TITLE: About the Hausdorff dimension of removable singularities of PDE
QUESTION [6 upvotes]: 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.

REPLY [2 votes]: I am not familiar with the current studies on the Hausdorff dimension of the singular set of solutions to PDE, but I know that in [1] a necessary and sufficient condition for the holding of Hartogs phenomenon for a linear system of partial differential operators was stated and proved. The author does not use the methods of geometric measure theory: however, he deals with general compact singularities for the solutions, including the ones with non-zero Lebesgue measure.
[1] Gaetano Fichera (1983), "Sul fenomeno di Hartogs per gli operatori lineari alle derivate parziali [Hartogs phenomenon for certain linear partial differential operators]" (Italian), Rendiconti dell' Istituto Lombardo di Scienze e Lettere. Scienze Matematiche e Applicazioni, Series A., 117: 199–211, MR0848259, Zbl 0603.35013.