ABSTRACT
The present paper is concerned with the regularity of a solution to the following problem: Find and satisfying
Here and hereafter the following notation is employed: Ω is a bounded domain in m=2 or 3. The boundary ∂Ω is composed of two connected components Γ0 and Γ. For the sake of simplicity, we assume that Γ0 and Γ of class C
2 and that The additional smoothness assumption on Γ will be specified later. We introduce
then denotes the solenoidal subspace of K 1(Ω). (·,·) denotes the inner product in L 2(Ω) or L 2 (Ω) m according as scalar-valued functions or vector-valued functions. We
(1.1)
set
for u=(u 1,…, u m) and v=(v 1,…, v m). Finally
where g is a given scalar function defined on Γ. As was described in Fujita and Kawarada [7], the variational inequality (1.1) arises in
the study of the steady motions of the viscous incompressible fluid under the frictional boundary condition, where u denotes the flow velocity, p the pressure and f the external forces acting on the fluid, and g is called the modulus function of friction. We now review the boundary condition of this type. Let σ (u, p) be the stress vector to Γ. That is, we let σ(u, p)=S(u, p)n, where S(u, p)=[−pδ i,j +e i,j (u)] stands for the stress tensor and n the unit outer normal to Γ. Then we pose on σ(u, p) that
and
almost everywhere on Γ. The classical form of the frictional boundary value problem for the Stokes equations dealt with in [7] consists of
together with (1.3) and (1.4). The inequality (1.1) is a weak form of this problem. Remark 1.1. To be more precise, j should be understood as the functional on H 1/2 (Γ)
However, for the sake of simplicity, we will regard j as the functional on H 1 (Ω) through
(1.2)
(1.3)
(1.4)
(1.5)
and write it as (1.2). The existence theorem was established in [7]. Assume that
Then (1.1) admits of a solution{u, p}. The velocity part u is unique, and the uniqueness of the pressure part p depends on cases. We shall give an example about this issue in §5.
