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

Abstract

We consider existence and uniqueness of homogeneous solutions u > 0 to certain PDE of p-Laplace type, p fixed,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 R n when n = 2, 3.2010 Mathematics Subject Classification. 35J60,31B15,39B62,52A40,35J20,52A20,35J92.

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

“…Lemma 2.6 remains valid when p ≥ n, for G properly defined. However for p > n there is a sign reversal in the inequality and so this Lemma can no longer be used to get an analogue of Lemma 2.7 when p ≥ n. Instead in [3] we use a different Rellich inequality derived from the work on Theorem B in [1] and Theorem B in [2] on a Minkowski existence problem. Armed with this inequality, the proof of Theorems 1.1, 3.10 are similar to the proof outlined for n − 1 < p < n. Finally we note that our interest in this eigenvalue problem stems from our study of regularity in a Minkowski problem, originally proved in Theorem 0.7 of [10] for harmonic functions and later generalized in Theorem 1.4 of [5] to p−harmonic functions when 1 < p < 2.…”

The Lp Minkowski problem for q-capacity

“…On the other hand, similar problems have also been solved for other important Borel measures in physics, such as Harmonic measure, capacity measure, A-capacity measure, the f irst Dirichlet eigenvalue measure and the torsion measure of the Laplacian and some of their L p generalizations, see for example [5][6][7]26 To the best knowledge of the author, there is no research on the L p Brunn-Minkowski theory for q-capacity for any fixed p > 1 and q > n. This leads us to focus on the L p Brunn-Minkowski theory for q-capacity for any fixed p > 1 and q > n in the present paper.…”