# Weak solution to PDE boundary value problems

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.

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Why not applying the elliptic regularity on $v = \Delta u$? –  Shuhao Cao May 18 '13 at 18:46
I have stated the wrong PDE. It should be $\Delta u + \lambda u =f$. –  Aris May 19 '13 at 13:46
@Aris, take a look in THeorem 2.20 and the following corollary: google.com/… –  Tomás May 20 '13 at 22:15