arXiv subjects

Kyeong Song

Publications and source records attributed to Kyeong Song.

11 recordsLinked to original sources

arXiv2026

Self-improving properties for the fractional $p$-Laplacian via nonlinear commutators

We investigate a class of nonlocal equations whose leading operator is modeled on either the fractional $p$-Laplacian or the regional fractional $p$-Laplacian, $p \in (1,\infty)$. We prove local self-improving properties of weak solutions to the fractional $p$-Laplacian in the case $p\in(1,\infty)$ with non-integrable right-hand side, as well as to the regional fractional $p$-Laplacian in the subquadratic case $1 < p < 2$, by extending the nonlinear commutator estimates developed by Schikorra (Math. Ann. 366 (1-2):695--720, 2016).

math.AP↗

arXiv2026

Gradient estimates for singular elliptic measure data problems with double phase

We consider elliptic measure data problems of the type \[ -\mathrm{div}\,(|Du|^{p-2}Du+a(x)|Du|^{q-2}Du) = \mu \] in a bounded domain in $\mathbb{R}^n$, where $p<q$ and $a(\cdot) \ge 0$. We prove local Calder\'on--Zygmund estimates in the singular case $2-1/n < p < 2$, under natural assumptions on $p$, $q$ and $a(\cdot)$.

math.AP↗

arXiv2026

Gradient estimates for degenerate elliptic measure data problems with double phase

We study nonlinear elliptic equations modeled on \[ -\mathrm{div}\,(|Du|^{p-2}Du+a(x)|Du|^{q-2}Du) = \mu, \] where $2\le p<q<\infty$, $a(\cdot) \ge 0$, and $\mu$ is a signed Borel measure with finite total mass. We prove local Calder\'on--Zygmund type gradient estimates for SOLA (Solutions Obtained as Limits of Approximations) by finding new and natural assumptions on $p$, $q$ and $a(\cdot)$.

math.AP↗

arXiv2025

Regularity for mixed-order nonlinear fractional equations with degenerate coefficients

We consider a class of nonlinear integro-differential equations whose leading operator is obtained as a superposition of $(-\Delta_{p})^{s}$ and $(-\Delta_{p})^{t}$, where $0<s<t<1<p<\infty$, weighted via two possibly degenerate coefficients $a(\cdot,\cdot),b(\cdot,\cdot) \ge 0$. We prove local boundedness and H\"older regularity of its weak solutions under natural assumptions on the coefficients $a(\cdot,\cdot)$, $b(\cdot,\cdot)$ and the powers $s,t$, and $p$. Moreover, when $a(\cdot,\cdot) \equiv 1$, we also prove a Harnack inequality for weak solutions.

math.AP↗

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

Nonlocal equations with kernels of general order

We consider a broad class of nonlinear integro-differential equations with a kernel whose differentiability order is described by a general function $\phi$. This class includes not only the fractional $p$-Laplace equations, but also borderline cases when the fractional order approaches $1$. Under mild assumptions on $\phi$, we establish sharp Sobolev-Poincar\'e type inequalities for the associated Sobolev spaces, which are connected to a question raised by Brezis (Russian Math. Surveys 57:693--708, 2002). Using these inequalities, we prove H\"older regularity and Harnack inequalities for weak solutions to such nonlocal equations. All the estimates in our results remain stable as the associated nonlocal energy functional approaches its local counterpart.

math.AP↗

arXiv2024

Wolff potentials and nonlocal equations of Lane-Emden type

We consider nonlocal equations of the type \[ (-\Delta_{p})^{s}u = \mu \quad \text{in}\;\; \Omega, \] where $\Omega \subset \mathbb{R}^{n}$ is either a bounded domain or the whole $\mathbb{R}^{n}$, $\mu$ is a Radon measure on $\Omega$, $0 < s < 1$ and $1 < p < n/s$. In particular, we extend the existence, regularity and Wolff potential estimates for SOLA (Solutions Obtained as Limits of Approximations), established by Kuusi, Mingione, and Sire (Comm. Math. Phys. 337(3):1317--1368, 2015), to the strongly singular case $1 < p \le 2-s/n$. Moreover, using Wolff potentials and Orlicz capacities, we present both a sufficient condition and a necessary condition for the existence of SOLA to nonlocal equations of the type \[ (-\Delta_{p})^{s}u = P(u) + \mu \quad \text{in}\;\; \Omega, \] where $P(\cdot)$ is either a power function or an exponential function.

math.AP↗

arXiv2024

Riesz potential estimates for mixed local-nonlocal problems with measure data

We study gradient regularity for mixed local-nonlocal problems modelled upon \[ -Δ_p u +(-Δ_p)^su=μ\qquad\text{for} \quad 2-\tfrac{1}{n}<p<\infty\quad \text{and}\quad s\in(0,1)\,,\] where $μ$ is a bounded Borel measure. We prove pointwise bounds for the gradient $Du$ in terms of the truncated 1-Riesz potential of $μ$.

math.AP↗

arXiv2023

Singular elliptic measure data problems with irregular obstacles

We investigate elliptic irregular obstacle problems with $p$-growth involving measure data. Emphasis is on the strongly singular case $1 < p \le 2-1/n$, and we obtain several new comparison estimates to prove gradient potential estimates in an intrinsic form. Our approach can be also applied to derive zero-order potential estimates.

math.AP↗

arXiv2023

Regularity results for mixed local and nonlocal double phase functionals

We investigate the De Giorgi-Nash-Moser theory for minimizers of mixed local and nonlocal functionals modeled after \[ v \mapsto \int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\dfrac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}}\,dxdy+\int_Ωa(x)|Dv|^{q}\,dx, \] where $0<s<1<p \le q$ and $a(\cdot) \ge 0$. In particular, we prove Hölder regularity and Harnack's inequality under possibly sharp assumptions on $s,p,q$ and $a(\cdot)$.

math.AP↗

arXiv2021

Hölder regularity for weak solutions to nonlocal double phase problems

We prove local boundedness and Hölder continuity for weak solutions to nonlocal double phase problems concerning the following fractional energy functional \[ \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{|v(x)-v(y)|^p}{|x-y|^{n+sp}} + a(x,y)\frac{|v(x)-v(y)|^q}{|x-y|^{n+tq}}\, dxdy, \] where $0<s\le t<1<p \leq q<\infty$ and $a(\cdot,\cdot) \geq 0$. For such regularity results, we identify sharp assumptions on the modulating coefficient $a(\cdot,\cdot)$ and the powers $s,t,p,q$ which are analogous to those for local double phase problems.

math.AP↗