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Ki-ahm Lee

Publications and source records attributed to Ki-ahm Lee.

13 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↗
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↗
arXiv2017

Homogenization of the boundary value for the Dirichlet Problem

In this paper, we give a mathematically rigorous proof of the averaging behavior of oscillatory surface integrals. Based on ergodic theory, we find a sharp geometric condition which we call irrational direction dense condition, abbreviated as IDDC, under which the averaging takes place. It should be stressed that IDDC does not imply any control on the curvature of the given surface. As an application, we prove homogenization for elliptic systems with Dirichlet boundary data, in $C^1$-domains.

math.AP↗
arXiv2016

An Elliptic Free Boundary Arising From the Jump of Conductivity

In this paper we consider a quasilinear elliptic PDE, $\text{div} (A(x,u) \nabla u) =0$, where the underlying physical problem gives rise to a jump for the conductivity $A(x,u)$, across a level surface for $u$. Our analysis concerns Lipschitz regularity for the solution $u$, and the regularity of the level surfaces, where $A(x,u)$ has a jump and the solution $u$ does not degenerate. In proving Lipschitz regularity of solutions, we introduce a new and unexpected type of ACF-monotonicity formula with two different operators, that might be of independent interest, and surely can be applied in other related situations. The proof of the monotonicity formula is done through careful computations, and (as a byproduct) a slight generalization to a specific type of variable matrix-valued conductivity is presented.

math.AP↗
arXiv2016

The evolution of complete non-compact graphs by powers of Gauss curvature

We study the evolution of convex complete non-compact graphs by positive powers of Gauss curvature. We show that if the initial complete graph has a local uniform convexity, then the graph evolves by any positive power of Gauss curvature for all time. In particular, the initial graph is not necessarily differentiable.

math.DG↗
arXiv2014

$α$-Gauss Curvature flows

In this paper, we study the deformation of the n-dimensional strictly convex hypersurface in $\mathbb R^{n+1}$ whose speed at a point on the hypersurface is proportional to $α$-power of positive part of Gauss Curvature. For $\frac{1}{n}<α\leq 1$, we prove that there exist the strictly convex smooth solutions if the initial surface is strictly convex and smooth and the solution hypersurfaces converge to a point. We also show the asymptotic behavior of the rescaled hypersurfaces, in other words, the rescaled manifold converges to a strictly convex smooth manifold. Moreover, there exists a subsequence whose the limit satisfies a certain equation.

math.AP↗
arXiv2013

Homogenization of fully nonlinear elliptic equations with oscillating dirichlet boundary data

This paper deals with the homogenization of fully nonlinear second order equation with an oscillating Dirichlet boundary data when the operator and boundary data are $\e$-periodic. We will show that the solution $u_\e$ converges to some function $\bar u(x)$ uniformly on every compact subset $K$ of the domain $D$. Moreover, $\bar u$ is a solution to some boundary value problem. For this result, we assume that the boundary of the domain has no (rational) flat spots and the ratio of elliptic constants $Λ/ λ$ is sufficiently large.

math.AP↗
arXiv2012

Fully Degenerate Monge Ampére Equations

In this paper, we consider the following nonlinear eigenvalue problem for the Monge-Ampére equation: find a non-negative weakly convex classical solution $f$ satisfying {equation*} {cases} \det D^2 f=f^p \quad &\text{in $Ω$} f=\vp \quad &text{on $\partialΩ$} {cases} {equation*} for a strictly convex smooth domain $Ω\subset\R^2$ and $0 0\}}$ and whose free boundary $\partial\{f=0\}$ is also smooth.

math.AP↗
arXiv2012

Highly Oscillating Thin Obstacles

The focus of this paper is on a thin obstacle problem where the obstacle is defined on the intersection between a hyper-plane $Γ$ in $\mathbb{R}^n$ and a periodic perforation $\mathcal{T}_\varepsilon$ of $\mathbb{R}^n$, depending on a small parameter $\varepsilon>0$. As $\varepsilon\to 0$, it is crucial to estimate the frequency of intersections and to determine this number locally. This is done using strong tools from uniform distribution. By employing classical estimates for the discrepancy of sequences of type $\{kα\}_{k=1}^\infty$, $α\in\R$, we are able to extract rather precise information about the set $Γ\cap\mathcal{T}_\varepsilon$. As $\varepsilon\to0$, we determine the limit $u$ of the solution $u_\varepsilon$ to the obstacle problem in the perforated domain, in terms of a limit equation it solves. We obtain the typical "strange term" behaviour for the limit problem, but with a different constant taking into account the contribution of all different intersections, that we call the averaged capacity. Our result depends on the normal direction of the plane, but holds for a.e. normal on the unit sphere in $\R^n$.

math.AP↗
arXiv2012

Asymptotic Behavior in Degenerate Parabolic Fully Nonlinear equations and its application to Elliptic Eigenvalue Problems

We study the asymptotic behavior of the nonlinear parabolic flows $u_{t}=F(D^2 u^m)$ when $t\ra \infty$ for $m\geq 1$, and the geometric properties for solutions of the following elliptic nonlinear eigenvalue problems: F(D^2 \vp) &+ μ\vp^{p}=0, \quad \vp>0\quad\text{in $Ω$} \vp&=0\quad\text{on $\pΩ$} posed in a (strictly) convex and smooth domain $Ω\subset\re^n$ for $0< p \leq 1,$ where $F(\cdot)$ is uniformly elliptic, positively homogeneous of order one and concave. We establish that $\log (\vp)$ is concave in the case $p=1$ and that the function $\vp^{\frac{1-p}{2}}$ is concave for $0<p<1.$

math.AP↗
arXiv2011

The viscosity Method for the Homogenization of soft inclusions

In this paper, we consider periodic soft inclusions $T_ε$ with periodicity $ε$, where the solution, $u_ε$, satisfies semi-linear elliptic equations of non-divergence in $Ω_ε=Ω\setminus \bar{T}_ε$ with a Neumann data on $\partial T^{\mathfrak a}$ . The difficulty lies in the non-divergence structure of the operator where the standard energy method based on the divergence theorem can not be applied. The main object is developing a viscosity method to find the homogenized equation satisfied by the limit of $u_ε$, called as $u$, as $ε$ approaches to zero. We introduce the concept of a compatibility condition between the equation and the Neumann condition on the boundary for the existence of uniformly bounded periodic first correctors. The concept of second corrector has been developed to show the limit, $u$, is the viscosity solution of a homogenized equation.

math.AP↗
arXiv2011

α-Gauss Curvature flows with flat sides

In this paper, we study the deformation of the 2 dimensional convex surfaces in $\R^{3}$ whose speed at a point on the surface is proportional to $α$-power of positive part of Gauss Curvature. First, for 1/2<α\leq 1$, we show that there is smooth solution if the initial data is smooth and strictly convex and that there is a viscosity solution with $C^{1,1}$-estimate before the collapsing time if the initial surface is only convex. Moreover, we show that there is a waiting time effect which means the flat spot of the convex surface will persist for a while. We also show the interface between the flat side and the strictly convex side of the surface remains smooth on $0 < t < T_0$ under certain necessary regularity and non-degeneracy initial conditions, where $ T_0$ is the vanishing time of the flat side.

math.AP↗