Note on an eigenvalue problem for an ODE originating from a homogeneous $p$-harmonic function

Abstract

We discuss what is known about homogeneous solutions u to the p− Laplace equation, p fixed, 1 < p < ∞, when (A) u is an entire p− harmonic function on Euclidean n space, R n , or (B) u > 0 is p− harmonic in the cone, K(α) = {x = (x 1 ,. .. , x n) : x 1 > cos α |x|} ⊂ R n , n ≥ 2, with continuous boundary value zero on ∂K(α) \ {0} when α ∈ (0, π]. We also outline a proof of our new result concerning the exact value, λ = 1 − (n − 1)/p, for an eigenvalue problem in an ODE associated with u when u is pharmonic in K(π) and p > n − 1. Generalizations of this result are stated. Our result complements work of Krol'-Maz'ya for 1 < p ≤ n − 1.

Fractional-Order Adaptive P -Laplace Equation-Based Art Image Edge Detection

“…The literature [2] also improves the PM model by proposing a tensortype diffusion model, who mainly uses the diffusion tensor to replace the diffusion coefficient function in the PM model, although this improved model mainly addresses images with flow-like structures like fingerprints, which can be well enhanced for this class of images. In the literature [3,4], it was found that the edge and slope regions of an image can be identified by differential curvature, and both constructed two adaptive diffusion coefficients to discriminate the whole image. In the literature [5], a nonlinear anisotropic combined diffusion model was designed; this model is mainly by choosing different diffusion methods when facing different regions of the image, homogeneous diffusion for smooth regions, and mean curvature diffusion for edge regions, and the effectiveness of this model was experimentally demonstrated.…”

Spectral properties of nonassociative algebras and breaking regularity for nonlinear elliptic type PDEs

“…An important structure property of any Jordan algebra V is that the spectrum of any idempotent is a subset of {0, 1 2 , 1}. In particular, V admits the so-called Peirce decomposition:…”

“…The eigenvalue 1 2 and the corresponding subspace is distinguished in many ways. For example, a Jordan algebra is simple if and only its 1 2 -eigenspace is nontrivial for any idempotent. Moreover, V 1 2 satisfies the Jordan fusion laws…”

“…A commutative algebra V over C is generic if and only if the spectrum of any idempotent in V does not contain 1 2 . Because the essential role 1 2 playing in the present context and also for the reader convenience, we give a sketch of the proof.…”

“…It is interesting whether the converse non-vanishing property holds true. This question naturally leads to homogeneous p-harmonic functions (also called quasi-radial solutions in [4]); see the paper of M. Akman, J. Lewis, A. Vogel [1] in the present volume for the modern status of the problem and further discussion. We only mention that in certain dimensions n ≥ 4 one can explicitly construct rational homogenous p-harmonic functions for some distinguished values p, 2 < p < n based on isoparametric forms and generalizing the angle solutions constructed by Krol' and Maz'ya, see [55].…”