Oblique and Nonlinear Boundary Value Problem for the Stokes Operator in a Class Domain C2

Abstract

In this chapter, we study the existence, uniqueness and regularity of the solution of problem (2.1.1) which is a nonlinear boundary value problem related to the perturbed Stokes system, and this in regular domains. We searched for the solutions (u, p) ∈ H2(Ω)n × H1(Ω) of this problem going through several steps, first we approached the problem (2.1.1) by (2.1.2) , in the sense of Yosida, then we introduced a new problem (2.2.1) to apply the fixed point method, and subsequently we made an a priori inequality based on the integration by parts formula with a passage to the limits with respect to λ, and which gives us the existence and uniqueness of a solution (u, p) ∈ H2(Ω)n × H1(Ω) for k satisfies certain condition.