Two Oblique Derivative Problems

This paper mainly includes two parts. The first part is the LP estimate for the conormal derivative problem. This Lp estimate has been given in [4] under the condition that domain is (δ, R) Reifenberg flat. The proof given in [4] needs an inclusion relation which is stated as (2.1.6) in the present paper. This inclusion relation is not proven in [4]. The author of the present paper believes that this inclusion relation is not always true. Because of the importance of the Lp estimate, we give a proof under a stronger condition called (δ, R) flatness (See Definition2.2.1) on the domains, using the method in [4]. The second part is the regularity estimate of the Poincare problem with codimension-1 degeneracy, which has been discussed in [11]. But there are two points in the proof in [11] that are not very clear to the author of the present thesis. So, in this paper, we construct a proof. Our proof consists of constructing an auxiliary function to reduce the problem in the form that has been treated in [7], and using the result in [7].