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Jinsol Seo

Publications and source records attributed to Jinsol Seo.

15 recordsLinked to original sources
arXiv2026

Fractional Parabolic Partial Differential Equations in Anisotropic Spectral Barron Spaces: Regularity and Neural Approximation

We study fractional parabolic initial-value problems with lower-order drift and potential terms in anisotropic spectral Barron spaces, defined by weighted space--time Fourier $L^1$ norms adapted to parabolic scaling. We prove existence, uniqueness, and maximal regularity with a gain of one derivative in time and $γ$ derivatives in space, where $γ>0$ is the order of the fractional Laplacian. The evolution is defined only for $t\geq0$, whereas the finite-time norm requires a global extension with sufficient temporal Fourier decay. We construct a finite reflected semigroup extension using a Vandermonde system to match derivatives at $t=0$, obtaining temporal Fourier estimates uniform in the semigroup parameter. Combined with Fourier multiplier estimates for the damped principal operator, it yields maximal regularity. Dimension-independent multiplication estimates support a finite regularity bootstrap, while interpolation and sufficient damping absorb the lower-order terms in the base estimate. The a priori estimate and the method of continuity yield maximal regularity without smallness assumptions on the lower-order coefficients. A frequency-localized counterexample shows that a uniform-in-time spatial Barron bound on the forcing does not imply the corresponding two-derivative solution bound, even for the one-dimensional heat equation. Using this regularity, Fourier sampling yields $n^{-1/2}$ approximation rates for the solution in mixed space--time Sobolev norms using shallow networks with suitable activations. Sampling in a product Hilbert space yields a population-level PINN consistency estimate for shallow cosine networks on a bounded cylinder. There exists a single width-$n$ network for which the sum of the squared mixed-Sobolev solution error, the squared $L^2$-norm of the residual for the whole-space fractional equation, and the squared initial-data error is $O(n^{-1})$.

math.AP↗
arXiv2025

Characterizations of weighted Besov and Triebel-Lizorkin spaces with variable smoothness

In this paper, we study different types of weighted Besov and Triebel-Lizorkin spaces with variable smoothness. The function spaces can be defined by means of the Littlewood-Paley theory in the field of Fourier analysis, while there are other norms arising in the theory of partial differential equations such as Sobolev-Slobodeckij spaces. It is known that two norms are equivalent when one considers constant regularity function spaces without weights. We show that the equivalence still holds for variable smoothness and weights, which is accomplished by making use of shifted maximal functions, Peetre's maximal functions, and the reverse Hölder inequality. Moreover, we obtain a weighted regularity estimate for time-fractional evolution equations and a generalized Sobolev embedding theorem without weights.

math.CA↗
arXiv2025

Weighted Sobolev space theory for Poisson's equation in non-smooth domains

We introduce a general $L_p$-solvability result for the Poisson equation in non-smooth domains $Ω\subset \mathbb{R}^d$, with the zero Dirichlet boundary condition. Our sole assumption on the domain $Ω$ is the Hardy inequality: There exists a constant $N>0$ such that $$ \int_Ω\Big|\frac{f(x)}{d(x,\partialΩ)}\Big|^2\,\mathrm{d} x\leq N\int_Ω|\nabla f|^2 \,\mathrm{d} x\quad\text{for any}\quad f\in C_c^{\infty}(Ω)\,. $$ To describe the boundary behavior of solutions in a general framework, we propose a weight system composed of a superharmonic function and the distance function to the boundary. Additionally, we explore applications across a variety of non-smooth domains, including convex domains, domains with exterior cone condition, totally vanishing exterior Reifenberg domains, and domains $Ω\subset\mathbb{R}^d$ for which the Aikawa dimension of $Ω^c$ is less than $d-2$. Using superharmonic functions tailored to the geometric conditions of the domain, we derive weighted $L_p$-solvability results for various non-smooth domains and specific weight ranges that differ for each domain condition. Furthermore, we provide an application to the Hölder continuity of solutions.

math.AP↗
arXiv2025

Weighted Sobolev space theory for the heat equation and the time-fractional heat equation in non-smooth domains

