Kellogg-Evans theorem

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Kellogg's lemma

The set of all irregular boundary points of an arbitrary domain $D$ in the Euclidean space $\mathbf{R}^n$, $n\ge 2$, with respect to a generalized solution of the Dirichlet problem for $D$ in the sense of Wiener–Perron (see Perron method) has capacity zero, is a polar set and has type $F_\sigma$. A corollary of the Kellogg–Evans theorem is that if $K$ is a compact set of positive capacity in $\mathbf{R}^n$ and $D$ is the connected component of the complement $\complement K$ containing the point at infinity, then there exists on the boundary $\partial D \subset K$ at least one regular point. The Kellogg–Evans theorem was stated by O.D. Kellogg [1] as a conjecture, and was first proved by G.C. Evans [2].


[1] O.D. Kellogg, "Foundations of potential theory" , F. Ungar (1929) (Re-issue: Springer, 1967)
[2] G.C. Evans, "Application of Poincaré's sweeping-out process" Proc. Nat. Acad. Sci. USA , 19 (1933) pp. 457–461
[3] M.V. Keldysh, "On the solvability and stability of the Dirichlet problem" Uspekhi Mat. Nauk , 8 (1941) pp. 171–231 (In Russian)
[4] M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959)
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