On the Neumann problems for certain degenerate elliptic operators
The aim of this paper is to study the homogeneous Dirichlet and Neumann problems for the degenerate elliptic equation \[ -\text{div} (\varphi^ \alpha \nabla u) + c(x) \varphi^{\alpha - 1} u = f \quad \text{ in } \quad \Omega, \] where \(\alpha \in C^ 2 (\Omega)\), and \(\varphi (x)\) is a given nonnegative smooth function on \(\Omega\) equivalent to a distance to the boundary: \(\Omega = \{\varphi (x) > 0\}\), \(\partial \Omega = \{\varphi (x) = 0\}\), \(\alpha (x) \in C^ 2 (\Omega)\). It is proved that if \(\alpha (x) < 0\) and \(\partial \alpha/ \partial n = 0\) on \(\partial \Omega\) and \(\inf_{x \in \Omega} c(x) > 0\), then for every \(f \in C^ \mu (\Omega)\) there exists a unique solution \(u = \varphi^{1 - \alpha} v\) to the Dirichlet problem and \(\varphi v \in C^{2 + \mu} (\overline \Omega)\). If \(\alpha (x) > - 1\) and \(\max \{-1 + \mu,- 1 + \mu-\alpha (x), - 1 - \alpha (x)/2\} < \tau < 0\) on \(\partial \Omega\), then the Neumann problem has a unique solution in the space \(\Lambda^{2 + \mu}_{2 + \tau} (\Omega)\) for every \(f\) such that \(f/ \varphi^ \alpha \subset \Lambda^ \mu_ \tau (\Omega)\). The proof of the theorem is based upon a priori estimates for solutions, the method of continuity and the Green function for the approximate problem. An example of an unbounded solution to the Neumann problem is given.
homogeneous Dirichlet and Neumann problems
scientific article; zbMATH DE number 597342