• Problems in unbounded domains

Most of the above questions and open problems can naturally be stated for unbounded domains and half-spaces. Several authors have considered the scalar case of such problems and the literature is vast. The reader may consult paper [8], and the references therein for these problems. ...

OpenYear of origin: 2020

Posted online: 2019-12-26 21:41:25Z by Henrik Shahgholian7

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• Analysis of PDEs
• Singular perturbation

In scalar case, the Bernoulli problem discussed earlier can also be obtained as a limit problem of the singular perturbation; literature are vast for this problem, so none mentioned none forgotten. A similar theory, yet to be developed, seems plausible for the system case, which seems to have much more possibilities and variations than its scalar counterpart. ...

OpenYear of origin: 2020

Posted online: 2019-12-26 21:41:25Z by Henrik Shahgholian6

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• Analysis of PDEs
• Serrin type symmetry problem

Contrary to convexity problems, the symmetry methods, such as moving plane technique works very well for free boundary value problems for systems, when there is a symmetry in the given equation, the bulk domain, and boundary values. Here, we present the most simple example, leaving several obvious generalizations towards other problems to the reader. The original Saint Venant problem, with an overdetermination of the boundary gradient condition (here expressed as system) is to show that whenever there is a solution vector $\mathbf{u}$ to the following problem $$\tag{1} \begin{cases} \Delta \mathbf{u} = - \mathbf{k} & \text{in } \Omega , \\ \mathbf{u} = \mathbf{0} & \text{on } \partial\Omega , \\ |\nabla \mathbf{u}| = 1 & \text{on } \partial\Omega , \\ \end{cases}$$ the domain $\Omega$ has to be a ball. For scalar case, James Serrin [5] gave a very nice proof of this based on moving plane technique (of A. D. Alexandrov). ...

OpenYear of origin: 2020

Posted online: 2019-12-26 21:41:25Z by Henrik Shahgholian7

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• Analysis of PDEs
• Obstacle type problem

The system case of the free boundary for general $F(x, \mathbf{p})$, even if $F$ is independent of $x$, and convex in $\mathbf{p}$-variables, is of course a non-trivial problem. Let us consider the particular case as in ...

OpenYear of origin: 2020

Posted online: 2019-12-26 21:41:25Z by Henrik Shahgholian4

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• Analysis of PDEs
• Bernoulli free boundary problem

Convexity results for the Bernoulli type problems, represented in the system of equations $$\begin{cases} \Delta \mathbf{u} = 0 & \text{in } \Omega \setminus \mathbf{D}, \\ |\nabla \mathbf{u}| = g(x) & \text{on } \partial\Omega . \\ \end{cases}$$ seems for now out of reach, at least with the methods we know of. It is also not straightforward what conditions we should impose on the domains $\mathbf{D}$ (besides being convex and maybe $D_i$'s being homothetic). ...

OpenYear of origin: 2020

Posted online: 2019-12-26 21:41:25Z by Henrik Shahgholian7

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• Analysis of PDEs
• Optimal Constants in Two Inequalities (and Additive Combinatorics)

This is a classical problem in additive combinatorics that can be equivalently phrased as a problem in real analysis (this equivalence is due to Javier Cilleruelo, Imre Z. Ruzsa, and Carlos Vinuesa): what random variable $X$ with compact support on $\mathbb{R}$ has the property that the distribution of $X+X$ is as flat as possible? ...

Unconfirmed

Posted online: 2019-06-07 18:06:35Z by Stefan Steinerberger55

• Classical Analysis and ODEs
• Combinatorics
• Number Theory
• Probability
• The Ulam Sequence

Stanislaw Ulam, in a 1964 book, described what is now called the "Ulam sequence". It is defined by setting a_1 = 1, a_2 =2 and then picking a_n in a greedy manner as the smallest integer that can be uniquely written as the sum of two distinct earlier elements of the sequence. This results in the sequence

1,2,3,4,6,8,11, ... ...

Open

Posted online: 2019-06-06 15:45:26Z by Stefan Steinerberger88

• Number Theory
• Combinatorics
1. 1
2. ...
3. 14
4. 15