There are some interesting phenomenons about removable singularities (or extension problems). In the theory of functions of several complex variables, we know the classical Hartogs theorem:

Let f be a holomorphic function on a set G\K, where G is an open subset of $\mathbb{C}^n$ (n ≥ 2) and K is a compact subset of G. If the complement G\K is connected, then f can be extended to a unique holomorphic function on G. As a corollary, roughly speaking, holomorphic function can be extended in codimension 2 case.

Naturally, I wonder what's the largest Hausdorff dimension of removable singularities to elliptic equations perhaps under some assumptions, like boundedness. Are there some results?

For example, we know a bounded harmonic function in a ball without the center can be extended harmonically to the whole ball. But what if the region is a ball without a closed set of Hausdorff dimension 1, 2 and so on? Is there an optimal dimension? In fact, I am more concerned about the problem for minimal surface equations and Monge-Ampere equations.