A Barrier Function for the Regularity of the Free Boundary in $div(a(x) \nabla u )=-(h(x)\gamma)_x$ with $h_x < 0$

Abstract

A barrier function \( \displaystyle{ w=\frac{\lambda}{2}[(y-f(x))^+]^2}\) is compared to the solution \(u\) near a free boundary point. The properties $div (a(x) \nabla u)\geqslant -(h)_x \chi ([u>0])$ and \(\nabla w=0\) on \( [y=f(x)]\) avoided the comparison of the gradients of \(u \) and \(v\) as in the case \(h_x \geqslant 0\). A regularity of the free boundary is established.