Let Ω be a domain in ℝd+1 whose boundary is given as a uniform Lipschitz graph xd+1=η(x) for x ∈ ℝd. For such a domain, it is known that the Helmholtz decomposition is not always valid in Lp(Ω) except for the energy space L2(Ω). In this paper we show that the Helmholtz decomposition still holds in certain anisotropic spaces which include vector fields decaying slowly in the xd+1 variable. In particular, these classes include some infinite energy vector fields. For the purpose, we develop a new approach based on a factorization of divergence form elliptic operators whose coefficients are independent of one variable.
ASJC Scopus subject areas
- 数学 (全般)