Existence and uniqueness of classical solutions for certain degenerated elliptic equations of the second order (Q1061305)

The paper considers an operator \(A=- \sum^{n}_{j,k=1}a_{jk}(x)\partial_ j\partial_ k+\sum^{n}_{j=1}a_ j(x)\partial_ j+a_ 0(x),\) and which is elliptic in the interior of a domain \(\Omega \subset R^ n\) and degenerated only in the normal direction at each point of the boundary. Under certain assumptions on the coefficients and the degeneration degree the existence and uniqueness of the classical solution of the equation \(Au=f\) in some Hölder spaces with a non-isotropic distance among the elements are proved. In the proofs a priori inequalities of Schauder type are essentially used. The operator \(L_{\alpha}=-x_ n\partial^ 2_ n-\sum^{n-1}_{j=1}\partial^ 2_ j+\alpha \partial_ n,\) \(\alpha <0\) degenerating for \(x_ n=0\), may serve as a model of the operators investigated in \(\Omega =R_+^ n\).