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Jongmyeong Kim

Publications and source records attributed to Jongmyeong Kim.

13 recordsLinked to original sources
arXiv2026

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

Let $n\ge 3$, $0 0$, for the fast diffusion equation $u_t=Δ(u^m/m)$ in $(\mathbb{R}^n\setminus\{0\})\times (0,\infty)$, where $f$ satisfies \begin{equation*} Δ(f^m/m) + αf + βx \cdot \nabla f =0 \quad \text{in} \; \mathbb{R}^n\setminus\{0\} \end{equation*} with $\lim_{|x| \to 0} |x|^{ \fracαβ}f(x)=A$ and $\lim_{|x| \to \infty}f(x) = D_A$ for some constants $A>0$, $D_A > 0$. We also obtain an asymptotic expansion of such singular radially symmetric solution $f$ near the origin. We will also prove the asymptotic large time behaviour of the singular solutions of the fast diffusion equation $u_t= Δ(u^m/m)$ in $(\mathbb{R}^n\setminus\{0\})\times (0,\infty)$, $u(x,0)=u_0(x)$ in $\mathbb{R}^n\setminus\{0\}$, satisfying the condition $A_1|x|^{-γ}\leq u_0(x)\leq A_2|x|^{-γ}$ in $\mathbb{R}^n\setminus\{0\}$, for some constants $A_2>A_1>0$ and $n\leγ<\frac{n-2}{m}$.

math.AP↗
arXiv2026

The fractional $p$-Laplacian on hyperbolic spaces

We present three equivalent definitions of the fractional $p$-Laplacian $(-Δ_{\mathbb{H}^{n}})^{s}_{p}$, $0 1$, with normalizing constants, on hyperbolic spaces. The explicit values of the constants enable us to study the convergence of the fractional $p$-Laplacian to the $p$-Laplacian as $s \to 1^{-}$.

math.AP↗
arXiv2025

Harnack inequality for fractional Laplacian-type operators on hyperbolic spaces

We establish the Krylov--Safonov theory for a large class of nonlocal operators of order $2s \in (0,2)$ on hyperbolic spaces $\mathbb{H}^{n}_κ$ with curvature $-κ<0$. We prove the Alexandrov--Bakelman--Pucci (ABP) estimates, Krylov--Safonov Harnack inequality, and Hölder estimates. Notably, the Harnack inequality is new even for the fractional Laplacian. The novelty of the results lies in the robustness of the regularity estimates as $s \to 1$ and $κ\to 0$: they recover the classical regularity estimates for second-order operators on $\mathbb{H}^{n}_κ$ as $s \to 1$, and for fractional-order operators on Euclidean spaces as $κ\to 0$. Since the operators on hyperbolic spaces exhibit qualitatively different behavior compared to their Euclidean counterparts, we introduce new scale functions which take the effect of negative curvatures into account.

math.AP↗
arXiv2024

Calabi-Yau structures on Rabinowitz Fukaya categories

In this paper, we prove that the derived Rabinowitz Fukaya category of a Liouville domain $M$ of dimension $2n$ is $(n-1)$-Calabi--Yau assuming the wrapped Fukaya category of $M$ admits an at most countable set of Lagrangians that generate it and satisfy some finiteness condition on morphism spaces between them.

math.SG↗
arXiv2024

Regularity for solutions of non-uniformly elliptic equations in non-divergence form

We prove the Aleksandrov--Bakelman--Pucci estimate for non-uniformly elliptic equations in non-divergence form. Moreover, we investigate local behaviors of solutions of such equations by developing local boundedness and weak Harnack inequality. Here we impose an integrability assumption on ellipticity representing degeneracy or singularity, instead of specifying the particular structure of ellipticity.

math.AP↗
arXiv2023

Cluster categories from Fukaya categories

We show that the derived wrapped Fukaya category $D^π\mathcal{W}(X_{Q}^{d+1})$, the derived compact Fukaya category $D^π\mathcal{F}(X_{Q}^{d+1})$ and the cocore disks $L_{Q}$ of the plumbing space $X_{Q}^{d+1}$ form a Calabi--Yau triple. As a consequence, the quotient category $D^π\mathcal{W}(X_{Q}^{d+1})/D^π\mathcal{F}(X_{Q}^{d+1})$ becomes the cluster category associated to $Q$. One of its properties is a Calabi--Yau structure. Also it is known that this quotient category is quasi-equivalent to the Rabinowitz Fukaya category due to the work of Ganatra--Gao--Venkatesh. We compute the morphism space of $L_{Q}$ in $D^π\mathcal{W}(X_{Q}^{d+1})/D^π\mathcal{F}(X_{Q}^{d+1})$ using the Calabi--Yau structure, which is isomorphic to the Rabinowitz Floer cohomology of $L_{Q}$.

math.SG↗
arXiv2022

Computation of categorical entropy via spherical functors

We study the relationship between the categorical entropy of the twist and cotwist functors along a spherical functor. In particular, we prove the categorical entropy of the twist functor coincides with that of the cotwist functor if the essential image of the right adjoint functor of the spherical functor contains a split-generator. We also see our results generalize the computations of the categorical entropy of spherical twists and $\mathbb{P}$-twists by Ouchi and Fan. As an application, we apply our results to the Gromov--Yomdin type conjecture by Kikuta--Takahashi.

math.AG↗
arXiv2022

Categorical entropy, (co-)t-structures and ST-triples

In this paper, we study a dynamical property of an exact endofunctor $Φ: \mathcal{D} \to \mathcal{D}$ of a triangulated category $\mathcal{D}$. In particular, we are interested in the following question: Given full triangulated subcategories $\mathcal{A},\mathcal{B} \subset \mathcal{D}$ such that $Φ(\mathcal{A}) \subset \mathcal{A}$ and $Φ(\mathcal{B}) \subset \mathcal{B}$, how the categorical entropies of $Φ|_\mathcal{A}$ and $Φ|_\mathcal{B}$ are related? To answer this question, we introduce new entropy-type invariants using bounded (co-)t-structures with finite (co-)hearts and prove their basic properties. We then apply these results to answer our question for the situation where $\mathcal{A}$ has a bounded t-structure and $\mathcal{B}$ has a bounded co-t-structure which are, in some sense, dual to each other.

math.SG↗
arXiv2021

On Gromov-Yomdin type theorems and a categorical interpretation of holomorphicity

In topological dynamics, the Gromov--Yomdin theorem states that the topological entropy of a holomorphic automorphism $f$ of a smooth projective variety is equal to the logarithm of the spectral radius of the induced map $f^*$. In order to establish a categorical analogue of the Gromov--Yomdin theorem, one first needs to find a categorical analogue of a holomorphic automorphism. In this paper, we propose a categorical analogue of a holomorphic automorphism and prove that the Gromov--Yomdin type theorem holds for them.

math.AG↗
arXiv2021

Entropy of the composition of two spherical twists

Given a categorical dynamical system, i.e. a triangulated category together with an endofunctor, one can try to understand the complexity of the system by computing the entropy of the endofunctor. Computing the entropy of the composition of two endofunctors is hard, and in general the result doesn't have to be related to the entropy of the single pieces. In this paper we compute the entropy of the composition of two spherical twists around spherical objects, showing that it depends on the dimension of the graded vector space of morphisms between them. As a consequence of these computations we produce new counterexamples to Kikuta-Takahashi's conjecture. In particular, we describe the first counterexamples in odd dimension and examples for the $d$-Calabi-Yau Ginzburg dg algebra associated to the $A_2$ quiver.

math.AG↗
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↗