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## Asymptotic properties of a functional defined on Sobolev space

I found this text in an article studying the Cauchy problem related to some PDE. where $\Omega \subset \mathbb{R}^3$ is bounded with smooth $\partial \Omega$ and $\alpha >0.$ I have to questions: 1.I wonder why the operator $(\alpha^2 \Delta^2 + \Delta)^{\frac 12}$ is bounded. The other eigenvalues $\lambda_j$ for which $(\alpha^2 \lambda_j^2 – \lambda_j)^{\frac 12}$ […]