On Juliusz Schauder’s Paper on Linear Elliptic Differential Equations

In what follows by Ω we denote a bounded domain in R n with n ≥ 2, ∂Ω ∈ C 2,α and 0 < α < 1. We let Lu(x) = n i,k=1 a i,k (x) · u xix k (x) + n i=1 a i (x) · u xi (x) + a(x) · u(x) = f (x) to be a linear elliptic differential equation of second order with coefficients a ik , a i , a, f ∈ C 0,α (Ω) and a(x) ≤ 0 for all x ∈ Ω. The paper of J. Schauder " ¨ Uber lineare elliptische Differentialgleichungen zweiter Ordnung " is treating the solvability of Dirichlet's problem for the above equation. The main results of Schauder's work are given in the following theorems. Key words and phrases. Second order elliptic equations, boundary value problems for second order elliptic equations.