arXiv subjects
Publications and source records attributed to Minhyun Kim.
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 $C^α$.
We study the boundary behavior of solutions to the Dirichlet problems for integro-differential operators with order of differentiability $s \in (0, 1)$ and summability $p>1$. We establish a nonlocal counterpart of the Wiener criterion, which characterizes a regular boundary point in terms of the nonlocal nonlinear potential theory.
We study robust regularity estimates for a class of nonlinear integro-differential operators with anisotropic and singular kernels. In this paper, we prove a Sobolev-type inequality, a weak Harnack inequality, and a local Hölder estimate.
We prove a full Harnack inequality for local minimizers, as well as weak solutions to nonlocal problems with non-standard growth. The main auxiliary results are local boundedness and a weak Harnack inequality for functions in a corresponding De Giorgi class. This paper builds upon a recent work on regularity estimates for such nonlocal problems by the same authors.
We study robust regularity estimates for local minimizers of nonlocal functionals with non-standard growth of $(p,q)$-type and for weak solutions to a related class of nonlocal equations. The main results of this paper are local boundedness and Hölder continuity of minimizers and weak solutions. Our approach is based on the study of corresponding De Giorgi classes.
The Minkowski problem for electrostatic capacity characterizes measures generated by electrostatic capacity, which is a well-known variant of the Minkowski problem. This problem has been generalized to $L_p$ Minkowski problem for $\mathfrak{p}$-capacity. In particular, the logarithmic case $p=0$ relates to cone-volumes and therefore has a geometric significance. In this paper we solve the discrete logarithmic Minkowski problem for $1<\mathfrak{p}<n$ in the case where the support of the given measure is in general position.
In this paper, we study local regularity properties of minimizers of nonlocal variational functionals with variable exponents and weak solutions to the corresponding Euler--Lagrange equations. We show that weak solutions are locally bounded when the variable exponent $p$ is only assumed to be continuous and bounded. Furthermore, we prove that bounded weak solutions are locally Hölder continuous under some additional assumptions on $p$. On the one hand, the class of admissible exponents is assumed to satisfy a log-Hölder-type condition inside the domain, which is essential even in the case of local equations. On the other hand, since we are concerned with nonlocal problems, we need an additional assumption on $p$ outside the domain.
In this paper, we study the existence of minimizers of the Sobolev quotient for a class of nonlocal operators with an orthotropic structure having different exponents of integrability and different orders of differentiability. Our method is based on the concentration-compactness principle which we extend to this class of operators. One consequence of our main result is the existence of a nontrivial nonnegative solution to the corresponding critical problem.
We establish the Krylov Safonov Harnack inequalities and Holder estimates for fully nonlinear nonlocal operators of non-divergence form on Riemannian manifolds with nonnegative sectional curvatures. To this end, we first define the nonlocal Pucci operators on manifolds that give rise to the concept of non-divergence form operators. We then provide the uniform regularity results for these operators which recover the classical results for second order local operators as limits.
We establish the generalized Evans--Krylov and Schauder type estimates for nonlocal fully nonlinear elliptic equations with rough kernels of variable orders. In contrast to the fractional Laplacian type operators having a fixed order of differentiability $σ\in (0,2)$, the operators under consideration have variable orders of differentiability. Since the order is not characterized by a single number, we consider a function $φ$ describing the variable orders of differentiability, which is allowed to oscillate between two functions $r^{σ_1}$ and $r^{σ_2}$ for some $0 < σ_1 \leq σ_2 < 2$. By introducing the generalized Hölder spaces, we provide $C^{φψ}$ estimates that generalizes the standard Evans--Krylov and Schauder type $C^{σ+α}$ estimates.
We consider fully nonlinear elliptic integro-differential operators with kernels of variable orders, which generalize the integro-differential operators of the fractional Laplacian type in \cite{CS}. Since the order of differentiability of the kernel is not characterized by a single number, we use the constant \begin{align*} C_φ= \left( \int_{\mathbb{R}^n} \frac{1-\cos y_1}{\vert y \vert^n φ(\vert y \vert)} \, dy \right)^{-1} \end{align*} instead of $2-σ$, where $φ$ satisfies a weak scaling condition. We obtain the uniform Harnack inequality and Hölder estimates of viscosity solutions to the nonlinear integro-differential equations.
We study the boundary regularity of solutions of the Dirichlet problem for the nonlocal operator with a kernel of variable orders. Since the order of differentiability of the kernel is not represented by a single number, we consider the generalized Hölder space. We prove that there exists a unique viscosity solution of $Lu = f$ in $D$, $u=0$ in $\mathbb{R}^n \setminus D$, where $D$ is a bounded $C^{1,1}$ open set, and that the solution $u$ satisfies $u \in C^V(D)$ and $u/V(d_D) \in C^α(D)$ with the uniform estimates, where $V$ is the renewal function and $d_D(x) = \mbox{dist}(x, \partial D)$.