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

# Samia Challal1

1Department of Mathematics, Glendon college - York university, 2275 Bayview Ave. Toronto ON M4N 3M6 Canada.

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.
Keywords: Variational methods, positive solutions, Linear elliptic equations.

Cite this article as: Samia Challal, A Barrier Function for the Regularity of the Free Boundary in $div(a(x) \nabla u )=-(h(x)\gamma)_x$ with $h_x < 0$, Int. J. Math. And Appl., vol. 9, no. 4, 2021, pp. 35-43.

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