Elliptic Problems with Normal Boundary Conditions

Abstract

Let G ⊂ ℝⁿ be a bounded domain. Suppose that the boundary ∂G is sufficiently smooth, i.e., belongs to the class C ^t with some t ≥ 1 (see Section 1.10). Let x ₀ ∈ ∂G and let U =U(x ₀) be a sufficiently small neighborhood of the point x ₀ in ℝⁿ. In $\bar G \cap U$, we introduce a system of special local coordinates (y ₁, …, y _n ) such that (y ₁,…, y _n-1,0) = (y′, 0) is a system of local coordinates in ∂G∩ U and y _n is equal to the distance between the point y and ∂G (see Section 1.10). Below, we consider only special local coordinates defined in a sufficiently small neighborhood U(x ₀) of every point x ₀ ∈ ∂G. If (y′,…, y′ _n ) is any other system of special coordinates in G ∩ V, then, in U ∩ V ∩) G, we have

$$ y'_j = y'_j (y_1 , \ldots ,y_{n - 1} )\,(j = 1, \ldots ,n - 1),\,y'_n = y_n $$

((5.1.1))

and the determinant of the Jacobi matrix det dy′/dy of this transformation is not equal to zero.