Porosity of the Free Boundary in a Class of Higher-Dimensional Elliptic Problems

Abstract

We investigate a class of n-dimensional 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. Our proof relies on the comparison of the solution with an appropriately constructed barrier function.