Summary: Let \(\Omega\) be a bounded subset of \(\mathbb R^N\) with smooth boundary \(\partial\Omega\) in \(C^4\), \(a\in C^4(\overline{\Omega})\) with \(a>0\) in \(\overline{\Omega}\), and let \(A\) be the fourth-order operator defined by \(Au:= \Delta(a\Delta u)\), respectively, \(Au:= B^2u\), where \(Bu:=\nabla\cdot (a\nabla u))\), with general Wentzell boundary condition of the type
\[ \begin{aligned} &Au+\beta \frac{\partial(a\Delta u)}{\partial n}+\gamma u=0 \quad\text{on }\partial\Omega,\\ \bigg(\text{respectively}\quad &Au+\beta\frac{\partial(Bu)}{\partial n}+ \gamma u=0 \quad\text{ on }\partial\Omega\bigg). \end{aligned} \]
We prove that, under additional boundary conditions, if \(\beta,\gamma\in c^{3+\varepsilon}(\partial\Omega)\), \(\beta>0\), then the realization of the operator \(A\) on a suitable Hilbert space of \(L^2\) type, with a suitable weight on \(\partial\Omega\), is essentially self-adjoint and bounded below.