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

Farcaseanu, Maria and Mihăilescu, Mihai and Stancu-Dumitru, Denisa (2023) A minimization problem related to the principal frequency of the p-Bilaplacian with coupled Dirichlet–Neumann boundary conditions. ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS, 2023 (51). pp. 1-9. ISSN 1417-3875

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Abstract

For each fixed integer N ≥ 2 let Ω ⊂ RN be an open, bounded and convex set with smooth boundary. For each real number p ∈ (1,∞) define M(p;Ω) = where W2,∞ C (Ω) := ∩1<p<∞{u ∈ W2,p (exp(|∆u|p) −1) dx inf u∈W2,∞ C (Ω)\{0} Ω Ω (exp(|u|p) − 1) dx , 0 (Ω) : ∆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.

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