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Note on an eigenvalue problem with applications to a Minkowski type regularity problem in Rⁿ

Akman, Murat and Lewis, John and Vogel, Andrew (2019) Note on an eigenvalue problem with applications to a Minkowski type regularity problem in Rⁿ. Working Paper. Arxiv. (Submitted)

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Abstract

We consider existence and uniqueness of homogeneous solutions u > 0 to certain PDE of p-Laplace type, p fixed, n - 1 < p <∞, n ≥ 2, when u is a solution in K(α) ⊂ Rⁿ where K(α) := { x = (x₁,..., x_n ) : x₁ > cos α | x| } for fixed α ∈ (0, π ], with continuous boundary value zero on ∂K (α) \ {0\}. In our main result we show that if u has continuous boundary value 0 on ∂K (π)$ 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 Rⁿ when n = 2; 3.

Item Type: Monograph (Working Paper)
Uncontrolled Keywords: Eigenvalue problem, homogeneous solutions to $\mathcal{A}$-harmonic PDEs, Potentials, capacities, $\mathcal{A}$-harmonic Green's function, Minkowski problem, regularity in Monge-Amp{\`e}re equation
Divisions: Faculty of Science and Health > Mathematical Sciences, Department of
Depositing User: Elements
Date Deposited: 12 Sep 2019 13:05
Last Modified: 12 Sep 2019 13:15
URI: http://repository.essex.ac.uk/id/eprint/25322

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