Open problems in Mathematics stemming from research papers

Interior regularity for nonlocal minimal surfaces

Up to which dimension are $s$-minimal surfaces smooth for any $s$? Equivalently, up to which dimension are $s$-minimal cones flat for any $s$?

By [Savin-Valdinoci, 2013], it is known that these properties hold true in dimension 2. ...
- E. Valdinoci
Open
Posted online: 2018-07-02 18:09:47Z by Enrico Valdinoci69

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- Analysis of PDEs
- Differential Geometry
On "small" homogeneities

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.

Let $n_0$ be the largest integer such that the set $\mathcal O_{n_0}$ is disjoint from the interval $(2,3)$. Find the best possible lower bound on $n_0$. ...
- Alessio Figalli
Open
Posted online: 2018-06-30 13:20:13Z by Alessio Figalli284

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- Analysis of PDEs
Discreteness of possible homogeneities

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 $L>0$? ...
- Alessio Figalli
Open
Posted online: 2018-06-30 13:20:13Z by Alessio Figalli1847

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- Analysis of PDEs
Random walks governed by BS

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]. ...
- Hayk Aleksanyan,
- Henrik Shahgholian
OpenYear of origin: 2017
Posted online: 2018-06-28 19:02:43Z by Henrik Shahgholian115

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- Analysis of PDEs
- Combinatorics
- Probability
Other type of processes

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?

...
- Hayk Aleksanyan,
- Henrik Shahgholian
OpenYear of origin: 2017
Posted online: 2018-06-28 19:02:43Z by Henrik Shahgholian88

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- Analysis of PDEs
- Combinatorics
- Probability
The limiting PDE of BS

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? ...
- Hayk Aleksanyan,
- Henrik Shahgholian
OpenYear of origin: 2017
Posted online: 2018-06-28 19:02:43Z by Henrik Shahgholian95

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- Analysis of PDEs
- Combinatorics
- Probability
Scaling limits of BS

Problem 1.

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

...
- Hayk Aleksanyan,
- Henrik Shahgholian
OpenYear of origin: 2017
Posted online: 2018-06-28 19:02:43Z by Henrik Shahgholian95

View the group
- Analysis of PDEs
- Combinatorics
- Probability

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