The Lavrentiev gap phenomenon for harmonic maps into spheres holds on a dense set of zero degree boundary data

We prove that for each positive integer $N$ the set of smooth, zero degree maps $\psi\colon\mathbb{S}^2\to \mathbb{S}^2$ which have the following three properties: (1) there is a unique minimizing harmonic map $u\colon \mathbb{B}^3\to \mathbb{S}^2$ which agrees with $\psi$ on the boundary of the unit ball; (2) this map $u$ has at least $N$ singular points in $\mathbb{B}^3$; (3) the Lavrentiev gap phenomenon holds for $\psi$, i.e., the infimum of the Dirichlet energies $E(w)$ of all smooth extensions $w\colon \mathbb{B}^3\to\mathbb{S}^2$ of $\psi$ is strictly larger than the Dirichlet energy $\int_{\mathbb{B}^3} |\nabla u|^2$ of the (irregular) minimizer $u$, is dense in the set of all smooth zero degree maps $\phi\colon \mathbb{S}^2\to\mathbb{S}^2$ endowed with the $W^{1,p}$-topology, where $1\le p < 2$. This result is sharp: it fails in the $W^{1,2}$ topology on the set of all smooth boundary data.Comment: 14 pages, 3 figures; minor typos etc. correcte

Potential implications of labour market opening in Germany and Austria on emigration from Poland

The aim of this study is to present the characteristic of present-day migrants and the potential for possible migration after the opening of the labour markets in Austria and Germany. The econometric analysis shows that differences in unemployment rates between sending and receiving countries were the most important for changes in the emigration from Poland in the period 2002-2009. Mostly due to persistence of these differences the intruduction of the open-door policy by two last EU countries in the spring of 2011 can intensify the further emigration flows from Poland. Data concerning the structure of the present emigration in Germany indicate that emigrants from Poland are mainly persons with vocational and secondary education, working primarily in the sections of services (e.g. health care and social assistance, accommodation and catering). There is also a relatively high percentage of persons employed in agriculture and the construction sector. These sectors will probably continue to be the most frequent workplace for emigrants, where the internal supply of work seems insufficient to meet the needs of this part of the German economy. The current limitations push better educated emigrants from Poland to work mainly as specialists in the sectors of economy preferred by Germany or as self-employed persons. The caps applied by German authorities concerning the number of Polish employees on secondment under the framework of the cross-border provision of services remain underused. Moreover, German data (which do not cover persons holding dual nationality) indicate that for the time being emigration from Poland is, to a large extent, circulatory by nature. Examples of other EU countries which already opened their labour markets indicate that the removal of barriers to access may increase emigration in the first year, but the differences and changes in unemployment rates among countries are a much more important factor for migratory flows, particularly at a later stage. The opening of labour markets in Germany and Austria may contribute to a change in the nature of the present short-term to a more permanent migration from Poland. The first part of the study presents information on the existing work limitations for Poles in Germany and the characteristics of the present emigrants from Poland to Germany and Austria. The second part discusses determinants of emigration in 2002-2009, putting a special emphasis on those countries which already managed to open their labour markets for the ‘new’ EU members. The third part delivers the estimates of possible emigration changes from Poland to Germany and Austria that are going to happen after 1 May 2011.labour migration, open-door policy, Poland, Germany, determinants of migration

The multi-state projection of Poland’s population by educational attainment for the years 2003–2030

The first population projection by education level of the Poland’s population until the year 2030 is presented. The projecton is based on the multi-state projection model LIPRO developed by E. van Imhoff and N. Keilman (1991) and the LIPRO 4.0 software. The initial population as well as the model parameters were calculated for the year 2002 on the basis of the National Population Census data and the Labour Force Survey data. The projection was prepared under an assumption on the constant parameters up to the year 2030.&nbsp;The differences in mortality and fertility by education attainment were also taken into consideration. The projection results for the years 2002–2006 were compared with the observed values to check the assumptions formulated. The projection results show that the increasing enrolment at the tertiary and secondary levels of education among the baby boomers born in the mid-1970s and the early 1980s will dramatically change the population composition by education in the next two decades. The percentage of people with tertiary education in the working age population (15-59/64) will increase from 12% in the year 2002 to 35% in the year 2030. Moreover, the analysis was performed to demonstrate changes in the life expectancy at birth (e0) and the total fertility rate (TFR) which can be attributed to the changing population composition by education and existing differences in mortality and fertility by education, to be kept in the future. Until 2030 the life expectancy increases by about 2.2 years for men and 2.0 years for women only because of the shifts in the education composition. On the other hand, this factor was found as not contributing to the fertility changes in the next years

