arXiv subjects

Ki-Ahm Lee

Publications and source records attributed to Ki-Ahm Lee.

At least 37 records · Page 2Linked to original sources

arXiv2021

Properties of Generalized Degenerate Parabolic Systems

In this paper, we consider the solution $\bold{u}=\left(u^1,\cdots,u^k\right)$ of the generalized parabolic system \begin{equation*} \left(u^i\right)_t=\nabla\cdot\left(mU^{m-1}\mathcal{A}\left(\nabla u^i,u^i,x,t\right)+\mathcal{B}\left(u^i,x,t\right)\right), \qquad \left(1\leq i\leq k\right) \end{equation*} in the range of exponents $m>\frac{n-2}{n}$ where the diffusion coefficient $U$ depends on the components of the solution $\bold{u}$. Under suitable structure conditions on the vector fields $\mathcal{A}$ and $\mathcal{B}$, we first show the uniform $L^{\infty}$ bound of the function $U$ for $t\geq τ>0$ and law of $L^1$ mass conservation of each component $u^i$, $(i=1,\cdots,k)$, with system version of Harnack type inequality. As the last result, we also deal with the local continuity of solution $\bold{u}=\left(u^1,\cdots,u^k\right)$ with the intrinsic scaling. If the ratio between $U$ and components $u^i$, $(i=1,\cdots,k)$, is uniformly bounded above and below, all components of the solution $\bold{u}$ have the same modulus of continuity.

math.AP↗

arXiv2021

System of Degenerate Parabolic $p$-Laplacian

In this paper, we study the mathematical properties of the solution $\bold{u}=\left(u^1,\cdots,u^k\right)$ to the degenerate parabolic system \begin{equation*} \bold{u}_t=\nabla\cdot\left(\left|\nabla\bold{u}\right|^{p-2}\nabla \bold{u}\right), \qquad \qquad \left(p>2\right). \end{equation*} More precisely, we show the uniqueness and existence of solution $\bold{u}$ and investigate a priori $L^{\infty}$ boundedness of the gradient of the solution. Assuming that the solution decays quickly at infinity, we also prove that the component $u^l$, $\left(1\leq l\leq k\right)$, converges to the function $c^l\mathcal{B}$ in space as $t\to\infty$. Here, the function $\mathcal{B}$ is the fundamental or Barenblatt solution of $p$-Laplacian equation and the constant $c^l$ is determined by the $L^1$-mass of $u^l$. The proof is based on the existence of entropy functional.\\ \indent As an application of the asymptotic large time behaviour, we establish a Harnack type inequality which makes the size of spatial average being controlled by the value of solution at one point.

math.AP↗

arXiv2021

Harnack inequality for Nonlocal operators on Manifolds with nonnegative curvature

We establish the Krylov Safonov Harnack inequalities and Holder estimates for fully nonlinear nonlocal operators of non-divergence form on Riemannian manifolds with nonnegative sectional curvatures. To this end, we first define the nonlocal Pucci operators on manifolds that give rise to the concept of non-divergence form operators. We then provide the uniform regularity results for these operators which recover the classical results for second order local operators as limits.

math.AP↗

arXiv2020

Optimal configuration and symmetry breaking phenomena in the composite membrane problem with fractional Laplacian

We consider the following eigenvalue optimization in the composite membrane problem with fractional Laplacian: given a bounded domain $Ω\subset \mathbb{R}^n$, $α>0$ and $0<A<|Ω|$, find a subset $D\subset Ω$ of area $A$ such that the first Dirichlet eigenvalue of the operator $(-Δ)^s+αχ_D$ is as small as possible. The solution $D$ is called as an optimal configuration for the data $(Ω,α,A)$. Looking at the well-known extension definition for the fractional Laplacian, in the case $s=1/2$ this is essentially the composite membrane problem for which the mass is concentrated at the boundary as one is trying to maximize the Steklov eigenvalue. We prove existence of solutions and study properties of optimal configuration $D$. This is a free boundary problem which could be formulated as a two-sided unstable obstacle problem. Moreover, we show that for some rotationally symmetric domains (thin annuli), the optimal configuration is not rotational symmetric, which implies the non-uniqueness of the optimal configuration $D$. On the other hand, we prove that for a convex domain $Ω$ having reflection symmetries, the optimal configuration possesses the same symmetries, which implies uniqueness of the optimal configuration $D$ in the ball case.

math.AP↗

arXiv2020

Generalized Evans--Krylov and Schauder type estimates for nonlocal fully nonlinear equations with rough kernels of variable orders

