arXiv research

Study eigenvalues of a generalized p-Laplacian on forms.

The p-Laplacian Transformer improves transformer models by assigning higher attention weights to tokens in close proximity.

The paper sets up eigenvalue comparison theorems for specific Laplacians on manifolds.

Let $M$ be an $n$-dimensional closed Riemannian manifold with metric $g$, $dμ=e^{-φ(x)}dν$ be the weighted measure and $Δ_{p,φ}$ be the weighted $p$-Laplacian. In this article we will investigate monotonicity for the first eigenvalue problem of the weighted $p$-Laplace operator acting on the space of functions along th…

The paper proves gradient estimates for a weighted p-Laplacian equation on Riemannian manifolds.

Sharp gradient estimates for a weighted p-Laplacian equation on metric measure spaces.

The paper explores inequalities between eigenvalues on Riemannian manifolds.

Combined with our previous work \cite{LW19eigenvalue}, we prove sharp lower bound estimates for the first nonzero eigenvalue of the weighted $p$-Laplacian with $1< p< \infty$ on a compact Bakry-Émery manifold $(M^n,g,f)$, without boundary or with a convex boundary and Neumann boundary condition, satisfying $\text{Ric}+…

New lower bounds of the first nonzero eigenvalue of the weighted $p$-Laplacian are established on compact smooth metric measure spaces with or without boundaries. Under the assumption of positive lower bound for the $m$-Bakry--Émery Ricci curvature, the Escober--Lichnerowicz--Reilly type estimates are proved; under the…

We prove Cheeger inequalities for p-Laplacians on finite and infinite weighted graphs. Unlike in previous works, we do not impose boundedness of the vertex degree, nor do we restrict ourselves to the normalized Laplacian and, more generally, we do not impose any boundedness assumption on the geometry. This is achieved …

Researchers find second-order estimates for $p$-Laplacian in RCD spaces.

We study Riemannian manifolds with boundary under a lower $N$-weighted Ricci curvature bound for $N$ at most $1$, and under a lower weighted mean curvature bound for the boundary. We examine rigidity phenomena in such manifolds with boundary. We conclude a volume growth rigidity theorem for the metric neighborhoods of …

New $p$-Laplacian GNN model tackles heterophilic graphs by improving node classification.

The paper defines new constants for $p$-Laplacian on manifolds.

The study finds bounds for the first eigenvalue of the p-Laplacian on submanifolds.

Study eigenvalues of p-Laplacian on quaternionic Kähler manifolds.

Proves a Liouville-type theorem for p-Laplacian on manifolds.

Study p-Laplacian equation on Riemannian manifolds with positive Ricci curvature.

Sharp lower bound for $p$-Laplacian eigenvalue on non-compact manifolds.

Study eigenvalues of p-Laplacian on Kähler manifolds, proving lower bounds.

We prove sharp lower bound estimates for the first nonzero eigenvalue of the weighted $p$-Lapacian operator with $1< p< \infty$ on a compact Bakry-Emery manifold $(M^n,g,f)$ satisfying $\Ric+\nabla^2 f \geq κ\, g$, provided that either $1<p \leq 2$ or $κ\leq 0$. Same conclusions hold when the manifold has nonempty boun…

We study Riemannian manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound. In our weighted setting, we prove several rigidity theorems for such manifolds with boundary. We conclude a rigidity theorem for the inscribed radii, a volume growth rigidity theorem for the metric neighborhoods of the boundar…

In this paper, we investigate submanifolds with locally bounded mean curvature in Hadamard manifolds, product manifolds $N\times\mathbb{R}$, submanifolds with bounded $\varphi$-mean curvature in the hyperbolic space, and successfully give lower bounds for the weighted fundamental tone and the first eigenvalue of the $p…

Let $M$ be an $n$-dimensional closed orientable submanifold in an $N$-dimensional space form. When $1<p \le \frac n2 + 1$, we obtain an upper bound for the first nonzero eigenvalue of the $p$-Laplacian in terms of the mean curvature of $M$ and the curvature of the space form. This generalizes the Reilly inequality for …

Study on p-Laplacian problems with critical exponent, focusing on existence of solutions.

Paper proves inequality for p-Laplacian eigenvalues on curved spaces.

Sharp Talenti-type comparison theorem for p-Laplacian on RCD(K,N) spaces.

Given a Finsler manifold $(M,F)$, it is proved that the first eigenvalue of the Finslerian $p$-Laplacian is bounded above by a constant depending on $\ p$, the dimension of $M$, the Busemann-Hausdorff volume and the reversibility constant of $(M,F)$. For a Randers manifold $(M,F:=\sqrt{g}+β)$, where $g$ is a Riemannian…

New inequalities for submanifolds in curved spaces.

In this paper, we establish gradient estimates for positive solutions to the following equation with respect to the $p$-Laplacian $Δ_{p}u=-λ|u|^{p-2}u$ with $p>1$ on a given complete Riemannian manifold. Consequently, we derive upper bound estimates of the first nontrivial eigenvalue of the $p$-Laplacian.

Study compares manifolds with boundary under weighted Ricci curvature bounds.

Study eigenvalues of p-Laplacian on manifolds with Robin boundary conditions.

Enhances GNN robustness against attacks.

The graph Laplacian plays key roles in information processing of relational data, and has analogies with the Laplacian in differential geometry. In this paper, we generalize the analogy between graph Laplacian and differential geometry to the hypergraph setting, and propose a novel hypergraph $p$-Laplacian. Unlike the …

We prove the sharp estimate on the first nonzero eigenvalue of the p-laplacian on a compact Riemannian manifold with nonnegative Ricci curvature and possibly with convex boundary (in this case we assume Neumann b.c. on the p-laplacian). The proof is based on a gradient comparison theorem. We will also charachterize the…

Researchers prove zero solutions for certain p-Laplacian equations in convex cones.

The paper examines geometric properties of domains for the p-Laplacian in Euclidean and hyperbolic spaces.

Study finds bounds for fundamental tone on special Riemannian manifolds.

In this paper, we study eigenvalues and eigenfunctions of $p$-Laplacians with Dirichlet boundary condition on graphs. We characterize the first eigenfunction (and the maximum eigenfunction for a bipartite graph) via the sign condition. By the uniqueness of the first eigenfunction of $p$-Laplacian, as $p\to 1,$ we ident…

The paper proves existence and growth estimates for inverse mean curvature flow and related $p$-Laplacian Green kernel decay.

The paper finds lower bounds for the first eigenvalue of p-Laplacian in specific manifolds.

We discuss the behavior of $(λ_{1. p}(M))^{1/p}$ with respect to the Gromov-Hausdorff topology and the variable $p$, where $λ_{1, p}(M)$ is the first positive eigenvalue of the $p$-Laplacian on a compact Riemannian manifold $M$. Applications include new estimates for the first eigenvalues of the $p$-Laplacian on Rieman…

We introduce higher-order Poincar'e constants for compact weighted manifolds and estimate them from above in terms of subsets. These estimates imply upper bounds for eigenvalues of the weighted Laplacian and the first nontrivial eigenvalue of the $p$-Laplacian. In the case of the closed eigenvalue problem and the Neuma…

Smooth solutions up to evolving free boundaries for degenerate equations.

This paper approximates $p$-resistance for multi-class graph clustering.

Develops a Barta theorem for p-Laplacian on manifolds.

Derives an inequality for submanifolds in spheres.

Paper solves Minkowski problem for anisotropic p-torsional rigidity.