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Minhyun Kim

Publications and source records attributed to Minhyun Kim.

At least 19 recordsLinked to original sources
arXiv2026

Capacities, Wiener criteria and fine continuity for nonlocal nonlinear equations

In this paper we study nonlocal nonlinear equations of $s$-fractional $p$-Laplacian type in open subsets of $\mathbf{R}^n$. We investigate how the boundary regularity of solutions depends on the parameters $s$ and $p$, including the local case $s=1$. Specifically, we show exactly when regularity for $(s_1, p_1)$ implies regularity for $(s_2, p_2)$. The proof relies on the equivalence between Wiener criteria formulated with condenser and Sobolev capacities. To establish this equivalence, we derive precise comparison estimates between the two capacities. The Wiener integral defines thinness and the fine topology. We show that every superharmonic function associated with a nonlocal nonlinear operator is finely continuous. Moreover, we prove that polar sets in this fractional setting coincide with sets of zero capacity.

math.AP↗
arXiv2026

Capacitary estimates for solutions to nonlocal Dirichlet problems

We study the boundary regularity of weak solutions to nonlocal nonlinear elliptic equations with bounded measurable coefficients. Our main result establishes that a capacity density condition is equivalent to the validity of a uniform boundary H\"older estimate for solutions with H\"older continuous exterior data. More generally, we derive a fine capacitary estimate on the modulus of continuity that captures how regularity is inherited from the exterior datum to the solution.

math.AP↗
arXiv2026

Orlicz Potential Theory: Balayage, Riesz Measures, and Very Weak Solutions

We develop a nonlinear potential theory for elliptic equations with Orlicz growth under general monotonicity and growth conditions, without any homogeneity or scaling assumptions. The lack of scaling invariance prevents the use of many classical tools from nonlinear potential theory. To overcome this difficulty, we establish a new framework that includes global H\"older regularity for obstacle problems, a balayage theory, the construction and analysis of Riesz measures associated with superharmonic functions, the identification of capacitary potentials, capacitary estimates for polar sets, and the quasicontinuity of superharmonic functions. As an application of this theory, we prove that the classes of superharmonic functions and renormalized solutions to elliptic measure data problems coincide. This extends the classical equivalence theory from the homogeneous $p$-growth setting to general Orlicz growth and is new even for power-growth operators without homogeneity assumptions.

math.AP↗
arXiv2026

Partial H\"older regularity for fully nonlinear nonlocal parabolic equations with integrable kernels

In this work, we consider solutions to (fully nonlinear) parabolic integro-differential equations with integrable interaction kernels. A typical equation would be that obtained by starting with, for $s\in(0,1)$, the $s$-fractional heat equation, but replacing the interaction kernel in the integro-differential term with one which has been truncated, for $\rho>0$, at the value $\rho^{-d-2s}$, hence integrable. We show that solutions to these equations have a partial regularity estimate which captures differences of the solution up to the scale at which the kernel has a truncation in its singularity. The estimates we provide are robust with respect to the truncation parameter, and they include the existing results for the original operators without truncation. There are some earlier results for linear and elliptic cases of this situation of integrable interaction kernels, and so our work is a generalization of those to the nonlinear and parabolic setting.

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↗
arXiv2025

Optimal boundary regularity and Green function estimates for nonlocal equations in divergence form

In this article we prove for the first time the $C^s$ boundary regularity for solutions to nonlocal elliptic equations with Hölder continuous coefficients in divergence form in $C^{1,α}$ domains. So far, it was only known that solutions are Hölder continuous up to the boundary, and establishing their optimal regularity has remained an open problem in the field. Our proof is based on a delicate higher order Campanato-type iteration at the boundary, which we develop in the context of nonlocal equations and which is quite different from the local theory. As an application of our results, we establish sharp two-sided Green function estimates in $C^{1,α}$ domains for the same class of operators. Previously, this was only known under additional structural assumptions on the coefficients and in more regular domains.

math.AP↗
arXiv2025

Liouville theorem for singular solutions to nonlocal equations

We study singular solutions to the fractional Laplace equation and, more generally, to nonlocal linear equations with measurable kernels. We establish Bôcher type results that characterize the behavior of singular solutions near the singular point. In addition, we prove Liouville theorems for singular solutions. To this end, we construct fundamental solutions for nonlocal linear operators and establish a localized comparison principle.

math.AP↗
arXiv2025

Singularities of solutions of nonlocal nonlinear equations

We study the local behavior of weak solutions, with possible singularities, of nonlocal nonlinear equations. We first prove that sets of capacity zero are removable for weak solutions under certain integrability conditions. We then characterize the asymptotic behavior of singular solutions near an isolated singularity in terms of the fundamental solution.

math.AP↗
arXiv2025

Semiregular and strongly irregular boundary points for nonlocal Dirichlet problems

