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Woocheol Choi

Publications and source records attributed to Woocheol Choi.

At least 37 records · Page 2Linked to original sources
arXiv2020

A sharp error analysis for the discontinuous Galerkin method of optimal control problems

In this paper, we are concerned with a nonlinear optimal control problem of ordinary differential equations. We consider a discretization of the problem with the discontinuous Galerkin method with arbitrary order $r \in \mathbb{N}\cup \{0\}$. Under suitable regularity assumptions on the cost functional and solutions of the state equations, we provide sharp estimates for the error of the approximate solutions. Numerical experiments are presented supporting the theoretical results.

math.NA↗
arXiv2019

On the splitting method for the nonlinear Schrödinger equation with initial data in $H^1$

In this paper, we establish a convergence result for the operator splitting scheme $Z_τ$ introduced by Ignat, with initial data in $H^1$, for the nonlinear Schrödinger equation : $$ \partial_t u = i Δu + iλ|u|^{p} u,\qquad u (x,0) =ϕ(x), $$ where $(x,t) \in \mathbb{R}^d \times [0,\infty)$, with $0< p < 4/(d-2)$ for $d\geq3$ and $0< p<\infty$ for $d=1,2$. We prove the $L^2$ convergence of order $\mathcal{O}(τ^{1/2})$ for this scheme with initial data in the space $H^1 (\mathbb{R}^d)$.

math.AP↗
arXiv2019

Privileged Coordinates and Nilpotent Approximation for Carnot Manifolds, II. Carnot Coordinates

This paper is a sequel of arxiv:1709.09045 and deals with privileged coordinates and nilpotent approximation of Carnot manifolds. By a Carnot manifold it is meant a manifold equipped with a filtration by subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. In this paper, we single out a special class of privileged coordinates in which the nilpotent approximation at a given point of a Carnot manifold is given by its tangent group. We call these coordinates Carnot coordinates. Examples of Carnot coordinates include Darboux coordinates on contact manifolds and the canonical coordinates of the first kind of Goodman and Rothschild-Stein. By converting the privileged coordinate of Bellaïche into Carnot coordinates we obtain an effective construction of Carnot coordinates, which we call $\varepsilon$-Carnot coordinates. They form the building block of all systems of Carnot coordinates. On a graded nilpotent Lie group they are given by the group law of the group. For general Carnot manifolds, they depend smoothly on the base point. Moreover, in Carnot coordinates at a given point, they are osculated in a very precise manner by the group law of the tangent group at the point.

math.DG↗
arXiv2019

Privileged Coordinates and Nilpotent Approximation of Carnot Manifolds, I. General Results

In this paper we attempt to give a systematic account on privileged coordinates and the nilpotent approximation of Carnot manifolds. By a Carnot manifold it is meant a manifold with a distinguished filtration of subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. This paper lies down the background for its sequel by clarifying a few points on privileged coordinates and the nilpotent approximation of Carnot manifolds. In particular, we give a description of all the systems of privileged coordinates at a given point. We also give an algebraic characterization of all nilpotent groups that appear as the nilpotent approximation at a given point. In fact, given a nilpotent group $G$ satisfying this algebraic characterization, we exhibit all the changes of variables that transform a given system of privileged coordinates into another system of privileged coordinates in which the nilpotent approximation is given by $G$.

math.DG↗
arXiv2018

Tangent Maps and Tangent Groupoid for Carnot Manifolds

This paper studies the infinitesimal structure of Carnot manifolds. By a Carnot manifold we mean a manifold together with a subbundle filtration of its tangent bundle which is compatible with the Lie bracket of vector fields. We introduce a notion of differential, called Carnot differential, for Carnot manifolds maps (i.e., maps that are compatible with the Carnot manifold structure). This differential is obtained as a group map between the corresponding tangent groups. We prove that, at every point, a Carnot manifold map is osculated in a very precise way by its Carnot differential at the point. We also show that, in the case of maps between nilpotent graded groups, the Carnot differential is given by the Pansu derivative. Therefore, the Carnot differential is the natural generalization of the Pansu derivative to maps between general Carnot manifolds. Another main result is a construction of an analogue for Carnot manifolds of Connes' tangent groupoid. Given any Carnot manifold $(M,H)$ we get a smooth groupoid that encodes the smooth deformation of the pair $M\times M$ to the tangent group bundle $GM$. This shows that, at every point, the tangent group is the tangent space in a true differential-geometric fashion. Moreover, the very fact that we have a groupoid accounts for the group structure of the tangent group. Incidentally, this answers a well-known question of Bellaïche.

math.DG↗
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

Asymptotic behavior of least energy solutions to the Lane-Emden system near the critical hyperbola

The Lane-Emden system is written as \begin{equation*} \begin{cases} -Δu = v^p &\text{in } Ω,\\ -Δv = u^q &\text{in } Ω,\\ u, v > 0 &\text{in } Ω,\\ u = v = 0 &\text{on } \partial Ω\end{cases} \end{equation*} where $Ω$ is a smooth bounded domain in the Euclidean space $\mathbb{R}^n$ for $n \ge 3$ and $0< p< q <\infty$. The asymptotic behavior of least energy solutions near the critical hyperbola was studied by Guerra \cite{G} when $p \geq 1$ and the domain is convex. In this paper, we cover all the remaining cases $p < 1$ and extend the results to any smooth bounded domain.

