A minimization problem related to the principal frequency of the p -Bilaplacian with coupled Dirichlet–Neumann boundary conditions

Abstract

For each fixed integer N ≥ 2 let Ω ⊂ R N be an open, bounded and convex set with smooth boundary. For each real number p ∈ ( 1 , ∞ ) define M ( p ; Ω ) = inf u ∈ W C 2 , ∞ ( Ω ) ∖ { 0 } ∫ Ω ( exp ⁡ ( | Δ u | p ) − 1 ) d x ∫ Ω ( exp ⁡ ( | u | p ) − 1 ) d x , where W C 2 , ∞ ( Ω ) := ∩ 1 < p < ∞ { u ∈ W 0 2 , p ( Ω ) : Δ u ∈ L ∞ ( Ω ) } . We show that if the radius of the largest ball which can be inscribed in Ω is strictly larger than a constant which depends on N then M ( p ; Ω ) vanishes while if the radius of the largest ball which can be inscribed in Ω is strictly less than 1 then M ( p ; Ω ) is a positive real number. Moreover, in the latter case when p is large enough we can identify the value of M ( p ; Ω ) as being the principal frequency of the p -Bilaplacian on Ω with coupled Dirichlet–Neumann boundary conditions.