Higher-Order Elliptic Equations in Non-Smooth Domains: a Partial Survey

Abstract

Recent years have brought significant advances in the theory of higher-order elliptic equations in non-smooth domains. Sharp pointwise estimates on derivatives of polyharmonic functions in arbitrary domains were established, followed by the higher-order Wiener test. Certain boundary value problems for higher-order operators with variable non-smooth coefficients were addressed, both in divergence form and in composition form, the latter being adapted to the context of Lipschitz domains. These developments brought new estimates on the fundamental solutions and the Green function, allowing for the lack of smoothness of the boundary or of the coefficients of the equation. Building on our earlier account of history of the subject (published in Concrete operators, spectral theory, operators in harmonic analysis and approximation). Operator Theory: Advances and Applications, vol. 236, Birkhauser/Springer, Basel, 2014, pp. 53–93), this survey presents the current state of the art, emphasizing the most recent results and emerging open problems.