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  "body": "\n## Counterexample\n\nIn $\\R^3$, write points as $x = (x_1, x_2, x_3)$ and consider the closed thin spike\n$$K = \\set{x \\in \\R^3 : 0 \u003c x_1 \\le 1, \\; x_2^2 + x_3^2 \\le e^{-2/x_1}} \\cup \\set{0}.$$\nIts cross-section at distance $x_1$ from the origin is a disc of radius $e^{-1/x_1}$.\nLet $\\Omega = B \\setminus K$, where $B$ is the [[open ball|0YE5868HZB]] of radius $2$ centred at $0$.\nThe origin $0$ is a boundary point of $\\Omega$, the tip of the spike, and it is not a [[regular boundary point|B7948M365K]].\nCheck with [[Wiener's criterion|STMT1K1865]], using the part $E_k$ of the complement in the shell $2^{-k-1} \\le \\abs{x} \\le 2^{-k}$.\nFor large $k$, $E_k$ lies in a thin cylinder of length about $2^{-k-1}$ and radius at most $e^{-2^k}$.\nA cylinder of length $\\ell$ and radius $r \\le \\ell/2$ has Newtonian [[capacity|4YZVFGJVSZ]] at most a constant times $\\ell / \\log(\\ell / r)$.\nSo $\\operatorname{cap}(E_k)$ is at most a constant times $2^{-k} / 2^k = 4^{-k}$, and $\\sum_k 2^k \\operatorname{cap}(E_k)$ converges.\nHence $0$ is irregular.\n\n## Remarks\n\nThe example is due to Lebesgue.\nThe spike is connected and has interior points, so a boundary point can be irregular even when the complement of $\\Omega$ near it is a connected set of positive volume.\nSuppose the spike is less sharp, for example with radius $x_1^m$ for some $m \u003e 1$.\nThen $E_k$ contains a cylinder whose capacity is at least a constant times $2^{-k}/k$.\nThe terms of Wiener's series are then at least a constant times $1/k$, the series diverges, and the tip is regular.\nIn the plane no such example exists: a boundary point that lies in a connected component of the complement with more than one point is always regular.\n\n## Explanations\n\n### L1 human\n\nPush a very sharp needle into a ball from the inside, so that the needle tip is a point of the boundary.\nIf the needle gets thin fast enough near its tip, the solution of the Dirichlet problem does not feel the tip.\nThen a prescribed boundary value at the tip can be ignored by the solution.\n",
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