Higher-order elliptic equations in non-smooth domains: history and recent results

TL;DR

This survey reviews recent advances in the theory of higher-order elliptic equations in non-smooth domains, highlighting new estimates, boundary value problem solutions, and open problems in the field.

Contribution

It provides a comprehensive overview of recent progress and emerging challenges in higher-order elliptic equations within non-smooth domains.

Findings

Established sharp pointwise estimates for derivatives of polyharmonic functions.

Developed the higher order Wiener test for non-smooth domains.

Addressed boundary value problems with variable non-smooth coefficients.

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, this survey presents the current state of the art, emphasizing the most recent results and emerging open problems.