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Kyeongbae Kim

Publications and source records attributed to Kyeongbae Kim.

15 recordsLinked to original sources
arXiv2026

Kinetic Fokker-Planck equations with Maxwell boundary conditions

We develop the boundary regularity theory for solutions to linear kinetic Fokker-Planck equations with Maxwell boundary conditions. These conditions interpolate between diffuse and specular reflection via an accommodation coefficient $\alpha \in [0,1]$. While existing literature is restricted to the extreme cases $\alpha = 0$ and $\alpha = 1$, we resolve the entire intermediate regime $\alpha \in (0,1)$. Specifically, we show that solutions are H\"older continuous if the coefficients are merely uniformly elliptic. Furthermore, for sufficiently smooth coefficients, we establish boundary regularity of order $\frac{3}{\pi} \arccos(\frac{\alpha}{2}) - 1$ up to the grazing set. This exponent is optimal. Beyond Maxwell conditions, we develop a unified approach that extends to a broad class of reflection boundary conditions, including super-elastic collisions.

math.AP↗
arXiv2026

Boundary Harnack estimates of optimal order for kinetic Fokker-Planck equations

We establish higher order boundary Harnack estimates for solutions to kinetic Fokker-Planck equations with absorbing incoming boundaries. Unlike classical elliptic and parabolic equations with Dirichlet data, we show that the quotient of two solutions for kinetic equations is not $C^{\infty}$ up to the boundary. Instead, we develop a general theory showing that, near the grazing set, the quotient of two solutions is $C^{3/2}$ if the domain and data are sufficiently smooth, and $C^{1,1}$ in the absence of source terms. These exponents are optimal.

math.AP↗
arXiv2026

Sharp regularity near the grazing set for kinetic Fokker-Planck equations

We prove optimal regularity results for solutions to linear kinetic Fokker-Planck equations in bounded domains. Our contributions are two-fold. First, we establish the sharp $C^{1/2}$ regularity for either diffuse reflection or prescribed in-flow boundary conditions. Previously, in this setting, it was only known that solutions are $C^{\alpha}$ for some small $\alpha > 0$. Second, we provide a complete characterization of the solution behavior near the grazing set by proving higher order expansions beyond the critical regularity threshold of $\frac{1}{2}$. These results demonstrate for the first time that solutions maintain higher smoothness up to the grazing set near the incoming boundary.

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arXiv2026

Fine regularity of fractional harmonic maps and applications

In this paper, we derive several regularity results for harmonic mappings into Euclidean spheres associated with rather general energies related to fractional Sobolev spaces. These maps generalize families of maps introduced by Da Lio, Rivi\`ere and Schikorra and are related to harmonic maps with free boundaries. In our context, there is in general no monotonicity formula, which prevents the use of some classical methods. Despite this limitation, under natural assumptions on a Gagliardo-type energy, we succeed in proving a variety of small energy regularity results and improve on known results, even in the isotropic case for which some monotonicity formula is available. To this end, we exploit recent developments in the regularity theory of nonlocal equations and as a by-product, we explain how these results apply to classes of harmonic maps with free boundary and lead to new potential-theoretic estimates. As another application, we obtain higher differentiability results for the fractional harmonic map heat flow.

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arXiv2025

Nonlinear nonlocal equations in Reifenberg flat domains

We consider nonhomogeneous fractional $p$-Laplace equations defined on a bounded nonsmooth domain which goes beyond the Lipschitz category. Under a sufficient flatness assumption on the domain in the sense of Reifenberg, we establish several fine boundary regularity results for solutions, and their gradient, near the boundary. To the best of our knowledge, each of our results is new even in the linear case.

math.AP↗
arXiv2025

Logarithmic continuity for the Nonlocal degenerate two-phase Stefan problem

We establish certain oscillation estimates for weak solutions to nonlinear, anomalous phase transitions modeled on the nonlocal two-phase Stefan problem. The problem is singular in time, is scaling deficient and influenced by far-off effects. We study the the problem in a geometry adapted to the solution and obtain oscillation estimates in intrinsically scaled cylinders. Furthermore, via certain uniform estimates, we construct a continuous weak solution to the corresponding initial boundary value problem with a quantitative modulus of continuity.

