Abstract

In this thesis we investigate global solutions of the classical obstacle problem. We give a partial result towards a conjecture by H. Shahgholian ('92) saying that the coincidence sets of global solutions of the obstacle problem are in the closure of ellipsoids, i.e. ellipsoids, paraboloids, cylinders with one of the two as basis, or half-spaces. We give a short proof of the known result that bounded coincidence sets of global solutions of the obstacle problem that have non-empty interior are ellipsoids. Our main and new result is that in dimensions greater or equal to 6 coincidence sets of global solutions, that are not constant in any direction and have a blow-down that is independent of exactly one direction, are paraboloids if they have non-empty interior. The proof rests on a careful analysis of the asymptotics of solutions at infinity, the Newton-potential expansion of the solution and a comparison argument that only requires two solutions to be compared on a sufficiently large portion of huge spheres.

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