We present a general $L_p$-solvability framework for both the classical and time-fractional heat equations in non-smooth domains under the zero Dirichlet boundary condition. We consider domains $Ω$ admitting the Hardy inequality: There exists a constant $N>0$ such that $$ \int_Ω\Big|\frac{f(x)}{d(x,\partialΩ)}\Big|^2\,\mathrm{d} x\leq N\int_Ω|\nabla f|^2 \,\mathrm{d} x\quad\text{for any}\quad f\in C_c^{\infty}(Ω)\,. $$ To illustrate the boundary behavior of solutions in a general framework, we employ a weight system composed of a superharmonic function and a distance function to the boundary. Further, we investigate applications to various non-smooth domains, including convex domains, domains with exterior cone condition, totally vanishing exterior Reifenberg domains, and domains $Ω\subset\mathbb{R}^d$ for which the Aikawa dimension of $Ω^c$ is less than $d-2$. By using superharmonic functions tailored to the geometric conditions of the domain, we derive weighted $L_p$-solvability results for various non-smooth domains, with specific weight ranges that differ for each domain condition. In addition, we provide an application to the Hölder continuity of solutions in domains with the volume density condition, as well as pointwise estimates for solutions in Lipschitz cones.

math.AP↗
arXiv2025

Maximal operators given by Fourier multipliers with dilation of fractional dimensions

In this paper, we investigate $L^p$ bounds of maximal Fourier multiplier operators with dilation of fractional dimensions. For Fourier multipliers, we suggest a criterion related to dimensions of dilation sets which guarantees $L^p$ bounds of the maximal operators for each $p$. Our criterion covers Mikhlin-type multipliers, multipliers with limited decay, and multipliers with slow decay.

math.CA↗
arXiv2025

A regularity theory for evolution equations with space-time anisotropic non-local operators in mixed-norm Sobolev spaces

In this article, we study the regularity of solutions to inhomogeneous time-fractional evolution equations involving anisotropic non-local operators in mixed-norm Sobolev spaces of variable order, with non-trivial initial conditions. The primary focus is on space-time non-local equations where the spatial operator is the infinitesimal generator of a vector of independent subordinate Brownian motions, making it the sum of subdimensional non-local operators. A representative example of such an operator is $(Δ_{x})^{β_{1}/2}+(Δ_{y})^{β_{2}/2}$. We establish existence, uniqueness, and precise estimates for solutions in corresponding Sobolev spaces. Due to singularities arising in the Fourier transforms of our operators, traditional methods involving Fourier analysis are not directly applicable. Instead, we employ a probabilistic approach to derive solution estimates. Additionally, we identify the optimal initial data space using generalized real interpolation theory.

math.AP↗
arXiv2023

On the trace theorem to Volterra-type equations with local or non-local derivatives

This paper considers traces at the initial time for solutions of evolution equations with local or non-local derivatives in vector-valued $L_p$ spaces with $A_p$ weight. To achieve this, we begin by introducing a generalized real interpolation method. Within the framework of generalized interpolation theory, we make use of stochastic process theory and two-weight Hardy's inequality to derive our trace and extension theorems. Our results encompass findings applicable to time-fractional equations with broad temporal weight functions.

math.AP↗
arXiv2023

Electronic-grade epitaxial (111) KTaO3 heterostructures

KTaO3 has recently attracted attention as a model system to study the interplay of quantum paraelectricity, spin-orbit coupling, and superconductivity. However, the high and low vapor pressures of potassium and tantalum present processing challenges to creating interfaces clean enough to reveal the intrinsic quantum properties. Here, we report superconducting heterostructures based on electronic-grade epitaxial (111) KTaO3 thin films. Electrical and structural characterizations reveal that two-dimensional electron gas at the heterointerface between amorphous LaAlO3 and KTaO3 thin film exhibits significantly higher electron mobility, superconducting transition temperature and critical current density than those in bulk single crystal KTaO3-based heterostructures owing to cleaner interface in KTaO3 thin films. Our hybrid approach may enable epitaxial growth of other alkali metal-based oxides that lie beyond the capabilities of conventional methods.

cond-mat.mtrl-sci↗
arXiv2023

Sobolev space theory for Poisson's and the heat equations in non-smooth domains via superharmonic functions and Hardy's inequality

We prove the unique solvability for the Poisson and heat equations in non-smooth domains $Ω\subset \mathbb{R}^d$ in weighted Sobolev spaces. The zero Dirichlet boundary condition is considered, and domains are merely assumed to admit the Hardy inequality: $$ \int_Ω\Big|\frac{f(x)}{d(x,\partialΩ)}\Big|^2\,\,\mathrm{d} x\leq N\int_Ω|\nabla f|^2 \,\mathrm{d} x\,\,\,\,,\,\,\,\, \forall f\in C_c^{\infty}(Ω)\,. $$ To describe the boundary behavior of solutions, we introduce a weight system that consists of superharmonic functions and the distance function to the boundary. The results provide separate applications for the following domains: convex domains, domains with exterior cone condition, totally vanishing exterior Reifenberg domains, conic domains, and domains $Ω\subset\mathbb{R}^d$ which the Aikawa dimension of $Ω^c$ is less than $d-2$.

