Reference for compution adjoint of the operator $L=\Delta^2$ I need to use adjoint operator of the partial differential operator $L=\Delta^2$, where $\Delta$ denotes the Laplacian. 
I do not want to put this computation in my thesis, because I feel is a bit distracting since it would probably involve the local and integral form of $L$. 
Is there some book where I can find this adjoint operator?
 A: Consider the Green's identity for smooth function first:
$$
\int_{\Omega} (\Delta^2 u )\,v = \int_{\Omega}\Delta u\Delta v + \int_{\partial \Omega} v\frac{\partial \Delta u}{\partial n} dS - \int_{\partial \Omega} \frac{\partial v}{\partial n} \Delta u\,dS.
$$
Hence we have:
$$
\int_{\Omega} (\Delta^2 u )\,v - \int_{\Omega} u\,(\Delta^2 v ) = 
\int_{\partial \Omega} v\frac{\partial \Delta u}{\partial n} dS 
-\int_{\partial \Omega} \frac{\partial v}{\partial n} \Delta u\,dS 
- \int_{\partial \Omega} u\frac{\partial \Delta v}{\partial n} dS 
+ \int_{\partial \Omega} \frac{\partial u}{\partial n} \Delta v\,dS.
$$
Therefore, when the spacial domain of interest $\Omega$ is bounded and open, $\Delta^2$ is self-adjoint operator when its domain is


*

*$\{w\in H^4(\Omega): w = \Delta w = 0 \text{ on }\partial \Omega\}.$

*$\{w\in H^4(\Omega): \dfrac{\partial w}{\partial n} = \dfrac{\partial \Delta w}{\partial n} = 0 \text{ on }\partial \Omega\}.$

*$\{w\in H^4(\Omega): w = \dfrac{\partial w}{\partial n}   = 0 \text{ on }\partial \Omega\}.$

*$\{w\in H^4(\Omega): \Delta w=  \dfrac{\partial \Delta w}{\partial n} = 0 \text{ on }\partial \Omega\}.$
Whenever the combination of these boundary conditions will make the boundary terms vanish ($u$, $v$ both satisfy the boundary conditions), and actually make physical sense, $\Delta^2$ is self-adjoint. 
If there is no boundary condition given, just plain $H^4(\Omega)$, the adjoint operator will have to take boundary terms into account, and it is most of the time impossible to write the explicit expression.
If the spacial domain of interest is $\mathbb{R}^n$, then $\Delta^2$ is self-adjoint on $H^4(\mathbb{R}^n)$ (when certain decaying properties are assumed/proved).
If $\Delta^2$'s domain is hard to analyze, you can construct a self-adjoint extension using Friedrichs extension theorem since it can be viewed a densely defined operator. For example, $\Delta^2$ is symmetric on $C^{\infty}_c(\mathbb{R}^n)$, which is dense in $H^4(\mathbb{R}^n)$.
