Schauder theory for Dirichlet elliptic operators in divergence form (Q358613)

The author investigates the strongly elliptic operator of order \(2m\) in divergence form \[ Au(x)=\sum_{|\alpha|\leq m,\,|\beta|\leq m}D^{\alpha}(a_{\alpha\beta}(x)D^{\beta}u(x)) \] with Hölder continuous coefficients of exponent \(\sigma\in (0,1)\) under Dirichlet boundary condition in a domain \(\Omega\subset\mathbb{R}^n\), where \(D=(D_1,\dots ,D_n)\), \(D_j=-\sqrt{-1}\partial/\partial x_j\,\,\,(j=1,\dots ,n)\). By using an operator theoretic approach, he shows that, for an operator \(A:C_0^{m+\sigma}(\Omega)\to C^{-m+\sigma}(\Omega)\), the inverse operator \((A-\lambda)^{-1}:C^{-m+\sigma}(\Omega)\to C_0^{m+\sigma}(\Omega)\) exists for \(\lambda\) in a suitable angular region of \(\mathbb{C}\) and estimates its operator norm.