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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?

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The theory of general elliptic and parabolic equations of arbitrary order with general boundary conditions (subject to certain quite general compatibility conditions) is indeed well-developed. I would suggest the following texts.

  • Partial Differential Equations by Wloka. This treats both elliptic and parabolic problems with thorough detail.
  • Non-homogeneous Boundary Value Problems and Applications by Lions and Magenes. Volume 1 handles elliptic theory, and Volume 2 handles parabolic problems.
  • Interpolation Theory, Function Spaces, Differential Operators by Triebel. Chapter 5 has a very general treatment of the elliptic theory you're after.
  • Sobolev Spaces, Their Generalizations, and Elliptic Problems in Smooth and Lipschitz Domains by Agranovich. This is limited to elliptic problems.
  • Elliptic Boundary Value Problems in Domains with Point Singularities by Kozlov, Maz'ya, and Rossman. The first part of the book actually deals with smooth boundaries, but then the latter part develops the theory in domains with worse boundary geometry.
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