Singularly perturbed elliptic boundary value problems

Summary The paper treats elliptic operators of the form L(ɛ∂₁, ..., ɛ∂_n), where L is a polynomial in a variables of order 2m₁, and ɛ is a small parameter. Solutionsu _ɛ of Lu=0 in a half space satisfyng conditions B_j(ɛ∂₁, ɛ∂₂, ..., ɛ∂_n)u=ɛγ_jϕ_j(x)(j=1, ..., m₁) on the boundary are constructed and estimated using H?lder norms, Poisson kernels, and an elaborate potential theory. Properties of the interior limit u₀=u _ɛ(κ) are studied. The paper is preparatory to a detailed investigation of Schauder estimates for such problems with variables coefficients. Supported in part by N. S. F. Grant GP-11660. Entrata in Redazione il 9 gennaio 1971.