Let $U$ be a open, bounded with boudary sufficiently regular domain. I want to show that the operator $$ A:= -\Delta^2 - \beta\Delta $$ defined on $U$, for some $\beta\in\mathbb{R}$ and Dirichlet+Neumann boundary conditions is sectorial.
What I did up to now:

Let $L^2(U)$ be the Hilbert ambient space

  1. $A:D(A)\to L^2(U)$ is defined on $H^4(U)\cap H^1_0(U)$, thus is densely defined. Moreover is closed and self-adjoint since $-\Delta$ is such. Hence, $\sigma(A)$ is real.

To state that $A$ is sectorial I still have to show that $||R(\lambda:A)||\leq \frac{M}{|\lambda|}$ for some constant $M$.

  1. $||R(\lambda:A)||=||(\Delta^2 + \beta\Delta +\lambda I)^{-1}||=||(\Delta+bI)^{-1}||||(\Delta+aI)^{-1}||\leq \frac{M}{|ab|}$. However, it is possible to show that exist $a,b\in\mathbb{C}$, such that the factorization above holds.

How can I be sure that $\lambda$ is in a sector? Any help will be appreciate!


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