arXiv subjects

Minha Yoo

Publications and source records attributed to Minha Yoo.

5 recordsLinked to original sources

arXiv2024

Homogenization of an obstacle problem with highly oscillating coefficients and obstacles

We develop the viscosity method for the homogenization of an obstacle problem with highly oscillating obstacles. The associated operator, in non-divergence form, is linear and elliptic with variable coefficients. We first construct a highly oscillating corrector, which captures the singular behavior of solutions near periodically distributed holes of critical size. We then prove the uniqueness of a critical value that encodes the coupled effects of oscillations in both the coefficients and the obstacles.

math.AP↗

arXiv2014

Regularity Conditions of 3D Navier-Stokes flow in terms of large spectral components

We develop Ladyzhenskaya-Prodi-Serrin type spectral regularity criteria for 3D incompressible Navier-Stokes equations in a torus. Concretely, for any $N>0$, let $w_N$ be the sum of all spectral components of the velocity fields whose all three wave numbers are greater than $N$ absolutely. Then, we show that for any $N>0$, the finiteness of the Serrin type norm of $w_N$ implies the regularity of the flow. It implies that if the flow breaks down in a finite time, the energy of the velocity fields cascades down to the arbitrarily large spectral components of $w_N$ and corresponding energy spectrum, in some sense, roughly decays slower than $κ^{-2}$

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

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↗

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↗