On Juliusz Schauder's paper on linear elliptic differential equations (Q1282201)

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^n\), \(n\geq 2\), \(\partial\Omega\in C_{2,\alpha}\) and \(0<\alpha< 1\). Let \[ Lu(x)= \sum^n_{i,k= 1} a_{i,k}(x)\cdot u_{x_ix_k}(x)+ \sum^n_{i= 1} a_i(x)\cdot u_{x_i}(x)+ a(x)\cdot u(x)= f(x) \] be a linear elliptic differential equation of second order with coefficients \(a_{ik}\), \(a_i\), \(a\), \(f\in C_{0,\alpha}(\overline\Omega)\) and \(a(x)\leq 0\) for all \(x\in\Omega\). In the paper of \textit{J. Schauder} [Math. Z. 38, 257-282 (1934; Zbl 0008.25502)] is treated the solvability of Dirichlet's problem for the above equation. In order to read the original work of Schauder, a strong background in potential theory is required so that the proofs in that paper could only be carried out by a few readers. In the literature, the results of above paper are used without going into the proofs. For this reason it is often overlooked that not only Theorems 1, 3 and 4 stem from Schauder but Theorem 2 as well. Our aim is to show that Theorem 2 is in fact the principal result of the Schauder's paper. If one has Theorem 2 the Theorems 1, 3 and 4 follow with much less work than is necessary in giving a direct proof.