Take $\lambda>0$ and let $\Delta$ be the Laplacian operator in dimension $n$ and let $\Delta^2:=\Delta\circ\Delta$. Consider the boundary value problem $\Delta^2 u+\lambda u=f$ on some open subset $U\subset\mathbb{R}^n$ with smooth boundary $\partial U$ and boundary conditions $u=\frac{\partial u}{\partial n}=0$ on $\partial U$. Here $\frac{\partial}{\partial n}$ is the directional derivative in the outward direction normal to $\partial U$. Furthermore $f\in L^2(U)$.
This is part of a more general parabolic PDE boundary/ininital-value problem which I am trying to solve by means of Semi Group theory and Hille-Yosida. Now I was able to find a weak solution $u$ of the above problem in the Sobolev (closed sub)space $H^2_0U)$. In order to make my argument complete I need to show that $u$ is actually also in $H^4$. This is the part I would like to have some support for.
It is a regularity problem. I am using Evans as a reference. But Evans only discusses second order Elliptic PDE's. In which case the typical result he discuses is that given certain conditions it holds that $f\in H^n\Rightarrow u\in H^{n+2}$.
It would be very convenient if these type of results also apply to higher order PD operators such $\Delta^2$. But I am not quite sure how to generalise this to my case.