A free boundary problem for a Hamilton-Jacobi equation arising in ion etching (Q749785)

Consider two regions in the plane that at time \(t=0\) are given by the inequalities \[ (I)\quad x<0,\quad g(x)<y<h(x),\quad (II)\quad x<0,\quad y<g(x)\quad or\quad x\geq 0,\quad y<h(x). \] Here g is Lipschitz continuous and strictly increasing, h is Lipschitz and strictly decreasing, and \(g(0)=h(0)=0\). The regions are occupied by two different materials. When bombarded with an ion beam, their common upper surface \(y=h(x)\) evolves into a surface \(y=u(x,t)\) at time t governed by the equations \[ u_ t+f_ 1(u_ x)=0\quad (x<s(t)),\quad u_ t+f_ 2(u_ x)=0\quad (x>s(t)), \] where the free boundary \(x=s(t)\) is given by \(u(s(t),t)=g(s(t)).\) The author defines continuous generalized solutions: The differential equations are to hold in the viscosity sense, and an entropy condition has to hold across the free boundary. For Lipschitz continuous positive \(f_ i\), a Lipschitz continuous strictly decreasing solution for the problem involving \(f_ 1\) together with a Lipschitz free boundary is constructed by means of a change of dependent variables, conversion to a conservation law, and polygonal approximation, and then a Lipschitz solution to the problem involving \(f_ 2\) is found via viscous approximation. Uniqueness of these solutions is also shown.