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Soojung Kim

Publications and source records attributed to Soojung Kim.

13 recordsLinked to original sources
arXiv2024

Translating surfaces under flows by sub-affine-critical powers of Gauss curvature

We classify the surfaces translating under the flows by sub-affine-critical powers of the Gauss curvature. This, in particular, lists all translating solitons possibly model Type II singularities for convex closed solutions in all positive powers. The surfaces are entire graphs, and therefore our result corresponds to the Liouville theorem for the degenerate Monge--Ampère equations $\det D^2 u=(1+|Du|^2)^{2-\frac{1}{2α}}$ on $\mathbb{R}^2$ in the range $0<α<1/4$. The result also reveals that the moduli spaces of solutions are homeomorphic to either Euclidean spaces or cylinders.

math.DG↗
arXiv2024

Continuous family of surfaces translating by powers of Gauss curvature

This paper shows the existence of convex translating surfaces under the flow by the $α$-th power of Gauss curvature for the sub-affine-critical regime $ 0 < α< 1/4$. The key aspect of our study is that our ansatz at infinity is the graph of homogeneous functions whose level sets are closed curves shrinking under the flow by the $\fracα{1-α}$-th power of curvature. For each ansatz, we construct a family of translating surfaces generated by the Jacobi fields with effective growth rates. Moreover, the construction shows quantitative estimate on the rate of convergence between different translators to each other, which is required to show the continuity of the family. As a result, the family is regarded as a topological manifold. The construction in this paper will become the ground of forthcoming research, where we aim to prove that every translating surface must correspond to one of the solutions obtained herein, classifying translating surfaces and identifying the topology of the moduli space.

math.DG↗
arXiv2024

Monge-Ampère equations with right-hand sides of polynomial growth

We study the regularity and the growth rates of solutions to two-dimensional Monge-Ampère equations with the right-hand side exhibiting polynomial growth. Utilizing this analysis, we demonstrate that the translators for the flow by sub-affine-critical powers of the Gauss curvature are smooth, strictly convex entire graphs. These graphs exhibit specific growth rates that depend solely on the power of the flow.

math.AP↗
arXiv2018

Vanishing time behavior of solutions to the fast diffusion equation

Let $n\geq 3$, $0< m<\frac{n-2}{n}$ and $T>0$. We construct positive solutions to the fast diffusion equation $u_t=Δu^m$ in $\mathbb{R}^n\times(0,T)$, which vanish at time $T$. By introducing a scaling parameter $β$ inspired by \cite{DKS}, we study the second-order asymptotics of the self-similar solutions associated with $β$ at spatial infinity. We also investigate the asymptotic behavior of the solutions to the fast diffusion equation near the vanishing time $T$, provided that the initial value of the solution is close to the initial value of some self-similar solution and satisfies some proper decay condition at infinity. Depending on the range of the parameter $β$, we prove that the rescaled solution converges either to a self-similar profile or to zero as $t\nearrow T$. The former implies asymptotic stabilization towards a self-similar solution, and the latter is a new vanishing phenomenon even for the case $n\ge3$ and $m=\frac{n-2}{n+2}\,$ which corresponds to the Yamabe flow on $\mathbb{R}^n$ with metric $g=u^{\frac{4}{n+2}}dx^2$.

math.AP↗
arXiv2016

Twisted solutions to a simplified Ericksen-Leslie equation

In this article we construct global solutions to a simplified Ericksen-Leslie system on $\mathbb{R}^3$. The constructed solutions are twisted and periodic along the $x_3$-axis with period $d = 2π\big/ μ$. Here $μ> 0$ is the twist rate. $d$ is the distance between two planes which are parallel to the $x_1x_2$-plane. Liquid crystal material is placed in the region enclosed by these two planes. Given a well-prepared initial data, our solutions exist classically for all $t \in [0, \infty)$. However these solutions become singular at all points on the $x_3$-axis and escape into third dimension exponentially while $t \rightarrow \infty$. An optimal blow up rate is also obtained.

math.AP↗
arXiv2015

Asymptotic large time behavior of singular solutions of the fast diffusion equation

