# Note on an eigenvalue problem with applications to a Minkowski type regularity problem in $\mathbb{R}$^n

## TL;DR

This paper investigates the existence, uniqueness, and homogeneity of solutions to a class of p-Laplace PDEs within specific conical domains, with applications to Minkowski-type regularity problems in low-dimensional Euclidean spaces.

## Contribution

It establishes conditions under which solutions are homogeneous of a specific degree, advancing understanding of PDE solutions in conical geometries and their geometric applications.

## Key findings

- Solutions are homogeneous of degree 1 - (n-1)/p when p > n - 1.
- Existence and uniqueness of solutions are proven for certain boundary conditions.
- Applications to Minkowski regularity problems in 2D and 3D are demonstrated.

## Abstract

We consider existence and uniqueness of homogeneous solutions $ u > 0 $ to certain PDE of $p$-Laplace type, $ p $ fixed, $ n - 1 <p< \infty, n \geq 2, $ when $ u $ is a solution in $K(\alpha)\subset\mathbb{R}^n$ where \[ K (\alpha) := \{ x = (x_1, \dots, x_n ): x_1 > \cos \alpha \, | x| \} \quad \mbox{for fixed}\, \, \alpha \in (0, \pi ], \] with continuous boundary value zero on $ \partial K ( \alpha ) \setminus \{0\}$. In our main result we show that if $ u $ has continuous boundary value $0$ on $ \partial K ( \pi )$ then $u$ is homogeneous of degree $ 1 - (n-1)/p $ when $ p > n - 1. $ Applications of this result are given to a Minkowski type regularity problem in $ \mathbb{R}^{n}$ when $n=2,3$.