arXiv research

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

Smooth solutions up to evolving free boundaries for degenerate equations.

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

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

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

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.

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

Paper solves Minkowski problem for anisotropic p-torsional rigidity.

In these notes we study the Dirichlet problem for critical points of a convex functional of the form \[ F(u)=\int_Ωφ\left( \left\vert \nabla u\right\vert \right) , \] where $Ω$ is a bounded domain of a complete Riemannian manifold $\mathcal{M}.$ We also study the asymptotic Dirichlet problem when $Ω=\mathcal{M}$ is a C…

Study shows conditions for nonexistence of solutions in Riemannian geometry.

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

New method of symmetrization applied to PDEs on spheres.

We study the Dirichlet problem at infinity on a Cartan-Hadamard manifold for a large class of operators containing in particular the p-Laplacian and the minimal graph operator.

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.

Assume $α\geq p>1$. Consider the following $p$-th Yamabe equation on a connected finite graph $G$: $Δ_p\varphi+h\varphi^{p-1}=λf\varphi^{α-1},$ where $Δ_p$ is the discrete $p$-Laplacian, $h$ and $f>0$ are fixed real functions defined on all vertices. We show that the above equation always has a positive solution $\va…

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

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

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

The paper develops Morse homology for a class of elliptic partial differential equations.

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.

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 paper classifies solutions to a Liouville equation on a half-space with a specific boundary condition.

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…

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

The paper uses Nash-Moser iteration to prove gradient estimates for nonlinear equations on Riemannian manifolds.

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…

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.

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 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.

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.