We study the asymptotic large time behavior of singular solutions of the fast diffusion equation $u_t=Δu^m$ in $({\mathbb R}^n\setminus\{0\})\times(0,\infty)$ in the subcritical case $0 A_1>0$ and $\frac{2}{1-m}<γ<\frac{n-2}{m}$, where $β:=\frac{1}{2-γ(1-m)}$, $α:=\frac{2β-1}{1-m},$ and the self-similar profile $f_i$ satisfies the elliptic equation $$ Δf^m+αf+βx\cdot \nabla f=0\quad \mbox{in ${\mathbb R}^n\setminus\{0\}$} $$ with $\lim_{|x|\to0}|x|^{\frac{ α}{ β}}f_i(x)=A_i$ and $\lim_{|x|\to\infty}|x|^{\frac{n-2}{m}}{f_i}(x)= D_{A_i} $ for some constants $D_{A_i}>0$. When $\frac{2}{1-m}<γ<n$, under an integrability condition on the initial value $u_0$ of the singular solution $u$, we prove that the rescaled function $$ \tilde u(y,τ):= t^{\,α} u(t^{\,β} y,t),\quad{ τ:=\log t}, $$ converges to some self-similar profile $f$ as $τ\to\infty$.

math.AP↗
arXiv2015

Harnack inequality for degenerate and singular operators of $p$-Laplacian type on Riemannian manifolds

We study viscosity solutions to degenerate and singular elliptic equations of $p$-Laplacian type on Riemannian manifolds. The Krylov-Safonov type Harnack inequality for the $p$-Laplacian operators with $1<p<\infty$ is established on the manifolds with Ricci curvature bounded from below based on ABP type estimates. We also prove the Harnack inequality for nonlinear $p$-Laplacian type operators assuming that a nonlinear perturbation of Ricci curvature is bounded below.

math.AP↗
arXiv2014

Regularity for fully nonlinear integro-differential operators with regularly varying kernels

In this paper, the regularity results for the integro-differential operators of the fractional Laplacian type by Caffarelli and Silvestre \cite{CS1} are extended to those for the integro-differential operators associated with symmetric, regularly varying kernels at zero. In particular, we obtain the uniform Harnack inequality and Hölder estimate of viscosity solutions to the nonlinear integro-differential equations associated with the kernels $K_{σ, β}$ satisfying $$ K_{σ,β}(y)\asymp \frac{ 2-σ}{|y|^{n+σ}}\left( \log\frac{2}{|y|^2}\right)^{β(2-σ)}\quad \mbox{near zero} $$ with respect to $σ\in(0,2)$ close to $2$ (for a given $β\in\mathbb R$), where the regularity estimates do not blow up as the order $ σ\in(0,2)$ tends to $2.$

math.AP↗
arXiv2014

Parabolic Harnack inequality of viscosity solutions on Riemannian manifolds

We consider viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on a Riemannian manifold $M$, with the sectional curvature bounded from below by $-κ$ for $κ\geq 0$. In the elliptic case, Wang and Zhang \cite{WZ} recently extended the results of \cite{Ca} to nonlinear elliptic equations in nondivergence form on such $M$, where they obtained the Harnack inequality for classical solutions. We establish the Harnack inequality for nonnegative {\it viscosity solutions} to nonlinear uniformly {\it parabolic equations} in nondivergence form on $M$. The Harnack inequality of nonnegative viscosity solutions to the elliptic equations is also proved.

math.AP↗
arXiv2012

Harnack inequality for nondivergent parabolic operators on Riemannian manifolds

We consider second-order linear parabolic operators in non-divergence form that are intrinsically defined on Riemannian manifolds. In the elliptic case, Cabré proved a global Krylov-Safonov Harnack inequality under the assumption that the sectional curvature of the underlying manifold is nonnegative. Later, Kim improved Cabré's result by replacing the curvature condition by a certain condition on the distance function. Assuming essentially the same condition introduced by Kim, we establish Krylov-Safonov Harnack inequality for nonnegative solutions of the non-divergent parabolic equation. This, in particular, gives a new proof for Li-Yau Harnack inequality for positive solutions to the heat equation in a manifold with nonnegative Ricci curvature.

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↗