Compact contact sets of sub-quadratic solutions to the thin obstacle problem

We study global solutions to the thin obstacle problem with at most quadratic growth at infinity. We show that every ellipsoid can be realized as the contact set of such a solution. On the other hand, if such a solution has a compact contact set, we show that it must be an ellipsoid.
Comment: A uniqueness result is added to Theorem 1.1, thanks to the suggestion of one referee. Funding information is added