IVAN BLANK 1 Sharp Results for the Regularity and Stability of the Free Boundary in the Obstacle Problem
نویسنده
چکیده
The obstacle problem is a linearized version of the problem of stretching an elastic membrane over an obstacle. One of its central features is that the solution will divide the original domain into a set of points where the membrane lies strictly above the obstacle and satisfies a partial differential equation, and its complement, where the membrane touches the obstacle. The fact that these sets are unknown at the outset is what defines a free boundary problem. The obstacle problem is one of the simplest free boundary problems, and it arises in many contexts including superconductivity, fluid filtration in porous media, optimal control, and financial mathematics, to name a few. Specifically, we search for the minimum superharmonic function u (the membrane) in B1, which lies above a function φ (the obstacle), and which vanishes on ∂B1. By defining the height of the membrane over the obstacle to be w, it is not hard to show that w is the unique minimizer of
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