An approximate solution to Mosolov-Myasnikov variational problem with Coulomb boundary friction (Q2729359)

Let \(\Omega\subset\mathbb R^2\) be a bounded domain with sufficiently smooth boundary \(\Gamma\). The following so-called Mosolov-Myasnikov variational problem with Coulomb boundary friction is considered: NEWLINE\[NEWLINE J(u) = \frac{1}{2}\int_{\Omega}|\nabla u|^2d\Omega - \int_{\Omega}fu\,d\Omega + \int_{\Omega}g_1|\nabla u|\,d\Omega + \int_{\Gamma}g_2| u|\,d\Gamma \to \min,\quad u\in W_2^1(\Omega), NEWLINE\]NEWLINE where \(g_1\), \(g_2\) are positive constants and \(f\in L_2(\Omega)\). First, the authors study the regularized problem NEWLINE\[NEWLINE J_{\varepsilon}(u) = \frac{1}{2}\int_{\Omega}|\nabla u|^2d\Omega - \int_{\Omega}fu\,d\Omega + \int_{\Omega}g_1\sqrt{|\nabla u|^2 + \varepsilon^2}d\Omega + \int_{\Gamma}g_2| u|\,d\Gamma \to \min,\quad u\in W_2^1(\Omega). NEWLINE\]NEWLINE The following theorems hold:NEWLINENEWLINE Theorem 1. Let \(\int_{\Gamma}g_2d\Gamma - \left|\int_{\Omega}fd\Omega\right| > 0\). Then the solution \(u\) to the regularized problem is unique in \(W_2^2(\Omega)\).NEWLINENEWLINE Theorem 2. Let the condition of theorem 1 holds. Then an arbitrary minimizing sequence \(\{u^n\}\) of the regularized problem converges in \(W_2^1(\Omega)\) to the unique solution \(u^*\) of the original Mosolov-Myasnikov problem.NEWLINENEWLINE To obtain an approximate solution of the Mosolov Myasnikov problem, the authors construct a method based on modified Newton algorithm with step regularization. The authors prove the stability of the method and give an example of its numerical realization.