A bstract QUALITATIVE PROPERTIES OF THE FREE-BOUNDARY OF THE REYNOLDS EQUATION IN LUBRICATION

The hidrodynamic lubrication of a cylindrical bearing is governed by the Reynolds equation that must be satisfied by the preassure of lubricatiog oil. When cavitation occurs we are carried to an elliptic free-boundary problem where the free-boundary separates the lubricated region from the cavited region. Some qualitative properties are obtained about the shape of the free-boundary as well as the localization of the cavited region.

We consider the following : Problem (P) .Find a pair of functions (p, -y) (1 .1)(p, y) E Vu x L°°(9) (1 .2)p>OandH(p)<-y<1such that a.e. in 9 (1 .3)fs2 h1,7p,71 = fs1 hy á where h = h(x) = 1+a cos x, with 0 < a < 1, and H is the Heaviside function .This problem is related to the lubrication with cavitation arising in bearings .The first unknow is the pressure distribution -pin a thin film of lubricant contained in the narrow gap between two circular cylinders of parallel axes (the shaft and the bearing) ; another unknow is the percentage -yof oil contained in an elementary volume.
VI E V, such that they and their first shaft and the bearing) ; another unknow is the percentage --yof oil contained in an elementary volume.
Introducing cylindrical coordinates, the gap h depends only on the angular coordinate, being a the eccentricity ratio of the bearing .
The equation (1 .3)derives from the Reynolds equation, div (h 3 Vp) = h', which must be satisfied for p on the region [p > 0], and from conservation laws of flow across the free boundary separing the regions [p > 0] and [p = 0] in 2. In the lubricated region (completely occuped for oil) y is equal to one, while over the cavited region ([p = 0]) y must satisfy 0 < y < 1.
The main goal of this paper is to give some qualitative properties of the free-boundary, The existente of solutions for Problem (P) was proved by Bayada and Chambat in [B-Ch] ; they prove also uniqueness of solutions under the assumption that the free-boundary is a Lipschitz-continuous function of x.A comparison result and uniqueness was proved by Carrillo and the author in [A-C], without any of the previous assumption related to the free-boundary.
For a more general treatment on physical aspects and the formulation of Problem (P), see [A], [B-Ch], [D-TI, [F] .
About existente and uniqueness, we recall the following results: Theorem 1.1
Remark .Theorem 1.2 gives a global comparison result in S2 for pl and P2, when we can compare their values on Po and P1 : this remain true to compare solutions of Problem (P) with solutions of a swiftly modified Problem, as we will precise later in Section 3.
We have: 2. Uniforme bounds for solutions in the x-variable In this section we shall give an upper bound and a lower bound, both independents of x, for solutions of Problem (P).
Let M = maximum h3(X) , and, for 0 < y < 1, let us define, In order to complet the boundedness of p, we have: Theorem 2 .2 .
If (p, -y) is the solution of Problem (P), with pa > M/2, then p > 0 in 9 and so there is not free-boundary .
The figures one and two illustrate functions v_ and v in the two differents cases: p,, < M/2 and p,, > M/2.where the free-boundary (when it exist) lies.The function U attain a maximum in y = z +pQ/M E (0,1) ; we shall prove later that, fixed x, p(x, .) is a non-decreasing monotone function up to this point.
Figure 2 corresponds to the case where there is not free-boundary ; when p,, » M/2 the solution is very close to the function w(y) = p,,y, which satisfies that div (h 37W) = 0, corresponding to the limit case when the eccentricity ratio a is equal to zero, and evidencing that this eccentricity is negligible when the pression on the supply line is very great .

Behaviour of the free-boundary in the y-variable
We consider in this section the case p,, < M/2, denoting by y, the value ym = z +p,, /M, where the function U, defined by (2 .1),attain a maximum.Let yo = 2p Q /M, and take yi any value in (y,, l).Finally, let SZl = (0, 27r) x (yo , yi ), denoting by ró and ri the lower and upper boundarys of Q1 respectively .(see Fig .3) .
I3efore to give the proof of Theorem ,4 .1 we shall first prove some previous results about p(x, y) .We remark that the technics to prove this theorem are the same that the ones used to prove uniqueness .They are based on ¡he construction of a class of test functions defined in a multidimensional domain .Moreover p :5 p on r;, because y = z and f2 > 1, and p < p,, < fl2pa = p on rl .0 We shall distinguish the x-variable for p and p, using the variables (xl, y) E 521 for (p, -y) and (x2, y) E 521 for (p, y) ; we set Q = (0, 27r) x (0, 27r) x (yo , y,), and let us consider J(r) and p(r), real functions such that : For small e > 0 we define p,(r) = (lle)p(rle), and finally for (xl, x2, let F(x1, x2, y) be defined by p is a pair function .In the new variables, the equation (3.4) becomes: where we omite the constant due to the coordinates transformation, and denote by Qtz the new domain .Proof.
J3 1 :51 J2 1 and J2 can be decomposed in two integrals having both of them limit equal to zero, when we pass to the limit first as 6 -> 0 and later as e ---> 0.

. Behaviour of y in the x-variable
We go to study some properties of y with geometrical consequences on the free-boundary, when x E (0, 7r) .
Let (p, y) be the solution of Problem (P), and let X be the characteristic function of the set [p > 0] ; then, (4 .1)(hy)x -h'X > 0 in D'(S2) .
Proof.: From the continuity of p, there exist QE _ (xo -E, xo -}-E) x (yo -E, yo + E) such that p > 0 in QE (see Fig .8) and y = 1 a.e. in Q, Like yx > 0 we get y = 1 a.e. in CE .Using the strong minimum principle, p can not attain the minimum value zero in CE and hence Remark.As a consequence of this Corollary the free-boundary can not Nave vertical oscillations in the interval (0, 7r) .
Taking account the Corollary 3.7, we conclude that the free-boundary is a monotone decreasing graph -y = F(x)-in the interval (0, 7r) (see Fig .9).Given y E (0, 1) there exist a region of positive measure in (0, 21r) where p>0 .