Characterizing W2,pW^{2,p}~submanifolds by pp-integrability of global curvatures

We give sufficient and necessary geometric conditions, guaranteeing that an immersed compact closed manifold $\Sigma^m\subset \R^n$ of class $C^1$ and of arbitrary dimension and codimension (or, more generally, an Ahlfors-regular compact set $\Sigma$ satisfying a mild general condition relating the size of holes in $\Sigma$ to the flatness of $\Sigma$ measured in terms of beta numbers) is in fact an embedded manifold of class $C^{1,\tau}\cap W^{2,p}$, where $p>m$ and $\tau=1-m/p$. The results are based on a careful analysis of Morrey estimates for integral curvature--like energies, with integrands expressed geometrically, in terms of functions that are designed to measure either (a) the shape of simplices with vertices on $\Sigma$ or (b) the size of spheres tangent to $\Sigma$ at one point and passing through another point of $\Sigma$. Appropriately defined \emph{maximal functions} of such integrands turn out to be of class $L^p(\Sigma)$ for $p>m$ if and only if the local graph representations of $\Sigma$ have second order derivatives in $L^p$ and $\Sigma$ is embedded. There are two ingredients behind this result. One of them is an equivalent definition of Sobolev spaces, widely used nowadays in analysis on metric spaces. The second one is a careful analysis of local Reifenberg flatness (and of the decay of functions measuring that flatness) for sets with finite curvature energies. In addition, for the geometric curvature energy involving tangent spheres we provide a nontrivial lower bound that is attained if and only if the admissible set $\Sigma$ is a round sphere.Comment: 44 pages, 2 figures; several minor correction

Tangent-point self-avoidance energies for curves

We study a two-point self-avoidance energy $E_q$ which is defined for all rectifiable curves in $R^n$ as the double integral along the curve of $1/r^q$. Here $r$ stands for the radius of the (smallest) circle that is tangent to the curve at one point and passes through another point on the curve, with obvious natural modifications of this definition in the exceptional, non-generic cases. It turns out that finiteness of $E_q(\gamma)$ for $q\ge 2$ guarantees that $\gamma$ has no self-intersections or triple junctions and therefore must be homeomorphic to the unit circle or to a closed interval. For $q>2$ the energy $E_q$ evaluated on curves in $R^3$ turns out to be a knot energy separating different knot types by infinite energy barriers and bounding the number of knot types below a given energy value. We also establish an explicit upper bound on the Hausdorff-distance of two curves in $R^3$ with finite $E_q$-energy that guarantees that these curves are ambient isotopic. This bound depends only on $q$ and the energy values of the curves. Moreover, for all $q$ that are larger than the critical exponent $2$, the arclength parametrization of $\gamma$ is of class $C^{1,1-2/q}$, with H\"{o}lder norm of the unit tangent depending only on $q$, the length of $\gamma$, and the local energy. The exponent $1-2/q$ is optimal.Comment: 23 pages, 1 figur

On some knot energies involving Menger curvature

We investigate knot-theoretic properties of geometrically defined curvature energies such as integral Menger curvature. Elementary radii-functions, such as the circumradius of three points, generate a family of knot energies guaranteeing self-avoidance and a varying degree of higher regularity of finite energy curves. All of these energies turn out to be charge, minimizable in given isotopy classes, tight and strong. Almost all distinguish between knots and unknots, and some of them can be shown to be uniquely minimized by round circles. Bounds on the stick number and the average crossing number, some non-trivial global lower bounds, and unique minimization by circles upon compaction complete the picture.Comment: 31 pages, 4 figures; version 2 with minor changes and modification

Compactness and isotopy finiteness for submanifolds with uniformly bounded geometric curvature energies

In this paper, we establish compactness for various geometric curvature energies including integral Menger curvature, and tangent-point repulsive potentials, defined a priori on the class of compact, embedded $m$-dimensional Lipschitz submanifolds in ${\mathbb{R}}^n$. It turns out that due to a smoothing effect any sequence of submanifolds with uniformly bounded energy contains a subsequence converging in $C^1$ to a limit submanifold. This result has two applications. The first one is an isotopy finiteness theorem: there are only finitely many isotopy types of such submanifolds below a given energy value, and we provide explicit bounds on the number of isotopy types in terms of the respective energy. The second one is the lower semicontinuity - with respect to Hausdorff-convergence of submanifolds - of all geometric curvature energies under consideration, which can be used to minimise each of these energies within prescribed isotopy classes.Comment: 44 pages, 5 figure

Integral Menger curvature for surfaces

We develop the concept of integral Menger curvature for a large class of nonsmooth surfaces. We prove uniform Ahlfors regularity and a $C^{1,\lambda}$-a-priori bound for surfaces for which this functional is finite. In fact, it turns out that there is an explicit length scale $R>0$ which depends only on an upper bound $E$ for the integral Menger curvature $M_p(\Sigma)$ and the integrability exponent $p$, and \emph{not} on the surface $\Sigma$ itself; below that scale, each surface with energy smaller than $E$ looks like a nearly flat disc with the amount of bending controlled by the (local) $M_p$-energy. Moreover, integral Menger curvature can be defined a priori for surfaces with self-intersections or branch points; we prove that a posteriori all such singularities are excluded for surfaces with finite integral Menger curvature. By means of slicing and iterative arguments we bootstrap the H\"{o}lder exponent $\lambda$ up to the optimal one, $\lambda=1-(8/p)$, thus establishing a new geometric `Morrey-Sobolev' imbedding theorem. As two of the various possible variational applications we prove the existence of surfaces in given isotopy classes minimizing integral Menger curvature with a uniform bound on area, and of area minimizing surfaces subjected to a uniform bound on integral Menger curvature.Comment: 64 pages, 7 figures. Submitted. Version 2: extended comments on the relation to Lerman's and Whitehouse's work on Menger curvature