Symmetric positive systems in corner domains

Abstract

Consider the symmetric positive system of n equations in m + 2 variables, A ∂u ∂x + B ∂u ∂y + ∑ i=1 m C i ∂u ∂z i + Du = ƒ in the corner domain x > 0, y > 0, − ∞ < z i < ∞, with homogeneous data on x = 0 and y = 0. The n × n matrices A, B, C i are symmetric and D is sufficiently positive. On the boundary surfaces the matrix coefficients A, B, C i satisfy certain “torsion” conditions. For ƒ with square integrable first-order derivatives, the strong solution with first-order strong derivatives is derived for the boundary value problem. For less restricted ƒ, the partially differentiable strong solution is established, provided more severe torsion conditions are satisfied on the boundaries. Also, the partially differentiable strong solution is obtained for the case that the torsion conditions are satisfied on one side of the boundary only.