The Decomposition Theorem

If K ⊂ Ω is compact and u is harmonic on Ω \ K, then u might be badly behaved near both ∂K and ∂Ω; see, for example, Theorem 11.18. In this chapter we will see that u is the sum of two harmonic functions, one extending harmonically across ∂K, the other extending harmonically across ∂Ω. More precisely, u has a decomposition of the form $$ % MathType!MTEF!2!1!+-% feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB% LrhDaibaieYlf9irVeeu0dXdh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhacqGH9a% qpcqaH9oGBcqGHRaWkcaWG3baaaa!3B9A!$$$$u = \nu + w$$ on Ω \ K, where v is harmonic on Ω and w is harmonic on Rn \ K. Furthermore, there is a canonical choice for w that makes this decomposition unique.