arXiv research

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

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.

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…

Study eigenvalues of a generalized p-Laplacian on forms.

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

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 …

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

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.

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

Smooth solutions up to evolving free boundaries for degenerate equations.

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

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

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.

We prove a Lichnerowicz type lower bound for the first nontrivial eigenvalue of the $p$-Laplacian on Kähler manifolds. Parallel to the $p = 2$ case, the first eigenvalue lower bound is improved by using a decomposition of the Hessian on Kähler manifolds with positive Ricci curvature.

Gradient estimates for solutions to a p-Laplacian equation on Riemannian manifolds.

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

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…

In this paper, we successfully generalize the eigenvalue comparison theorem for the Dirichlet $p$-Laplacian ($1<p<\infty$) obtained by Matei [A.-M. Matei, First eigenvalue for the $p$-Laplace operator, Nonlinear Anal. TMA 39 (8) (2000) 1051--1068] and Takeuchi [H. Takeuchi, On the first eigenvalue of the $p$-Laplacian …

The paper constructs Morse homology for functionals involving the p-Laplacian in Banach spaces.

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 …

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

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}+…

Paper discusses solving generalized Hessian inequalities with various operators.

We show that a Weyl law holds for the variational spectrum of the $p$-Laplacian. More precisely, let $(λ_i)_{i=1}^\infty$ be the variational spectrum of $Δ_p$ on a closed Riemannian manifold $(X,g)$ and let $N(λ) = \#\{i:\, λ_i < λ\}$ be the associated counting function. Then we have a Weyl law $N(λ) \sim c \operatorna…

Sphere theorems for p-Laplacian eigenvalues established.

We complete the picture of sharp eigenvalue estimates for the p-Laplacian on a compact manifold by providing sharp estimates on the first nonzero eigenvalue of the nonlinear operator $Δ_p$ when the Ricci curvature is bounded from below by a negative constant. We assume that the boundary of the manifold is convex, and p…

The paper explores inequalities between eigenvalues on Riemannian manifolds.

Most network-based machine learning methods assume that the labels of two adjacent samples in the network are likely to be the same. However, assuming the pairwise relationship between samples is not complete. The information a group of samples that shows very similar pattern and tends to have similar labels is missed.…