Bol. Soc. Paran. Mat. (3s.) v. 30 2 (2012): 79–83. c SPM –ISSN-2175-1188 on line ISSN-00378712 in press SPM: www.spm.uem.br/bspm doi:10.5269/bspm.v30i2.14502 Strict monotonicity and unique continuation of the biharmonic operator Najib Tsouli, Omar Chakrone, Mostafa Rahmani and Omar Darhouche abstract: In this paper, we will show that the strict monotonicity of the eigen- values of the biharmonic operator holds if and only if some unique continuation property is satisfied by the corresponding eigenfunctions. Key Words:variational methods, eigenvalues, biharmonic operator, unique continuation. Contents 1 Introduction 79 2 Strict monotonicity and unique continuation 80 1. Introduction Consider the Navier boundary value problem involving the biharmonic operator    find (λ, u) ∈ R × ( (H 1 0 (Ω) ∩ H 2 (Ω))\{0} ) such that Δ 2 u = λmu on Ω, u =Δu =0 in ∂ Ω (1) where Ω ⊂ R N (N ≥ 1) is an open bounded with a boundary ∂ Ω and Δ 2 denotes the biharmonic operator defined by Δ 2 u = Δ(Δu). The weight function m(x) is assumed to be in L ∞ (Ω) and meas {x ∈ Ω; m(x) =0} > 0. It is very well known (cf. [7]), that the spectrum σ(Δ 2 ,m) of the problem (1) contains a sequence of positive eigenvalues 0 <λ 1 (m) <λ 2 (m) ≤ λ 3 (m) ≤···→ +∞. Moreover, λ k (m) has the variational characterization : 1 λ k (m) = sup F k inf u∈F k  Ω mu 2 ,  Ω |Δu| 2 =1  , (2) where F k varies over all k-dimensional subspaces of H 1 0 (Ω) ∩ H 2 (Ω) and λ k (m) is repeated with its order of multiplicity. For convenience, we now give the following definitions : 2000 Mathematics Subject Classification: 35G15, 35J35 79 Typeset by B S P M style. c  Soc. Paran. de Mat.