On solvability of the Reynolds equation for gas lubrication (Q2773598)

Let \(\Omega\) be a connected bounded domain in \(\mathbb R^2\) with smooth boundary \(\partial\Omega\subset C^{1,\gamma}\). The following quasilinear elliptic problem is considered in \(\Omega\): NEWLINE\[NEWLINE \begin{gathered} \text{div}\bigl(h^3(x)p(x)\nabla p(x) - \Lambda h(x)p(x)\mathbf v\bigr)= 0, \quad x = (x_1,x_2) \in\Omega, \\ \left.p\right|_{\partial\Omega} = p_a,\quad p(x) \geq 0, \quad x \in \Omega, \end{gathered} NEWLINE\]NEWLINE where \(p_a\) is a positive number, \(\Lambda > 0\) is a numerical parameter, \(\mathbf v\) is a fixed vector directed along the \(x_1\)-axis. It is assumed that \(h\in C^{1,\gamma}(\bar\Omega)\), \(h \geq h_0\) for all \(x\in\bar\Omega\). The equation is known as the Reynolds equation which arises in the theory of gas lubrication.NEWLINENEWLINENEWLINEThe main result of the article reads as follows: Assume that the above-mentioned conditions on the data of the problem are fulfilled. Then, there exists a unique solution \(p\in C^{2,\gamma}\) such that \(p \geq p_0 > 0\).