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 $0$. ...

Let $\Omega$ be a bounded simply connected and sufficiently smooth plane domain. Let $u$ be the first (positive) Dirichlet eigenfunction in $\Omega$ for the Laplace operator. It is known that the critical points of $u$ are isolated and the number of its maximum points is one more than that of the saddle points. If $\Omega$ is convex, $u$ is log-convex and hence $u$ has only one critical (maximum) point. Problem: find an estimate of the number $N$ of maximum points of $u$ in terms of geometric information about $\Omega$. For instance, a conjecture is that $N$ does not exceed the number of (connected components of local) maximum points of the distance function from the boundary of $\Omega$. ...

Consider the homogenization problem of the elliptic system \begin{equation} - \nabla \cdot A \left( \frac{x}{\varepsilon} \right) \nabla u (x) = 0, \ \ x \in D, \tag{1} \end{equation} in a domain $D\subset \mathbb{R}^d$, ($d\geq 2$), and with oscillating Dirichlet boundary data \begin{equation} u(x) = g \left(x , \frac{x}{\varepsilon} \right), \ \ x \in \partial D. \tag{2} \end{equation} ...

Problem. Is there a subset $S\subset \mathbb{R}^2$ such that $\dim_H S > 1$ and a quasiconformal map $f : \mathbb{R}^n\to\mathbb{R}^n, n\geq 3$, such that $f(\{x\}\times (0,1))$ contains no nontrivial rectifiable arc for any $x\in S$? \\ ...

In the complex plane the Schwarz function of an arc $\Gamma$ is a holomorphic function $S(z) $ near $\Gamma$ and such that $S(z)=\bar{z}$ on the arc $\Gamma$. The Schwarz function of a straight line is a linear function $az+b$. The presuppose is that the Schwarz function of non-linear arcs develops singularities. And indeed, a beautiful theorem of Davis confirms this hypotheses [3]. Namely, suppose the Schwarz function $S(z)$ of an arc $\Gamma$ is an entire function in the complex plane, then $\Gamma$ is a straight line. ...

The complexity of algorithmic problems of group theoretic problems has been of great importance after many new applications in cryptography. Polycyclic Groups are the most successful one among others. In the reference there are several problems proposed for cryptography. However the complexity of some of them is still an open question. ...