in a half space with the Dirichlet or Neumann boundary condition when the coefficients do not have any regularity in a tangential direction.

math.AP↗
arXiv2014

GEANT4 and PHITS simulations of the shielding of neutrons from $^{252}$Cf source

Neutron shielding simulations by using GEANT4 and PHITS code are performed. As a neutron source, $^{252}$Cf is considered and the energy distribution of the neutrons emitted from $^{252}$Cf is assumed the Watt fission spectrum. The neutron dose equivalent rates with and without the shield are estimated for shielding materials such as graphite, iron, polyethylene, NS-4-FR and KRAFTON-HB. For the neutron shielding simulations by using GEANT4, high precision (G4HP) model with G4NDL 4.2 based on ENDF-VII data are used. And for PHITS simulations, JENDL-4.0 library are used for the same purpose. It is found that differences between the shielding calculations by using GEANT4 with G4NDL 4.2 and PHITS with JENDL-4.0 library are not significant for all cases considered in this work. We investigate the accuracy of the neutron dose equivalent rates obtained from GEANT4 and PHITS by comparing our simulation results with experimental data and other values calculated earlier. Calculated neutron dose equivalent rates agree well with the experimental dose equivalent rates within 20% errors except for polyethylene. For polyethylene material, discrepancy between the calculations and the experiments are up to 40% but all simulations show consistent features.

physics.ins-det↗
arXiv2014

Elliptic and parabolic equations with measurable coefficients in weighted Sobolev spaces

We consider both divergence and non-divergence parabolic equations on a half space in weighted Sobolev spaces. All the leading coefficients are assumed to be only measurable in the time and one spatial variable except one coefficient, which is assumed to be only measurable either in the time or the spatial variable. As functions of the other variables the coefficients have small bounded mean oscillation (BMO) semi-norms. The lower-order coefficients are allowed to blow up near the boundary with a certain optimal growth condition. As a corollary, we also obtain the corresponding results for elliptic equations.

math.AP↗
arXiv2013

Boundary value problems for parabolic operators in a time-varying domain

We prove the existence of unique solutions to the Dirichlet boundary value problems for linear second-order uniformly parabolic operators in either divergence or non-divergence form with boundary blowup low-order coefficients. The domain is possibly time varying, non-smooth, and satisfies the exterior measure condition.

math.AP↗
arXiv2013

Parabolic equations in simple convex polytopes with time irregular coefficients

We prove the $W^{1,2}_p$-estimate and solvability for the Dirichlet problem of second-order parabolic equations in simple convex polytopes with time irregular coefficients, when $p\in (1,2]$. We also consider the corresponding Neumann problem in a half space when $p\in [2,\infty)$. Similar results are obtained for equations in a half space with coefficients which are measurable in a tangential direction and have small mean oscillations in the other directions.

math.AP↗
arXiv2013

Schauder estimates for a class of non-local elliptic equations

We prove Schauder estimates for a class of non-local elliptic operators with kernel $K(y)=a(y)/|y|^{d+σ}$ and either Dini or Hölder continuous data. Here $0 < σ< 2$ is a constant and $a$ is a bounded measurable function, which is not necessarily to be homogeneous, regular, or symmetric. As an application, we prove that the operators give isomorphisms between the Lipschitz--Zygmund spaces $Λ^{α+σ}$ and $Λ^α$ for any $α>0$. Several local estimates and an extension to operators with kernels $K(x,y)$ are also discussed.

math.AP↗
arXiv2012

The conormal derivative problem for higher order elliptic systems with irregular coefficients

We prove $L_p$ estimates of solutions to a conormal derivative problem for divergence form complex-valued higher-order elliptic systems on a half space and on a Reifenberg flat domain. The leading coefficients are assumed to be merely measurable in one direction and have small mean oscillations in the orthogonal directions on each small ball. Our results are new even in the second-order case. The corresponding results for the Dirichlet problem were obtained recently in [15].

math.AP↗
arXiv2011

On $L_p$-estimates for a class of non-local elliptic equations

We consider non-local elliptic operators with kernel $K(y)=a(y)/|y|^{d+σ}$, where $0 < σ< 2$ is a constant and $a$ is a bounded measurable function. By using a purely analytic method, we prove the continuity of the non-local operator $L$ from the Bessel potential space $H^σ_p$ to $L_p$, and the unique strong solvability of the corresponding non-local elliptic equations in $L_p$ spaces. As a byproduct, we also obtain interior $L_p$-estimates. The novelty of our results is that the function $a$ is not necessarily to be homogeneous, regular, or symmetric. An application of our result is the uniqueness for the martingale problem associated to the operator $L$.

math.AP↗