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Jinmyoung Seok

Publications and source records attributed to Jinmyoung Seok.

15 recordsLinked to original sources
arXiv2026

Measure-valued free-energy minimizers for trapped bosons with repulsive Coulomb interaction

We study a free-energy minimization problem for a trapped semiclassical Bose gas with repulsive Coulomb self-interaction. Because the bosonic entropy density has zero recession slope, the natural relaxation is posed over nonnegative finite Radon measures on phase space, and a minimizer may have a singular component. We interpret this component as condensation within the relaxed semiclassical model. We prove existence and uniqueness at every positive temperature, derive the Euler-Lagrange contact condition and an obstacle-type formula for the condensed density, and establish sharp phase boundaries under only the standing continuity assumptions on the trap. At fixed mass there is a unique finite positive critical temperature, with condensation precisely below it. At fixed temperature there is a sharp critical mass, possibly infinite, separating normal and condensed minimizers. For harmonic traps we obtain a sharp confinement-strength dichotomy. We also prove an abstract conditional variational-stability statement for measure-valued curves that conserve mass and satisfy the free-energy inequality.

math.AP↗
arXiv2024

Non-ergodic linear convergence property of the delayed gradient descent under the strongly convexity and the Polyak-Łojasiewicz condition

In this work, we establish the linear convergence estimate for the gradient descent involving the delay $τ\in\mathbb{N}$ when the cost function is $μ$-strongly convex and $L$-smooth. This result improves upon the well-known estimates in Arjevani et al. \cite{ASS} and Stich-Karmireddy \cite{SK} in the sense that it is non-ergodic and is still established in spite of weaker constraint of cost function. Also, the range of learning rate $η$ can be extended from $η\leq 1/(10Lτ)$ to $η\leq 1/(4Lτ)$ for $τ=1$ and $η\leq 3/(10Lτ)$ for $τ\geq 2$, where $L >0$ is the Lipschitz continuity constant of the gradient of cost function. In a further research, we show the linear convergence of cost function under the Polyak-Łojasiewicz\,(PL) condition, for which the available choice of learning rate is further improved as $η\leq 9/(10Lτ)$ for the large delay $τ$. The framework of the proof for this result is also extended to the stochastic gradient descent with time-varying delay under the PL condition. Finally, some numerical experiments are provided in order to confirm the reliability of the analyzed results.

math.OC↗
arXiv2024

Semiclassical equivalence of two white dwarf models as ground states of the relativistic Hartree-Fock and Vlasov-Poisson energies

We are concerned with the semi-classical limit for ground states of the relativistic Hartree-Fock energies (HF) under a mass constraint, which are considered as the quantum mean-field model of white dwarfs \cite{LeLe}. In Jang and Seok \cite{JS}, fermionic ground states of the relativistic Vlasov-Poisson energy (VP) are constructed as a classical mean-field model of white dwarfs, and are shown to be equivalent to the classical Chandrasekhar model. In this paper, we prove that as the reduced Planck constant $\hbar$ goes to the zero, the $\hbar$-parameter family of the ground energies and states of (HF) converges to the fermionic ground energy and state of (VP) with the same mass constraint.

math.AP↗
arXiv2021

Solitary waves for the nonlinear Schrödinger-Poisson system with positron-electron interaction

In this paper, we study the existence of positive solutions to the nonlinear elliptic system, which is derived from taking the nonrelativistic limit of the nonlinear Maxwell-Klein-Gordon equations under the decomposition of waves functions into positron and electron parts. We characterize the existence and nonexistence of positive vector solutions, depending on parameters $p$ and $μ_{ij}$.

math.AP↗
arXiv2021

Semi-classical limit of quantum free energy minimizers for the gravitational Hartree equation

For the gravitational Vlasov-Poisson equation, Guo and Rein constructed a class of classical isotropic states as minimizers of free energies (or energy-Casimir functionals) under mass constraints. For the quantum counterpart, that is, the gravitational Hartree equation, isotropic states are constructed as free energy minimizers by Aki, Dolbeault and Sparber. In this paper, we are concerned with the correspondence between quantum and classical isotropic states. Precisely, we prove that as the Planck constant $\hbar$ goes to zero, free energy minimizers for the Hartree equation converge to those for the Vlasov-Poisson equation in terms of potential functions as well as via the Husimi transform and the Töplitz quantization.

math.AP↗
arXiv2020

Nonrelativistic limit of solitary waves for nonlinear Maxwell-Klein-Gordon equations

We study the nonrelativistic limit of solitary waves from Nonlinear Maxwell-Klein-Gordon equations (NMKG) to Nonlinear Schrodinger-Poisson equations (NSP). It is known that the existence or multiplicity of positive solutions depends on the choices of parameters the equations contain. In this paper, we prove that for a given positive solitary wave of NSP, which is found in Ruiz's work \cite{R}, there corresponds a family of positive solitary waves of NMKG under the nonrelativistic limit. Notably, our results contain a new result of existence of positive solutions to (NMKG) with lower order nonlinearity.

math.AP↗
arXiv2018

Convergence rate of the finite element approximation for extremizers of Sobolev inequalities

In this paper, we are concerned with the convergence rate of a FEM based numerical scheme approximating extremal functions of the Sobolev inequality. We prove that when the domain is polygonal and convex in $\R^2$, the convergence of a finite element solution to an exact extremal function in $L^2$ and $H^1$ norms has the rates $O(h^2)$ and $O(h)$ respectively, where $h$ denotes the mesh size of a triangulation of the domain.

