# References for regularity of solution to fourth order elliptic/parabolic PDEs

After searching, all I could find was regularity of solutions to second-order PDEs like in Partial Differential Equations by Evans and Functional Analysis, Sobolev Spaces and Partial Differential Equations by Brezis. A weak solution $$u$$ is found in $$H^1_0$$ then with improved regularity on the data, they can prove that $$u \in H^2$$.

I was wondering if there is anything similar for fourth-order equations, i.e. you prove a solution in $$H^2 \cap H^1_0$$ and then show that it is in $$H^4$$ for example.

This probably can work if we take a simple biharmonic equation. For sufficiently nice $$\Omega \in \mathbb{R}^n$$ \begin{align*} & \Delta^2 u = f, \ \ \ \text{ in } \Omega \\ & u =0, \ \ \ \Delta u =0 \ \ \ \text{ on } \Omega \end{align*} We can show that $$u \in H^2 \cap H^1_0$$, then we take $$z= \Delta u$$ so now we have $$\Delta z = f$$ and we apply regularity theorem on $$z$$ to get that $$z \in H^2 \cap H^1_0$$ and thus $$u \in H^4 \cap H^1_0$$.

But what about abstract (general) fourth-order equations where we also have coefficients? Are there any analogues theorems for fourth-order PDEs? Can you recommend books?