Porosity of the free boundary in a class of higher-dimensional elliptic problems

Abstract

We investigate a class of $n$-dimensional ($n\ge 2$) free boundary elliptic problems, which includes the dam problem, the aluminum problem, and the lubrication problem. We establish that the free boundary in this class is a porous set, which implies its Hausdorff dimension being less than $n$, which in turn leads to its Lebesgue measure being zero. As a corollary, we obtain the uniqueness of the so-called reservoirs-connected solution of the dam problem in higher dimension. Our proof of the porosity relies on the Lipschitz continuity of the solution and an appropriately constructed barrier function.