We establish the generalized Evans--Krylov and Schauder type estimates for nonlocal fully nonlinear elliptic equations with rough kernels of variable orders. In contrast to the fractional Laplacian type operators having a fixed order of differentiability $σ\in (0,2)$, the operators under consideration have variable orders of differentiability. Since the order is not characterized by a single number, we consider a function $φ$ describing the variable orders of differentiability, which is allowed to oscillate between two functions $r^{σ_1}$ and $r^{σ_2}$ for some $0 < σ_1 \leq σ_2 < 2$. By introducing the generalized Hölder spaces, we provide $C^{φψ}$ estimates that generalizes the standard Evans--Krylov and Schauder type $C^{σ+α}$ estimates.

math.AP↗

arXiv2019

System of Porous Medium Equations

We investigate the evolution of population density vector, $\bold{u}=\left(u^1,\cdots,u^k\right)$, of $k$-species whose diffusion is controlled by its absolute value $\left|\bold{u}\right|$. More precisely we study the properties and asymptotic large time behaviour of solution $\bold{u}=\left(u^1,\cdots,u^k\right)$ of degenerate parabolic system \begin{equation*} \left(u^i\right)_t=\nabla\cdot\left(\left|\bold{u}\right|^{m-1}\nabla u^i\right) \qquad \mbox{for $m>1$ and $i=1,\cdots,k$}. \end{equation*} Under some regularity assumption, we prove that the function $u^i$ which describes the population density of $i$-th species with population $M_i$ converges to $\frac{M_i}{\left|\bold{M}\right|}\mathcal{B}_{\left|\bold{M}\right|}$ in space with two different approaches where $\mathcal{B}_{\left|\bold{M}\right|}$ is the Barenblatt solution of the porous medium equation with $L^1$-mass $\left|\bold{M}\right|=\sqrt{M_1^2+\cdots+M_k^2}$. \indent As an application of the asymptotic behaviour, we establish a suitable harnack type inequality which makes the spatial average of $u^i$ under control by the value of $u^i$ at one point. We also find an 1-directional travelling wave type solutions and the properties of solutions which has travelling wave behaviour at infinity.

math.AP↗

arXiv2019

Higher Order Convergence Rates in Theory of Homogenization II: Oscillatory Initial Data

We establish higher order convergence rates in periodic homogenization of fully nonlinear uniformly parabolic Cauchy problems accompanied with rapidly oscillating initial data. Such result is new even for linear problems. Here we construct higher order initial layer and interior correctors, which describe the oscillatory behavior near the initial and interior time zone of the domain. To construct higher order correctors, we develop a regularity theory in macroscopic scales, and prove an exponential decay estimate for initial layer correctors. The higher order expansion requires an iteration process: successively correcting the initial layer, then the interior. This leads to a more complicated asymptotic expansion, as compared to the non-oscillating data case, and this complexity is present even in the linear case. A notable observation for fully nonlinear operators is that even if the given operator is space-time periodic, the interior correctors become aperiodic in the time variable as we proceed with the iteration process. Moreover, each interior corrector of higher order is paired with a space-time periodic version, and the difference between the two decays exponentially fast with time.

math.AP↗

arXiv2019

Anisotropic flow of convex hypersurfaces by the square root of the scalar curvature

We show the existence of a smooth solution for the flow deformed by the square root of the scalar curvature multiplied by a positive anisotropic factor $ψ$ given a strictly convex initial hypersurface in Euclidean space suitably pinched. We also prove the convergence of rescaled surfaces to a smooth limit manifold which is a round sphere. In dimension two, it is shown that, with a volume preserving rescaling, the limit profile satisfies a soliton equation.

math.DG↗

arXiv2019

Local Continuity and Asymptotic Behaviour of Degenerate Parabolic Systems

We study the local Hölder continuity and the asymptotic behaviour of solution, $\mathbf{u}=(u^1,\cdots, u^k)$, of the degenerate system \begin{equation*} u^i_t=\nabla\cdot\left(m\,U^{m-1}\nabla u^i\right) \qquad \text{for $m>1$ and $i=1,\cdots,k$ } \end{equation*} which describes the populations density of $k$-species whose diffusion is determined by their total population density $U=u^1+\cdots+u^k$. For the local Hölder continuity, we adopt the intrinsic scaling and iteration arguments of DeGiorgi, Moser, and Dibenedetto. Under some regularity conditions, we also prove that the population density function of $i$-th species with the population $M_i$ converges in $C_s^{\infty}$ to $\frac{M_i}{M}\mathcal{B}_M(x,t)$ as $t\to \infty$ where $\mathcal{B}_M$ is the Barenblatt profile of the standard porous medium equation with $L^1$ mass $M=M_1+\cdots+M_k$. As a consequence of asymptotic behaviour, it is shown that each density function becomes a concave function after a finite time.

