A boundary point is regular for the Dirichlet problem when the Perron solution approaches the prescribed continuous boundary value there. The existence of a barrier implies regularity.
A domain satisfies the exterior sphere condition at a boundary point when a ball contained in the complement is tangent to the boundary only at that point. Fundamental solutions of the Laplace equation centred at the ball's centre provide a harmonic barrier, so such a point is regular.

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  1. Barrier for the Dirichlet problem
  2. Perron method for the Dirichlet problem
  3. Dirichlet problem
  4. Analysis
  5. Area of mathematics
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