Akman, Murat and Lewis, John and Vogel, Andrew (2019) 'Note on an eigenvalue problem with applications to a Minkowski type regularity problem in Rⁿ.' Calculus of Variations and Partial Differential Equations, 59 (2). ISSN 09442669 (In Press)

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Abstract
We consider existence and uniqueness of homogeneous solutions u > 0 to certain PDE of pLaplace 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  (n1)/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:  Article 

Uncontrolled Keywords:  Eigenvalue problem; homogeneous solutions to $\mathcal{A}$harmonic PDEs; Potentials, capacities, $\mathcal{A}$harmonic Green's function, Minkowski problem, regularity in MongeAmp{\`e}re equation 
Divisions:  Faculty of Science and Health Faculty of Science and Health > Mathematical Sciences, Department of 
SWORD Depositor:  Elements 
Depositing User:  Elements 
Date Deposited:  12 Sep 2019 13:05 
Last Modified:  26 Feb 2022 10:40 
URI:  http://repository.essex.ac.uk/id/eprint/25322 
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