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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.

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    $\begingroup$ 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. $\endgroup$
    – Liviu Nicolaescu
    Commented Jul 1, 2020 at 10:23

3 Answers 3

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This paper has a very tidy discussion of this issue. Should be a better known paper IMHO.

Harvey, Reese; Polking, John; Removable singularities of solutions of linear partial differential equations. Acta Math. 125 1970 39–56.

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To go straight for the minimal surface equation you need to see the paper by Leon Simon, "On a theorem of de Giorgi and Stampacchia".

If you want minimal surface system then see Harvey and Lawson: "Extending minimal varieties".

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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.

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