Let $A\subset\mathbb{R}^{n}$ be a closed set and $\phi(x)=e^{-||x||^{2}}$, $x\in\mathbb{R}^{n}$. We shall say that $A$ is an "equilibrium set", if

$x\in\partial A$ $\Leftrightarrow\int_{A}\phi(x-u)du=\int_{\mathbb{R} ^{n}\backslash A}\phi(x-u)du$.

For example, the infinite chessboard defines an equilibrium set in $\mathbb{R}^{2}$. I obtained such "equilibrium sets" as stationary points (equilibrium states) of some dynamical system. So, I am asking have someone seen such equilibrium property, or something similar to, somewhere in the literature.

This definition rises some 'not real' geometrical questions:

1) Are there any interesting (curious) examples of equilibrium sets in the plane?

2) Which are the equilibrium sets in $\mathbb{R}^{1}$?

3) And finally, is it possible to describe these sets in purely geometrical terms?

Another interesting question could be the description of equilibrium sets in $\mathbb{S}^{2}$ for an appropriate bump function insread of $\phi(x)$.

Note also that the equilibrium condition may be simplified by requiring that for some fixed $\alpha>0$

$x\in\partial A$ $\Leftrightarrow m(A\cap B(x,\alpha))=m((\mathbb{R} ^{n}\backslash A)\cap B(x,\alpha))$,

where $B(x,\alpha)$ is the ball with center $x$ and radius $\alpha$ and $m$ is the Lebesgue measure. Then the class of 'equilibrium sets' seems much wider and depends on $\alpha$. For example, the infinite chessboards with different sides define equilibrium sets for small $\alpha$.


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