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A quasi-variational inequality arising in elastohydrodynamics. (English) Zbl 0718.35101
A quasi-variational inequality arising in elastohydrodynamic lubrication is studied. In case of constant viscosity the two-dimensional model of a thin fluid film between an elastic ball and a plane is given by: \[ (*)\quad -\nabla (h^ 3\cdot \nabla u)\geq -\lambda \cdot \partial h/\partial x,\quad (x,y)\in \Omega \subset {\mathbb{R}}^ 2;\quad u\geq 0,\quad (x,y)\in \Omega \] \[ u\cdot [-\nabla (h^ 3\nabla u)+\lambda \cdot \partial h/\partial x]=0,\quad (x,y)\in \Omega;\quad u=0,\quad (x,y)\in \partial \Omega \] \[ h(x,y)=K+x^ 2+y^ 2+\int_{\Omega}(u(s,t)/((x-s)^ 2+(y-t)^ 2))ds dt. \] Based on some useful a priori estimates for the solution of (*) derived by the author, the existence of \(C^{1,1}\) solution is proved by the fixed point theorem. The estimate on the support of the solution and the uniqueness of the solution are established when \(\Omega\) is a large disc or \(\lambda\) is sufficiently small respectively. In the case where the viscosity \(\mu =\mu_ 0\cdot \exp (\alpha u)\), the existence and the uniqueness of the solution are also obtained under the restriction of sufficiently small \(\alpha\).
The one-dimensional model is also discussed in this paper. The author indicates that there is at most one free boundary point in this case.
The main open problem is how to remove the restriction on the smallness of the constant \(\lambda\) or \(\alpha\).

35R35 Free boundary problems for PDEs
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
35B45 A priori estimates in context of PDEs
76B45 Capillarity (surface tension) for incompressible inviscid fluids
47H10 Fixed-point theorems
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