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arXiv2025

Local Hölder Regularity For Nonlocal Porous Media And Fast Diffusion Equations With General Kernel

We show that locally bounded, local weak solutions to certain nonlocal, nonlinear diffusion equations modeled on the fractional porous media and fast diffusion equations given by \begin{align*} \partial_t u + (-Δ)^s(|u|^{m-1}u) = 0 \quad \mbox{ for } \quad 0 0 \end{align*} are locally Hölder continuous. We work with bounded, measurable kernels and provide the corresponding $L^{\infty}_{loc} \rightarrow C^{0,α}_{loc}$ De Giorgi-Nash-Moser theory for the equation via a delicate analysis of the set of singularity/degeneracy in a geometry dictated by the solution itself and a careful analysis of far-off effects. In particular, our results are in the spirit of interior regularity, requiring the equation to hold only locally, and thus are new even for positive solutions of the equation with constant coefficients.

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arXiv2025

Gradient estimates for nonlinear kinetic Fokker-Planck equations

In this work, we provide a comprehensive gradient regularity theory for a broad class of nonlinear kinetic Fokker-Planck equations. We achieve this by establishing precise pointwise estimates in terms of the data in the spirit of nonlinear potential theory, leading to fine gradient regularity results under borderline assumptions on the data. Notably, our gradient estimates are novel already in the absence of forcing terms and even for linear kinetic Fokker-Planck equations in divergence form.

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arXiv2024

Gradient estimates for parabolic nonlinear nonlocal equations

The primary objective of this work is to establish pointwise gradient estimates for solutions to a class of parabolic nonlinear nonlocal measure data problems, expressed in terms of caloric Riesz potentials of the data. As a consequence of our pointwise estimates, we obtain that the first-order regularity properties of solutions to such general parabolic nonlinear nonlocal equations, both in terms of size and oscillations of the spatial gradient, closely resemble the ones of the fractional heat equation even at highly refined scales. Along the way, we show that solutions to homogeneous parabolic nonlinear nonlocal equations have Hölder continuous spatial gradients under optimal assumptions on the nonlocal tails.

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arXiv2024

Nonlinear nonlocal potential theory at the gradient level

The aim of this work is to establish numerous interrelated gradient estimates in the nonlinear nonlocal setting. First of all, we prove that weak solutions to a class of homogeneous nonlinear nonlocal equations of possibly arbitrarily low order have Hölder continuous gradients. Using these estimates in the homogeneous case, we then prove sharp higher differentiability as well as pointwise gradient potential estimates for nonlinear nonlocal equations of order larger than one in the presence of general measure data. Our pointwise estimates imply that the first-order regularity properties of such nonlinear nonlocal equations coincide with the sharp ones of the fractional Laplacian.

math.AP↗
arXiv2024

Higher differentiability for the fractional $p$-Laplacian

In this work, we study the higher differentiability of solutions to the inhomogeneous fractional $p$-Laplace equation under different regularity assumptions on the data. In the superquadratic case, we extend and sharpen several previous results, while in the subquadratic regime our results constitute completely novel developments even in the homogeneous case. In particular, in the local limit our results are consistent with well-known higher differentiability results for the standard inhomogeneous $p$-Laplace equation. All of our main results remain valid in the vectorial context of fractional $p$-Laplace systems.

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arXiv2024

Calderón-Zygmund theory of nonlocal parabolic equations with discontinuous coefficients

We prove Calderón-Zygmund type estimates of weak solutions to non-homogeneous nonlocal parabolic equations under a minimal regularity requirement on kernel coefficients. In particular, the right-hand side is presented by a sum of fractional Laplacian type data and a non-divergence type data. Interestingly, even though the kernel coefficients are discontinuous, we obtain a significant increment of fractional differentiability for the solutions, which is not observed in the corresponding local parabolic equations.

math.AP↗
arXiv2023

$L^{q}$ estimates for nonlocal p-Laplacian type equations with BMO kernel coefficients in divergence form

We study $s$-fractional $p$-Laplacian type equations with discontinuous kernel coefficients in divergence form to establish $W^{s+σ,q}$ estimates for any choice of pairs $( σ,q)$ with $q\in(p,\infty)$ and $σ\in\left(0,\min\left\{\frac{s}{p-1},1-s\right\}\right)$ under the assumption that the associated kernel coefficients have small BMO seminorms near the diagonal. As a consequence, we find in the literature an optimal fractional Sobolev regularity of such a non-homogeneous nonlocal equation when the right-hand side is presented by a suitable fractional operator. Our results are new even in the linear case.

math.AP↗
arXiv2023

Regularity results for a class of nonlocal double phase equations with VMO coefficients

We study a class of nonlocal double phase problems with discontinuous coefficients. A local self-improving property and a higher Hölder continuity result for weak solutions to such problems are obtained under the assumptions that the associated coefficient functions are of type VMO (vanishing mean oscillation) and that the principal coefficient depends not only on the variables but also on the solution itself.

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