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.

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.

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 Cn (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.In collary,Roughly speaking, holomorphic function can be extended in codimenson 2 case. Naturally,I wonder what's the largest Hausdorff dimension of removable singularities to elliptic equations perhaps under some assumptions  ,like boundedness.Is there some results？ For example,we know a bounded harmonic function in a ball without the center can be extended harmonicly to the whole ball. But what if the region is a ball without a closed set of Hausdorff diminsion 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.

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.