that ellipsoidal shells do not exert gravitational force inside the cavity of the shell was known to newton, laplace, and ivory.

in early 30’s p. dive proved the inverse of this theorem. in this talk, i shall recall the (partially geometric) proof of this fact and then extend this result to unbounded domains.

since ellipsoids, and any limit of a sequence of ellipsoids, are the so-called coincidence sets for the obstacle problem, there is a close link between the ellipsoidal potential theory and global solutions to the obstacle problem.

in this talk we present a complete classification (in terms of limit domains of ellipsoids) for global solutions to the obstacle problem in dimensions greater than five. the interesting ramification of this result is a new interpretation of the structure of the regular free boundary close to singular points.

this is a joint work with s. eberle, and g.s. weiss.

for further details and references see: https://www.scilag.net/problem/p-200218.1