Knowlpedia

Definition

Harmonic distribution

A distribution annihilated by the distributional Laplacian.

A distribution hh on an open set Ω⊂Rn\Omega\subset\mathbb R^n is harmonic in distributions if Δh=0\Delta h=0, meaning

⟨h,Δφ⟩=0(φ∈Cc∞(Ω)).\langle h,\Delta\varphi\rangle=0\qquad(\varphi\in C_c^\infty(\Omega)).

The Laplacian is formed using .

Ambiguity of Poisson solutions

If −Δu=f=−Δv-\Delta u=f=-\Delta v, then u−vu-v is harmonic in distributions. A nonzero constant is harmonic, so harmonicity alone does not force the difference to vanish. A global norm condition can remove this ambiguity. Every classical harmonic function gives an example of a harmonic distribution.