Non-transversal intersection of free and fixed boundary for fully nonlinear elliptic operators in two dimensions

In the study of classical obstacle problems, it is well known that in many configurations the free boundary intersects the fixed boundary tangentially. The arguments involved in producing results of this type rely on the linear structure of the operator. In this paper we employ a different approach and prove tangential touch of free and fixed boundary in two dimensions for fully nonlinear elliptic operators. Along the way, several $n$-dimensional results of independent interest are obtained such as BMO-estimates, $C^{1,1}$ regularity up to the fixed boundary, and a description of the behavior of blow-up solutions.Comment: 17 page

Synthesis of 1 alpha,25-Dihydroxyvitamin D Analogues Featuring a S-2-symmetric CD-ring Core

Three analogues of 1 alpha,25-dihydroxyvitamin D-3 (calcitriol), featuring a transfused decalin C, D-core with local S-2-symmetry, and possessing identical side-chain and seco-B,A-ring structures, have been synthesized starting from readily available (4aR, 8aS)octahydronaphthalene-1,5-dione(7). The very short sequences involve the simultaneous introduction of the side-chain and seco-B, A-ring fragments via Suzuki and Sonogashira coupling reactions. The analogues are devoid of relevant biological activity

Viscosity Solutions of Balanced Quasi-Monotone Fully Nonlinear Weakly Coupled Systems

We introduce so called balanced quasi-monotone systems. These are systems $F(x,r,p,X)=(F_1(x,r,p,X),\ldots,F_m(x,r,p,X))$, where $x$ belongs to a domain $\Omega$, $r=u(x)\in\mathbb{R}^m$, $p=Du(x)$ and $X=D^2u(x)$, that can be arranged into two categories that are mutually competitive but internally cooperative. More precisely, for all $i\neq j$ in the set $\{1,2,\ldots,m\}$, $F_j$ is monotone non-decreasing (non-increasing) in $r_i$ if and only if $F_i$ is monotone non-decreasing (non-increasing) in $r_j$ and $F_j$ is a monotone function in $r_i$. We prove the existence and uniqueness of viscosity solutions to systems of this type. For uniqueness we need to require that $F_j$ is monotone increasing in $r_j$, at an at least linear rate. This should be compared to the quasi-monotone systems studied by Ishii and Koike, where they assume that $F(x,r,p,X)\ge F(x,s,p,X)$ if $r\le s$.Comment: 10 page

Raising teacher supply: An assessment of three options for increasing wages

A shortage of teachers in primary and secondary schools in the Netherlands is expected in the near future as a large part of the current workforce will retire. Recently, the Dutch government has decided to increase wages of teachers. This paper qualitatively assesses the impact of three options for increasing teaching wages on teacher supply in the medium term: increasing wages at the start of the career, increasing wages at mid career or increasing wages at the end of the career.

Skill gaps in the EU: role for education and training policies

Skill gaps are widely seen as a problem that lowers aggregate productivity growth. A question for the European Commission is whether and how governments should take action with education and training policies to reduce skill gaps and make Europe the best performing region in the world. European citizens can best decide for themselves on the type of education. Distribution of information on occupation prospects is effective to influence their choice of education. Moreover, it is important that the education system is sufficiently flexible to absorb unexpected shocks in skill needs of employees. Policies stimulating education targeted at government-assigned sectors are risky policies. Intensification of general education at the cost of specific education, and intensification of training of employees find little support.

Successful knowledge policies

Knowledge policy is widely considered to be an important subject. The Dutch government conducts intensive policy on the foundations of the knowledge economy: education, research and innovation. In the literature and in policy circles, proposals for additional knowledge policies or reforms of existing policies are currently being discussed. Examples are the Knowledge Investment Agenda of the Innovation Platform, new innovation policies of the Ministry of Economic Affairs, an advice of the Council of Economic Advisors about research, education and entrepreneurship and an advice of the Education Council. Little is known, however, about the effects of all these policies. The question whether knowledge policy works remains difficult to answer. Despite much research effort in the past decades, the manner in which the creation and application of knowledge comes about and the way policy can influence that process are still not well understood. It is difficult, therefore, to form a good judgement about the optimal size and form of knowledge policy. This article discusses several policy options in the fields of education, research, and innovation that are likely to have beneficial, neutral, or negative effects on overall welfare in the Netherlands. For some options, the effects are unknown. Beneficial education policies are, for instance, policies aimed at increasing teachers' quality and early childhood education programs. Additional R&D tax credits for new firms have favourable effects on innovation. A further increase in the research incentives to universities is expected to raise scientific output.

Regularity of solutions in semilinear elliptic theory

We study the semilinear Poisson equation \begin{equation} \label{pro} \Delta u = f(x, u) \hskip .2 in \text{in} \hskip .2 in B_1. \end{equation} Our main results provide conditions on $f$ which ensure that weak solutions of this equation belong to $C^{1,1}(B_{1/2})$. In some configurations, the conditions are sharp