math.AP↗
arXiv2023

Maximal operators associated with Fourier multipliers and applications

In this paper, we introduce a criterion for maximal operators associated with Fourier multipliers to be bounded on $L^p(\mathbb{R}^d)$. Noteworthy examples satisfying the criterion are multipliers of the Mikhlin type or limited decay which are not necessarily radial. To do so, we make use of modified square function estimates and bilinear interpolation. In result, we obtain convergence results for fractional half-wave equations and surface averages as well as the $L^p$ boundedness for the maximal operators.

math.CA↗
arXiv2022

Sobolev space theory and Hölder estimates for the stochastic partial differential equations on conic and polygonal domains

We establish existence, uniqueness, and Sobolev and Hölder regularity results for the stochastic partial differential equation $$ du=\left(\sum_{i,j=1}^d a^{ij}u_{x^ix^j}+f^0+\sum_{i=1}^d f^i_{x^i}\right)dt+\sum_{k=1}^{\infty}g^kdw^k_t, \quad t>0, \,x\in \mathcal{D} $$ given with non-zero initial data. Here $\{w^k_t: k=1,2,\cdots\}$ is a family of independent Wiener processes defined on a probability space $(Ω, \mathbb{P})$, $a^{ij}=a^{ij}(ω,t)$ are merely measurable functions on $Ω\times (0,\infty)$, and $\mathcal{D}$ is either a polygonal domain in $\mathbb{R}^2$ or an arbitrary dimensional conic domain of the type \begin{equation} \label{conic} \mathcal{D}(\mathcal{M}):=\left\{x\in \mathbb{R}^d :\,\frac{x}{|x|}\in \mathcal{M}\right\}, \quad \quad \mathcal{M}\in S^{d-1}, \quad (d\geq 2) \end{equation} where $\mathcal{M}$ is an open subset of $S^{d-1}$ with $C^2$ boundary. We measure the Sobolev and Hölder regularities of arbitrary order derivatives of the solution using a system of mixed weights consisting of appropriate powers of the distance to the vertices and of the distance to the boundary. The ranges of admissible powers of the distance to the vertices and to the boundary are sharp.

math.PR↗
arXiv2021

Oxide two-dimensional electron gas with high mobility at room-temperature

The prospect of 2-dimensional electron gases (2DEGs) possessing high mobility at room temperature in wide-bandgap perovskite stannates is enticing for oxide electronics, particularly to realize transparent and high-electron mobility transistors. Nonetheless only a small number of studies to date report 2DEGs in BaSnO3-based heterostructures. Here, we report 2DEG formation at the LaScO3/BaSnO3 (LSO/BSO) interface with a room-temperature mobility of 60 cm2/V s at a carrier concentration of 1.7x1013 cm-2. This is an order of magnitude higher mobility at room temperature than achieved in SrTiO3-based 2DEGs. We achieved this by combining a thick BSO buffer layer with an ex-situ high-temperature treatment, which not only reduces the dislocation density but also produces a SnO2-terminated atomically flat surface, followed by the growth of an overlying BSO/LSO interface. Using weak-beam dark field imaging and in-line electron holography technique, we reveal a reduction of the threading dislocation density, and provide direct evidence for the spatial confinement of a 2DEG at the BSO/LSO interface. Our work opens a new pathway to explore the exciting physics of stannate-based 2DEGs at application-relevant temperatures for oxide nanoelectronics.

cond-mat.mtrl-sci↗
arXiv2021

A weighted Sobolev regularity theory of the parabolic equations with measurable coefficients on conic domains in $R^d$

We establish existence, uniqueness, and arbitrary order Sobolev regularity results for the second order parabolic equations with measurable coefficients defined on the conic domains $D$ of the type $$ D(M):=\left\{x\in R^d :\,\frac{x}{|x|}\in M\right\}, \quad \quad M \subset S^{d-1}. $$ We obtain the regularity results by using a system of mixed weights consisting of appropriate powers of the distance to the vertex and of the distance to the boundary. We also provide the sharp ranges of admissible powers of the distance to the vertex and to the boundary.

math.AP↗