Let $n\geq 2$, and let $\mathcal O_n$ denote the set of possible homogeneities in dimension $n$ for global solutions to the Signorini problem.

Is it true that the set $\mathcal O_n\cap [0,L]$ is finite for any $L>0$? ...

Problem 4.

Are there rules for random walks that can result in BS, both in discrete or in its scaling limit?

The well-known Internal DLA is an example of random walks that results in the so-called Divisible sandpile, as done in the work of L. Levine [3], and Levine-Peres [4]. ...

Problem 3. Is there a boundary sandpile type process, leading to that the total mass of the system is being redistributed onto the combinatorial free boundary (possibly with a different background rules) which has a sphere as its scaling limit?

Problem 2.

Assume the answer to Problem 1 is yes. Then, what would be the limiting PDE problem and what can be said about the geometry of the limiting shape? In particular, will it be convex? ...

Problem 1.

Does the boundary sandpile process, for initial distributions concentrated at a point, have a scaling limit?

Let $D$ be a given convex and bounded domain in ${\mathbb R}^n$. Then there is always a solution to the following semilinear PDE $$ \Delta u = \chi_{\{0< u < 1\}}, \quad \hbox{in } {\mathbb R}^n \setminus D, \qquad u= 2 \quad \hbox{on } D , $$ and $u$ continuous in ${\mathbb R}^n$, with compact support.

1) Is the solution unique?

2) Is the set $\Omega:= \{u>0 \}$ convex? ...

Let $\Omega$ be a bounded convex domain in Euclidean space. Consider in $\Omega$ the solution $u$ of the heat equation, which is initially equal to $1$ in $\Omega$ and is equal to $0$ on the boundary of $\Omega$ for all times. We know that $u$ has only one critical point in $\Omega$ at each time. This point is called the hot spot and in general moves in time. Let $0\in\Omega$; it is known that, if $\Omega$ is invariant under the action of an essential group $G$ of orthogonal transformations with center at $0$, the hot spot does not move and stays at $0$. We say that $G$ is essential if, for any $x\ne 0$, there is $g\in G$ such that $g\,x\ne x$. Chamberland and Siegel conjectured in 1997 that the converse is true: if the hot spot does not move with time, then $\Omega$ is invariant under the action of $G$. The conjecture is true if $\Omega$ is a triangle or a quadrangle. It is also true for pentagons and hexagons whose sides have the same distance from $0$. ...