arXiv subjects

Jinwan Park

Publications and source records attributed to Jinwan Park.

6 recordsLinked to original sources

arXiv2022

The Regularity Theory for the Double Obstacle Problem for Fully Nonlinear Operator

In this paper, we prove the existence and uniqueness of $W^{2,p}$ ($n<p<\infty$) solutions of a double obstacle problem with $C^{1,1}$ obstacle functions. Moreover, we show the optimal regularity of the solution and the local $C^1$ regularity of the free boundary. In the study of the regularity of the free boundary, we deal with a general problem, the no-sign reduced double obstacle problem with an upper obstacle $ψ$, $F(D^2 u,x) =fχ_{Ω(u) \cap\{ u< ψ\} } + F(D^2ψ,x) χ_{Ω(u)\cap \{u=ψ\}}, u\le ψ\text { in } B_1$, where $Ω(u)=B_1 \setminus \left( \{u=0\} \cap \{ \nabla u =0\}\right)$.

math.AP↗

arXiv2021

Asymptotic behaviour of singular solution of the fast diffusion equation in the punctured Euclidean space

For $n\ge 3$, $0 0$. As a consequence we prove the existence and uniqueness of solutions of Cauchy problem for the fast diffusion equation $u_t=\frac{n-1}{m}Δu^m$ in $(\mathbb{R}^n\setminus\{0\})\times (0,\infty)$ with initial value $u_0$ satisfying $f_{λ_1}(x)\le u_0(x)\le f_{λ_2}(x)$, $\forall x\in\mathbb{R}^n\setminus\{0\}$, which satisfies $U_{λ_1}(x,t)\le u(x,t)\le U_{λ_2}(x,t)$, $\forall x\in \mathbb{R}^n\setminus\{0\}, t\ge 0$, for some constants $λ_1>λ_2>0$. We also prove the asymptotic behaviour of such singular solution $u$ of the fast diffusion equation as $t\to\infty$ when $n=3,4$ and $\frac{n-2}{n+2}\le m<\frac{n-2}{n}$ holds. Asymptotic behaviour of such singular solution $u$ of the fast diffusion equation as $t\to\infty$ is also obtained when $3\le n<8$, $1-\sqrt{2/n}\le m<\min\left(\frac{2(n-2)}{3n},\frac{n-2}{n+2}\right)$, and $u(x,t)$ is radially symmetric in $x\in\mathbb{R}^n\setminus\{0\}$ for any $t>0$ under appropriate conditions on the initial value $u_0$.

math.AP↗

arXiv2018

Quantum dots formed in three-dimensional Dirac semimetal Cd$_3$As$_2$ nanowires

We demonstrate quantum dot (QD) formation in three-dimensional Dirac semimetal Cd$_{3}$As$_{2}$ nanowires using two electrostatically tuned p$-$n junctions with a gate and magnetic fields. The linear conductance measured as a function of gate voltage under high magnetic fields is strongly suppressed at the Dirac point close to zero conductance, showing strong conductance oscillations. Remarkably, in this regime, the Cd$_{3}$As$_{2}$ nanowire device exhibits Coulomb diamond features, indicating that a clean single QD forms in the Dirac semimetal nanowire. Our results show that a p$-$type QD can be formed between two n$-$type leads underneath metal contacts in the nanowire by applying gate voltages under strong magnetic fields. Analysis of the quantum confinement in the gapless band structure confirms that p$-$n junctions formed between the p$-$type QD and two neighboring n$-$type leads under high magnetic fields behave as resistive tunnel barriers due to cyclotron motion, resulting in the suppression of Klein tunneling. The p$-$type QD with magnetic field-induced confinement shows a single hole filling. Our results will open up a route to quantum devices such as QDs or quantum point contacts based on Dirac and Weyl semimetals.

cond-mat.mes-hall↗

arXiv2018

Regularity Results of the Thin Obstacle Problem for the $p(x)$-Laplacian

We study thin obstacle problems involving the energy functional with $p(x)$-growth. We prove higher integrability and Hölder regularity for the gradient of minimizers of the thin obstacle problems under the assumption that the variable exponent $p(x)$ is Hölder continuous.

math.AP↗

arXiv2017

Nondivergence elliptic and parabolic problems with irregular obstacles

We prove the natural weighted Calderón and Zygmund estimates for solutions to elliptic and parabolic obstacle problems in nondivergence form with discontinuous coefficients and irregular obstacles. We also obtain Morrey regularity results for the Hessian of the solutions and Hölder continuity of the gradient of the solutions.

math.AP↗

arXiv2017

The Regularity Theory for the Double Obstacle Problem

In this paper, we prove local $C^{1}$ regularity of free boundaries for the double obstacle problem with an upper obstacle $ψ$, \begin{align*} Δu &=fχ_{Ω(u) \cap\{ u< ψ\} }+ Δψχ_{Ω(u)\cap \{u=ψ\}}, \qquad u\le ψ\quad \text { in } B_1, \end{align*} where $Ω(u)=B_1 \setminus \left( \{u=0\} \cap \{ \nabla u =0\}\right)$ under a thickness assumption for $u$ and $ψ$.

math.AP↗