math.NA↗
arXiv2018

Limit profiles and uniqueness of ground states to the nonlinear Choquard equations

Consider nonlinear Choquard equations \begin{equation*} \left\{\begin{array}{rcl} -Δu +u & = &(I_α*|u|^p)|u|^{p-2}u \quad \text{in } \mathbb{R}^N, \\ \lim_{x \to \infty}u(x) & = &0, \end{array}\right. \end{equation*} where $I_α$ denotes Riesz potential and $α\in (0, N)$. In this paper, we investigate limit profiles of ground states of nonlinear Choquard equations as $α\to 0$ or $α\to N$. This leads to the uniqueness and nondegeneracy of ground states when $α$ is sufficiently close to $0$ or close to $N$.

math.AP↗
arXiv2017

Semilinear elliptic equations with the pseudo-relativistic operator on a bounded domain

We study the Dirichlet problem for the semilinear equations involving the pseudo-relativistic operator on a bounded domain, (\sqrt{-Δ+ m^2} - m)u =|u|^{p-1}u \quad \textrm{in}~Ω, with the Dirichlet boundary condition $u=0$ on $\partial Ω$. Here, $p \in (1,\infty)$ and the operator $(\sqrt{-Δ+ m^2} - m)$ is defined in terms of spectral decomposition. In this paper, we investigate existence and nonexistence of a nontrivial solution, depending on the choice of $p$, $m$ and $Ω$. Precisely, we show that $(i)$ if $p$ is not $H^1$ subcritical ($p \geq \frac{n+2}{n-2}$) and $Ω$ is star-shaped, the equation has no nontrivial solution for all $m > 0$; $(ii)$ if $p$ is not $H^{1/2}$ supercritical ($1 0$ and any bounded domain $Ω$; $(iii)$ finally, in the intermediate range ($\frac{n+1}{n-1}<p<\frac{n+2}{n-2}$), the problem has a nontrivial solution, provided that $m$ is sufficiently large and the problem -Δu = |u|^{p-1}u \quad \textrm{in}~Ω, \qquad u =0\quad \textrm{on}~\partial Ωadmits a non-degenerate nontrivial solution, for example, when $Ω$ is a ball or an annulus.

math.AP↗
arXiv2017

Uniqueness and symmetry of ground states for higher-order equations

We establish uniqueness and radial symmetry of ground states for higher-order nonlinear Schrödinger and Hartree equations whose higher-order differentials have small coefficients. As an application, we obtain error estimates for higher-order approximations to the pseudo-relativistic ground state. Our proof adapts the strategy of Lenzmann using local uniqueness near the limit of ground states in a variational problem. However, in order to bypass difficulties from lack of symmetrization tools for higher-order differential operators, we employ the contraction mapping argument in our earlier work to construct radially symmetric real-valued solutions, as well as improving local uniqueness near the limit.

math.AP↗
arXiv2017

Nonlinear Choquard equations: doubly critical case

Consider nonlinear Choquard equations \begin{equation*} \left\{\begin{array}{rcl} -Δu +u & = &(I_α*F(u))F'(u) \quad \text{in } \mathbb{R}^N, \\ \lim_{x \to \infty}u(x) & = &0, \end{array}\right. \end{equation*} where $I_α$ denotes Riesz potential and $α\in (0, N)$. In this paper, we show that when $F$ is doubly critical, i.e. $F(u) = \frac{N}{N+α}|u|^{\frac{N+α}{N}}+\frac{N-2}{N+α}|u|^{\frac{N+α}{N-2}}$, the nonlinear Choquard equation admits a nontrivial solution if $N \geq 5$ and $α+ 4 < N$.

math.AP↗
arXiv2017

On critical and supercritical pseudo-relativistic nonlinear Schrödinger equations

In this paper, we investigate existence and non-existence of a nontrivial solution to the pseudo-relativistic nonlinear Schrödinger equation $$\left( \sqrt{-c^2Δ+ m^2 c^4}-mc^2\right) u + μu = |u|^{p-1}u\quad \textrm{in}~\mathbb{R}^n~(n \geq 2)$$ involving an $H^{1/2}$-critical/supercritical power-type nonlinearity, i.e., $p \geq \frac{n+1}{n-1}$. We prove that in the non-relativistic regime, there exists a nontrivial solution provided that the nonlinearity is $H^{1/2}$-critical/supercritical but it is $H^1$-subcritical. On the other hand, we also show that there is no nontrivial bounded solution either $(i)$ if the nonlinearity is $H^{1/2}$-critical/supercritical in the ultra-relativistic regime or $(ii)$ if the nonlinearity is $H^1$-critical/supercritical in all cases.

math.AP↗
arXiv2016

Optimal convergence rate of nonrelativistic limit for the nonlinear pseudo-relativistic equations

In this paper, we are concerned with the nonrelativistic limit of the following pseudo-relativistic equation with Hartree nonlinearity or power type nonlinearity \[ \left(\sqrt{-\hbar^2c^2 Δ+m^2c^4} - mc^2 \right) u + μu = \mathcal{N}(u), \] where $c$ denotes the speed of light. We prove that the ground states of this equation converges to the ground state of its nonrelativistic counterpart \[ -\frac{\hbar^2}{2m}Δu + μu = \mathcal{N}(u) \] with an explicit convergence rate $1/c^2$ in arbitrary order as $c \to \infty$. Moreover, we show that this rate is optimal.

math.AP↗