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Publications (12)
We consider a flow by powers of Gauss curvature under the obstruction that the flow cannot penetrate a prescribed region, so called an obstacle. For all dimensions and positive powers, we prove the optimal curvature bounds of solutions and all time existence with its long time behavior. We also prove the C1\documentclass[12pt]{minimal} \usepackage{...
In this paper, we investigate energy-minimizing curves with fixed endpoints $p$ and $q$ in a constrained space. We prove that when one of the endpoints, say $p$, is fixed, the set of points $q$ for which the energy-minimizing curve is not unique has no interior points.
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...
We study an eigenvalue problem for prescribed $\sigma_k$-curvature equations of star-shaped, $k$-convex, closed hypersurfaces. We establish the existence of a unique eigenvalue and its associated hypersurface, which is also unique, provided that the given data is even. Moreover, we show that the hypersurface must be strictly convex. A crucial aspec...
We propose a new centrality incorporating two classical node-level centralities, the degree centrality and the information centrality, which are considered as local and global centralities, respectively. These two centralities have expressions in terms of the graph Laplacian L, which motivates us to exploit its fractional analog Lγ with a fractiona...
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 suf...
In this paper, we study the obstacle problem for the parabolic Monge-Ampère equation with the forcing term f(x,t,u,Du). We establish existence, uniqueness, and optimal regularity under some structure conditions via the penalization method and a priori estimates. Moreover, we discuss the regularity of the free boundary.
As a consequence of our appro...
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...
We consider the obstacle problem for the Gauss curvature flow with an exponent \(\alpha \). Under the assumption that both the obstacle and the initial hypersurface are strictly convex closed hypersurfaces and that the obstacle is enclosed by the initial hypersurface, uniform estimates are obtained for several curvatures via a penalty method. We al...
In this paper, we study the existence and optimal regularity of the solution and the regularity of the free boundary of the obstacle problem for the Monge–Ampère equation with the lower obstacle function, which arises in the prescribed Gauss curvature with an obstacle.
The main feature of this paper is that we consider the obstacle problem for Mong...
We consider the following eigenvalue optimization in the composite membrane problem with fractional Laplacian: given a bounded domain Ω⊂Rn, α>0 and 0<A<|Ω|, find a subset D⊂Ω of area A such that the first Dirichlet eigenvalue of the operator (−Δ)s+αχD is as small as possible. The solution D is called as an optimal configuration for the data (Ω,α,A)...
We consider the following eigenvalue optimization in the composite membrane problem with fractional Laplacian: given a bounded domain $\Omega\subset \mathbb{R}^n$, $\alpha>0$ and $0<A<|\Omega|$, find a subset $D\subset \Omega$ of area $A$ such that the first Dirichlet eigenvalue of the operator $(-\Delta)^s+\alpha \chi_D$ is as small as possible. T...
