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. ...

The exponential transform of a domain $\Omega$ in $\mathbb{R}^n$ ($n\geq 2$) is defined by $$ E(x)= \exp [-\frac{2}{|S^{n-1}|}\int_\Omega \frac{dy}{|x-y|^{n}}], \quad x\in\mathbb{R}^n\setminus\overline{\Omega}, $$ where $|S^{n-1}|$ denotes the area of the unit sphere. The exponential transform is a positive function in the exterior of $\Omega$, with zero boundary values at all reasonably regular points of $\partial\Omega$. The problem is to study in more detail the relationship between the regularity of $\partial\Omega$ and the boundary behavior of $E(x)$. More specifically we conjecture that if $\partial\Omega$ is smooth real analytic then $E(x)$ has a real analytic continuation across $\partial\Omega$. ...

Let $\mu$ be a positive measure with compact support in $\mathbb{R}^n$, $n\geq 3$, and let $K$ be the convex hull of the support of $\mu$. Let $u\geq 0$ be the smallest lower semicontinuous function satisfying $$ u\geq 0, \quad \mu+\Delta u\leq 1, $$ and let $\Omega=\{u>0\}$ be the non-coincidence set for this obstacle problem. The problem is to show that $\Omega$ can be written as a union of balls with centers in $K$. This amounts to saying that $\partial\Omega$ is subject to a natural, and sharp, upper bound for the curvature. ...