Solution Estimates For The Degenerate Oblique Derivative Problems

In this note, we consider the degenerate oblique derivative problem. This note covers the following two topics:One is about the elliptic gain of derivatives from the boundary term of the oblique derivative problem with degenerate boundary. We provide an estimate of the solution. Under the conditions given, the solution can get 3/2 -εderivatives from the boundary for all e> 0. The main tool we use is the theory of pseudo-differential operator. The other one is about degenerate oblique derivative problem contains singular perturbation term in function. We want to study the oblique derivative problem with degenerate equation and boundary by studying the solution estimate of our problem. This problem is uniformly elliptic singular perturbation function and the elliptic constant depends on the small parameterε. We provide an estimate of the second derivatives and the relationship between the constant and the parameterε.