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A mixed Dirichlet-Neumann problem for a nonlinear Reynolds equation in elastohydrodynamic piezoviscous lubrication. (English) Zbl 0857.35044
Given a rectangular $$\Omega=\{(x_1,x_2):|x_i|<1\}$$, the first goal is to find a nonnegative weak solution $$p\in W^{1,2}(\Omega)$$ of the equation $\text{div}(h^3 [p]\exp(-\alpha p)\nabla p)={\partial\over{\partial x_1}}c(H_\varepsilon(p) h[p])$ with the boundary conditions $$p=0$$ at $$\{|x_2|=1\}\cup\{x_1=1\}$$ and $${\partial\over{\partial x_1}}p=c(H_\varepsilon(p)- \theta_0)$$ at $$\{x_1=-1\}$$. Here $$c$$ and $$\alpha$$ are positive constants, $$h[p]=h_0(x)+ p*1/|x|$$, $$h_0$$ is a smooth positive function, $$H_\varepsilon(p)$$, $$\varepsilon\to 0$$, is an approximation of the Heaviside function. Then the estimates $$|p|_{L^\infty}\leq c$$ and $$|p|_{W^{1,2}}\leq c$$ are obtained uniformly in $$\varepsilon$$ to prove the convergence $$p\to\overline{p}$$ to a weak solution $$\overline{p}$$ of the limit equation with $$H_\varepsilon(p)$$ substituted by $$\theta(x,t)=w \text{-lim }H_\varepsilon(p)$$. It is claimed that the pair $$\overline{p}$$ and $$\theta$$ solves the lubricant pressure problem modelled by the Reynolds equation.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 76D08 Lubrication theory 35R35 Free boundary problems for PDEs
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##### References:
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