Sub-Elliptic Boundary Value Problems for Quasilinear Elliptic Operators

Abstract

Classical solvability and uniqueness in the Ho¨lder space C2+α(Ω) is proved for the oblique derivative problem aij(x)Diju + b(x, u, Du) = 0 in Ω, ∂u/∂R = ϕ(x) on ∂Ω in the case when the vector field R(x) = (R1(x), ... , Rn(x)) is tangential to the boundary ∂Ω at the points of some non-empty set S ⊂ ∂Ω, and the nonlinear term b(x, u, Du) grows quadratically with respect to the gradient Du.