On Wiener's type regularity of a boundary point for higher order elliptic equations (Q2707690)

The famous criterion for the so called regularity of a boundary point by \textit{N. Wiener} [J. Math. Phys. 3, 127-146 (1924)] states that a point \(O\) of a domain \(\Omega \subset \mathbb R^n\), \(n\geq 2\), is called regular if solutions of the Dirichlet problem for the Laplace equation in \(\Omega \) with the Dirichlet data continuous at \(O\) are also continuous at \(O\). \textit{Wiener} [ibid. 24-51] also introduced the notion of harmonic capacity \(\operatorname {cap}(K)\) and proved that a necessary and sufficient condition for \(O\in \partial \Omega \) to be regular is \(\sum _{k\geq 1}2^{(n-2)k}\operatorname {cap}(B_{2^{-k}}\setminus \Omega)=\infty \). This can be rewritten in the integral form \(\int _0\operatorname {cap}(B_\sigma \setminus \Omega)/ \operatorname {cap}(B_\sigma)\frac {d\sigma }\sigma =\infty \) where \(B_\sigma \) is the ball of radius \(\sigma \) centered in \(O\). The result has been extended in various ways for more general elliptic operators. The author [Vestnik Leningrad. Univ. 25, No. 13 (Mat. Meh. Astron. No. 3), 42-55 (1970; Zbl 0252.35024)] found a counterpart to the \(p\)-Laplacian and proved its sufficiency. Recently, \textit{J. Malý} and \textit{W. P. Ziemer} [Mathematical Surveys and Monographs 51, Providence, RI: Am. Math. Soc. (1997; Zbl 0882.35001)] proved its necessity for all \(p>1\). NEWLINENEWLINENEWLINEThe topic can be extended to include other equations, systems, boundary conditions and function spaces. The paper presents a comprehensive survey of corresponding and related results as seen from the list of titles of the sections: Regular points for arbitrary even order elliptic equations; The Hölder regularity and the \(k\)-regularity; Regularity of the vertex of a cone; Weighted positivity of \((-\Delta)^m\); Regularity of a boundary point as a local property; A local estimate; Local estimates stated in terms of the \(m\)-harmonic capacity; A pointwise estimate for a function, \(m\)-harmonic in \(\Omega \setminus B_\rho \); Estimates for the Green function. Many of the results are due to the author whose explanation spans both the history and the state-of-the-art of the topics, including the formulation of nine important open problems.NEWLINENEWLINEFor the entire collection see [Zbl 0952.00033].