Strict Monotonicity and Unique Continuation for the Third-Order Spectrum of Biharmonic Operator

Abstract

We will study the spectrum for the biharmonic operator involving the laplacian and the gradient of the laplacian with weight, which we call third-order spectrum. We will show that the strict monotonicity of the eigenvalues of the operator ${\mathrm{\Delta }}^{2}u+2\beta ·\nabla (\mathrm{\Delta }u)+|\beta {|}^2\mathrm{\Delta }u$, where $\beta \in {ℝ}^N$, holds if some unique continuation property is satisfied by the corresponding eigenfunctions.