The Boundary Value Problems for the Bi-Laplace-Beltrami Equation

摘要

The purpose of the present paper is to investigate the boundary value problems for the bi-Laplace–Beltrami equation ?2 C φ = f on a smooth hypersurface C with the boundary Γ = ?C . The unique solvability of the BVP is proved on the basis of Green’s formula and Lax–Milgram Lemma. We also prove the invertibility of the perturbed operator in the Bessel potential spaces ?2 C +H I : Hs+2 p (S ) → Hs?2 p (S ) for a smooth closed hypersurface S without boundary for arbitrary 1 < p < ∞ and ?∞ < s < ∞, provided H is a smooth function, has non-negative real part Re H (t) > 0 for all t ∈ S and non-trivial support mes supp Re H ?= 0.