TY - JOUR

T1 - Note on an eigenvalue problem with applications to a Minkowski type regularity problem in Rn

AU - Akman, Murat

AU - Lewis, John

AU - Vogel, Andrew

N1 - Funding Information:
Part of this research was done while the second author was visiting TIFR in Bangalore India. The second author thanks TIFR for their gracious hospitality.

PY - 2020/4/1

Y1 - 2020/4/1

N2 - 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,⋯,xn):x1>cosα|x|}forfixedα∈(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.

AB - 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,⋯,xn):x1>cosα|x|}forfixedα∈(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.

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U2 - 10.1007/s00526-020-1697-7

DO - 10.1007/s00526-020-1697-7

M3 - Article

AN - SCOPUS:85079177880

VL - 59

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

IS - 2

M1 - 47

ER -