math.AP↗

arXiv2019

The scalar curvature flow with a flat side

We study the near-the-interface behavior of a compact convex scalar curvature flow with a flat side. Under suitable initial conditions on the flat side, we show that the interface propagates with a finite and non-degenerate speed until the flat side vanishes. Then we get optimal derivative estimates of the pressure-like function, optimal decay estimates of curvatures near the interface, and an Aronson-Bénilan-type curvature lower bound, from which we obtain the Hölder regularity of the ratio of the curvature to the optimal decay rate up to the free boundary. In the end, we obtain the short-time and all-time existence of the solution, smooth up to the interface.

math.AP↗

arXiv2018

Translating solutions to the Gauss curvature flow with flat sides

We derive local $C^{2}$ estimates for complete non-compact translating solitons of the Gauss curvature flow in $\mathbb{R}^3$ which are graphs over a convex domain $Ω$. This is closely is related to deriving local $C^{1,1}$ estimates for the degenerate Monge-Ampére equation. As a result, given a weakly convex bounded domain $Ω$, we establish the existence of a $C^{1,1}_{\text{loc}}$ translating soliton. In particular, when the boundary $\partial Ω$ has a line segment, we show the existence of flat sides of the translator from a local a'priori non-degeneracy estimate near the free-boundary.

math.DG↗

arXiv2018

Regularity for fully nonlinear integro-differential operators with kernels of variable orders

We consider fully nonlinear elliptic integro-differential operators with kernels of variable orders, which generalize the integro-differential operators of the fractional Laplacian type in \cite{CS}. Since the order of differentiability of the kernel is not characterized by a single number, we use the constant \begin{align*} C_φ= \left( \int_{\mathbb{R}^n} \frac{1-\cos y_1}{\vert y \vert^n φ(\vert y \vert)} \, dy \right)^{-1} \end{align*} instead of $2-σ$, where $φ$ satisfies a weak scaling condition. We obtain the uniform Harnack inequality and Hölder estimates of viscosity solutions to the nonlinear integro-differential equations.

math.AP↗

arXiv2018

Boundary regularity for nonlocal operators with kernels of variable orders

We study the boundary regularity of solutions of the Dirichlet problem for the nonlocal operator with a kernel of variable orders. Since the order of differentiability of the kernel is not represented by a single number, we consider the generalized Hölder space. We prove that there exists a unique viscosity solution of $Lu = f$ in $D$, $u=0$ in $\mathbb{R}^n \setminus D$, where $D$ is a bounded $C^{1,1}$ open set, and that the solution $u$ satisfies $u \in C^V(D)$ and $u/V(d_D) \in C^α(D)$ with the uniform estimates, where $V$ is the renewal function and $d_D(x) = \mbox{dist}(x, \partial D)$.

math.AP↗

arXiv2017

Higher Order Convergence Rates in Theory of Homogenization III: viscous Hamilton-Jacobi Equations

In this paper, we establish the higher order convergence rates in periodic homogenization of viscous Hamilton-Jacobi equations, which is convex and grows quadratically in the gradient variable. We observe that although the nonlinear structure governs the first order approximation, the nonlinear effect is absorbed as an external source term of a linear equation in the second and higher order approximation. Moreover, we find that the geometric shape of the initial data has to be chosen carefully according to the effective Hamiltonian, in order to achieve the higher order convergence rates.

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

Nodal Sets for "Broken" Quasilinear PDEs

We study the local behavior of the nodal sets of the solutions to elliptic quasilinear equations with nonlinear conductivity part, \begin{equation*} \operatorname{div}(A_s(x,u)\nabla u)=\operatorname{div}{\vec f}(x), \end{equation*} where $A_s(x,u)$ has "broken" derivatives of order $s\geq 0$, such as \begin{equation*} A_s(x,u) = a(x) + b(x)(u^+)^s, \end{equation*} with $(u^+)^0$ being understood as the characteristic function on $\{u>0\}$. The vector ${\vec f}(x)$ is assumed to be $C^α$ in case $s=0$, and $C^{1,α}$ (or higher) in case $s>0$. Using geometric methods, we prove almost complete results (in analogy with standard PDEs) concerning the behavior of the nodal sets. More exactly, we show that the nodal sets, where solutions have (linear) nondegeneracy, are locally smooth graphs. Degenerate points are shown to have structures that follow the lines of arguments as that of the nodal sets for harmonic functions, and general PDEs.

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

Higher Order Convergence Rates in Theory of Homogenization I: Equations of Non-divergence Form

We establish higher order convergence rates in the theory of periodic homogenization of both linear and fully nonlinear uniformly elliptic equations of non-divergence form. The rates are achieved by involving higher order correctors which fix the errors occurring both in the interior and on the boundary layer of our physical domain. The proof is based on a viscosity method and a new regularity theory which captures the stability of the correctors with respect to the shape of our limit profile.

math.AP↗