Fourth order operators with general Wentzell boundary conditions

Abstract

Let Ω be a bounded subset of RN with smooth boundary ∂Ω in C4, a ∈ C4 (Ω̄) with a > 0 in Ω̄, and let A be the fourth order operator defined by Au: = Δ(aΔu), respectively Au: = B2u, where Bu: = ∇ ·(a∇u)), with general Wentzell boundary condition of the type Au + β∂(aΔu)/∂n + γu = 0 on ∂Ω, (respectively Au + β∂(Bu)/∂n + γu = 0 on ∂Ω). We prove that, under additional boundary conditions, if β, γ ∈ C 3+ε(∂Ω), β > 0, then the realization of the operator A on a suitable Hilbert space of L2 type, with a suitable weight on ∂Ω, is essentially self-adjoint and bounded below. Copyright © 2008 Rocky Mountain Mathematics Consortium.

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Rocky Mountain Journal of Mathematics

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