# Kellogg-Evans theorem

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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].