In this paper we study nonlocal nonlinear equations of fractional $(s,p)$-Laplacian type on $\mathbf{R}^n$. We show that the irregular boundary points for the Dirichlet problem can be divided into two disjoint classes: semiregular and strongly irregular boundary points, with very different behaviour. Two fundamental tools needed to show this are the Kellogg property (from our previous paper) and a new removability result for solutions in the $V^{s,p}$ Sobolev type space, which we deduce more generally also for supersolutions of equations with a right-hand side. Semiregular and strongly irregular points are also characterized in various ways. Finally, it is explained how semiregularity depends on $s$ and $p$.

math.AP↗
arXiv2025

Removable singularities for nonlocal minimal graphs

We prove the removable singularity theorem for nonlocal minimal graphs. Specifically, we show that any nonlocal minimal graph in $\Omega \setminus K$, where $\Omega \subset \mathbb{R}^n$ is an open set and $K \subset \Omega$ is a compact set of $(s, 1)$-capacity zero, is indeed a nonlocal minimal graph in all of $\Omega$.

math.DG↗
arXiv2024

Perron solutions and boundary regularity for nonlocal nonlinear Dirichlet problems

For nonlinear operators of fractional \p-Laplace type, we consider two types of solutions to the nonlocal Dirichlet problem: Sobolev solutions based on fractional Sobolev spaces and Perron solutions based on superharmonic functions. These solutions give rise to two different concepts of regularity for boundary points, namely Sobolev and Perron regularity. We show that these two notions are equivalent and we also provide several characterizations of regular boundary points. Along the way, we give a new definition of Perron solutions, which is applicable to arbitrary exterior Dirichlet data $g: \Omega^c \to [-\infty,\infty]$. We obtain resolutivity results for these Perron solutions, and show that the Sobolev and Perron solutions coincide for a large class of exterior Dirichlet data. This also implies invariance of the Perron solutions under perturbations on sets of zero fractional capacity. A uniqueness result for the Dirichlet problem is also obtained for the class of bounded solutions taking prescribed continuous exterior data quasieverywhere on the boundary.

math.AP↗
arXiv2023

Wolff potential estimates and Wiener criterion for nonlocal equations with Orlicz growth

We prove the Wolff potential estimates for nonlocal equations with Orlicz growth. As an application, we obtain the Wiener criterion in this framework, which provides a necessary and sufficient condition for boundary points to be regular. Our approach relies on the fine analysis of superharmonic functions in view of nonlocal nonlinear potential theory.

math.AP↗
arXiv2023

Supersolutions and superharmonic functions for nonlocal operators with Orlicz growth

We study supersolutions and superharmonic functions related to problems involving nonlocal operators with Orlicz growth, which are crucial tools for the development of nonlocal nonlinear potential theory. We provide several fine properties of supersolutions and superharmonic functions, and reveal the relation between them. Along the way we prove some results for nonlocal obstacle problems such as the well-posedness and (both interior and boundary) regularity estimates, which are of independent interest.

math.AP↗
arXiv2023

Gradient Riesz potential estimates for a general class of measure data quasilinear systems

We study the gradient regularity of solutions to measure data elliptic systems with Uhlenbeck-type structure and Orlicz growth. For any bounded Borel measure, pointwise estimates for the gradient of solutions are provided in terms of the truncated Riesz potential. This allows us to show a precise transfer of regularity from data to solutions on various scales.

math.AP↗
arXiv2023

Curvature bound for $L_p$ Minkowski problem

We establish curvature estimates for anisotropic Gauss curvature flows. By using this, we show that given a measure $\mu$ with a positive smooth density $f$, any solution to the $L_p$ Minkowski problem in $\mathbb{R}^{n+1}$ with $p \le -n+2$ is a hypersurface of class $C^{1,1}$. This is a sharp result because for each $p\in [-n+2,1)$ there exists a convex hypersurface of class $C^{1,\frac{1}{n+p-1}}$ which is a solution to the $L_p$ Minkowski problem for a positive smooth density $f$. In particular, the $C^{1,1}$ regularity is optimal in the case $p=-n+2$ which includes the logarithmic Minkowski problem in $\mathbb{R}^3$.

math.DG↗
arXiv2022

The fractional $p$-Laplacian on hyperbolic spaces

We present three equivalent definitions of the fractional $p$-Laplacian $(-\Delta_{\mathbb{H}^{n}})^{s}_{p}$, $0 1$, 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↗
arXiv2022

Diameter estimate for planar $L_p$ dual Minkowski problem

In this paper, given a prescribed measure on $\mathbb{S}^1$ whose density is bounded and positive, we establish a uniform diameter estimate for solutions to the planar $L_p$ dual Minkowski problem when $0<p<1$ and $q\ge 2$. We also prove the uniqueness and positivity of solutions to the $L_p$ Minkowski problem when the density of the measure is sufficiently close to a constant in $C^\alpha$.

math.DG↗