How solvability and regularity theory for elliptic operators of order 2m (the biharmonic and polyharmonic operators, and their divergence-form and composition-form variable-coefficient generalizations) behaves when the domain has only a Lipschitz or arbitrary rough boundary. Students learn why the second-order toolkit fails at higher order: the Agmon-Miranda maximum principle can break down above dimension three, sharp pointwise bounds on derivatives of polyharmonic functions and their Green functions replace it, and boundary regularity is characterized by a higher-order Wiener test built on a polyharmonic capacity. The topic also covers the well-posedness ranges of the Lp-Dirichlet, regularity, and Neumann problems on Lipschitz domains, and why even defining Neumann data is non-unique for operators of order four and above.
This arXiv paper by Ariel Barton and Svitlana Mayboroda is a comprehensive survey covering boundary value problems for linear higher-order elliptic PDEs (including polyharmonic equations and fourth-o…