# About the Hausdorff dimension of removable singularities of PDE

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 $$G \setminus K$$, where $$G$$ is an open subset of $$\mathbb{C}^n$$ ($$n \ge 2$$) and $$K$$ is a compact subset of $$G$$. If the complement $$G \setminus K$$ 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 the 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–Ampère equations.

• I do not have references in mind right now, but the size of removable singularities is typically measured by Sobolev capacities. if these are small you can remove them. In turn, Hausdorff dimension provides an upper bound for the capacity. Jul 1, 2020 at 10:23