12 papers · 1 filter
Two-point estimates for the logarithmic p-flux of the first Dirichlet p-eigenfunction
Rui Chen
Let \(u>0\) be the first Dirichlet \(p\)-eigenfunction on a bounded convex domain \(Ω\subset\mathbb R^N\), and set \[ X_Ω:=|\nabla\log u|^{p-2}\nabla\log u. \] We study whether the…
Fundamental Gaps for the Dirichlet p-Laplacian on Convex Domains
Rui Chen, Daniel Hauer
We study the fundamental gap of the Dirichlet -Laplacian on bounded convex domains. We first prove the sharp one-dimensional estimate , $I_D…
Sharp Lifespan Dichotomies and Threshold Phenomena for Semilinear Heat Equations Driven by the Logarithmic Laplacian
Huyuan Chen, Rui Chen, Daniel Hauer +1
We investigate nonnegative mild solutions of in , with initial datum , . Unlike the classical and frac…
Nonlocal Characterizations of Stochastic Completeness on Complete Riemannian Manifolds
Rui Chen, Bobo Hua
In this paper, we first prove that the following generalized conservation principle holds on complete Riemannian manifolds: for every \(0<s<1\) and \(t>0\), \[ T_t^{(s)}\mathbf 1+\…
The Conformal Fractional--Logarithmic Laplacian on the Sphere: Yamabe Problems and Sharp Inequalities
Huyuan Chen, Rui Chen, Daniel Hauer
In this paper, we introduce the conformal fractional--logarithmic Laplacian on the unit sphere, defined as the derivative of the conformal fractional Laplacian with respect to the…
The Fractional-Logarithmic Laplacian: Potentials, Regularity, and Critical Compact Embeddings
Rui Chen
We develop potential-theoretic and \(L^p\)-regularity results for the fractional--logarithmic Laplacian \((-Δ)^{s+\ln}\) and its inhomogeneous counterpart \((λI-Δ)^{s+\ln}\), \(λ>1…
Large-Time Asymptotics for Heat and Fractional Heat Equations on the Lattice and General Finite Subgraphs
Rui Chen, Bo Li
In this paper, we study large-time asymptotics for heat and fractional heat equations in two discrete settings: the full lattice \(\mathbb Z^d\) and finite connected subgraphs with…
The Fractional-Logarithmic Laplacian: Fundamental Properties and Eigenvalues
Huyuan Chen, Rui Chen, Daniel Hauer
In this paper, we introduce the fractional--logarithmic Laplacian \( (-Δ)^{s+\log} \), defined as the derivative of the fractional Laplacian \( (-Δ)^t \) at \(t=s\). It is a singul…
On Poincaré--Sobolev Levels for Fractional GJMS Operators on Hyperbolic Space
Huyuan Chen, Rui Chen
We study the monotonicity, attainment, and strict-gap regions of the Poincaré--Sobolev levels associated with fractional GJMS operators on hyperbolic space, where .…
The Logarithmic Laplacian on General Graphs
Rui Chen, Wendi Xu
We establish, for the first time, a Bochner-type integral representation for the logarithmic Laplacian on weighted graphs. Assuming stochastic completeness of the underlying graph,…
The Logarithmic Laplacian on Complete Riemannian Manifolds
Rui Chen
We introduce, for the first time, a Bochner representation of the logarithmic Laplacian on general complete Riemannian manifolds via spectral calculus and the heat semigroup. More…
Existence and Convergence of Least-Energy Solutions Involving the Logarithmic Schrödinger Operator
Huyuan Chen, Rui Chen, Bobo Hua
In this paper, we study critical semilinear nonlocal elliptic equations involving the logarithmic Schrödinger operator and its fractional pseudo-relativistic counterpart, both aris…