Perron–Wiener–Brelot solution

Summary

The Perron–Wiener–Brelot solution is the generalized solution of the Dirichlet problem given by the Perron method with both upper and lower classes. The upper solution is the infimum of superharmonic functions lying above the boundary data. The lower solution is the supremum of subharmonic functions lying below it. When the two agree and are harmonic, the boundary function is called resolutive, and the common value is the PWB solution. Wiener showed that every continuous boundary function is resolutive, and Brelot characterized resolutive functions as those integrable for harmonic measure.