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

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

The Malgrange-Ehrenpreis theorem for nonlocal Schrödinger operators with certain potentials

In this paper, we prove the Malgrange-Ehrenpreis theorem for nonlocal Schrödinger operators $L_K+V$ with nonnegative potentials $V\in L^q_{\loc}(\BR^n)$ for $q>\f{n}{2s}$ with $0<s<1$ and $n\ge 2$; that is to say, we obtain the existence of a fundamental solution $\fe_V$ for $L_K+V$ satisfying \begin{equation*}\bigl(L_K+V\bigr)\fe_V=\dt_0\,\,\text{ in $\BR^n$ }\end{equation*} in the distribution sense, where $\dt_0$ denotes the Dirac delta mass at the origin. In addition, we obtain a decay of the fundamental solution $\fe_V$.

math.CA↗
arXiv2016

$L^p$ mapping properties for nonlocal Schrödinger operators with certain potential

In this paper, we consider nonlocal Schrödinger equations with certain potentials $V$ given by an integro-differential operator $L_K$ as follows; \begin{equation*}L_K u+V u=f\,\,\text{ in $\BR^n$ }\end{equation*} where $V\in\rh^q$ for $q>\f{n}{2s}$ and $0<s<1$. We denote the solution of the above equation by $\cS_V f:=u$, which is called {\it the inverse of the nonlocal Schrödinger operator $L_K+V$ with potential $V$}; that is, $\cS_V=(L_K+V)^{-1}$. Then we obtain a weak Harnack inequality of weak subsolutions of the nonlocal equation \begin{equation}\begin{cases}L_K u+V u=0\,\,&\text{ in $\Om$,} \quad u=g\,\,&\text{ in $\BR^n\s\Om$,} \end{cases}\end{equation} where $g\in H^s(\BR^n)$ and $\Om$ is a bounded open domain in $\BR^n$ with Lipschitz boundary, and also get an improved decay of a fundamental solution $\fe_V$ for $L_K+V$. Moreover, we obtain $L^p$ and $L^p-L^q$ mapping properties of the inverse $\cS_V$ of the nonlocal Schrödinger operator $L_K+V$.

math.CA↗
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↗
arXiv2016

Minimal energy solutions to the fractional Lane-Emden system, I: Existence and singularity formation

This is the first of two papers which study asymptotic behavior of minimal energy solutions to the fractional Lane-Emden system in a smooth bounded domain $Ω$ \[(-Δ)^s u = v^p, \quad (-Δ)^s v = u^q \text{ in } Ω\quad \text{and} \quad u = v = 0 \text{ on } \pa Ω\quad \text{for } 0 < s < 1\] under the assumption that the subcritical pair $(p,q)$ approaches to the critical Sobolev hyperbola. If $p = 1$, the above problem is reduced to the subcritical higher-order fractional Lane-Emden equation with the Navier boundary condition \[(-Δ)^s u = u^{\frac{n+2s}{n-2s}-\ep} \text{ in } Ω\quad \text{and} \quad u = (-Δ)^{s \over 2} u = 0 \quad \text{for } 1 < s < 2.\] The main objective of this paper is to deduce the existence of minimal energy solutions, and to examine their (normalized) pointwise limits provided that $Ω$ is convex. As a by-product of our study, a new approach for the existence of an extremal function for the Hardy-Littlewood-Sobolev inequality is provided.

math.AP↗
arXiv2016

The Lane-Emden system near the critical hyperbola on nonconvex domains

In this paper we study the asymptotic behavior of minimal energy solutions to the Lane-Emden system $-Δu = v^p$ and $-Δv = u^q$ on bounded domains as the index $(p,q)$ approaches to the critical hyperbola from below. Precisely, we remove the convexity assumption on the domain in the result of Guerra \cite{G}. The main task is to get the uniform boundedness of the solutions near the boundary because it is difficulty to adapt the moving plane method for the system on nonconvex domains if $\max \{p,q\} > \frac{n+2}{n-2}$. For the purpose, we shall derive a contradiction by exploiting carefully the Pohozaev type identity if the maximum point approaches to the boundary.

math.AP↗
arXiv2015

Classification of finite energy solutions to the fractional Lane-Emden-Fowler equations with slightly subcritical exponents

We study qualitative properties of solutions to the fractional Lane-Emden-Fowler equations with slightly subcritical exponents where the associated fractional Laplacian is defined in terms of either the spectra of Dirichlet Laplacian or the integral representation. As a consequence, we classify the asymptotic behavior of all finite energy solutions. Our method also provides a simple and unified approach to deal with the classical (local) Lane-Emden-Fowler equation for any dimension greater than 2.

math.AP↗