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In this paper, firstly we give an atomic decomposition of the local Hardy spaces hтP(Ω)(0 ＜ p≤1) and their dual spaces, where the domain Ω is exterior regular in Rn(n≥ 3). Then for given data f ∈ hтp(Ω), we discuss the inhomogeneous Dirichlet problems: Lu= f inΩ u = 0 on δΩwhere the operator L is uniformly elliptic. Also we obtain the estimation of Green potential in the local Hardy spaces hrp(Ω).