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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\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.

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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 Haussdorff 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]", Rendiconti dell' Istituto Lombardo di Scienze e Lettere. Scienze Matemàtiche e Applicazioni, Series A. (in Italian), 117: 199–211, MR 0848259, Zbl 0603.35013.

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