On the minimization of Dirichlet eigenvalues

Results are obtained for two minimization problems: $$I_k(c)=\inf \{\lambda_k(\Omega): \Omega\ \textup{open, convex in}\ \mathbb{R}^m,\ \mathcal{T}(\Omega)= c \},$$ and $$J_k(c)=\inf\{\lambda_k(\Omega): \Omega\ \textup{quasi-open in}\ \mathbb{R}^m, |\Omega|\le 1, \mathcal {P}(\Omega)\le c \},$$ where $c>0$, $\lambda_k(\Omega)$ is the $k$'th eigenvalue of the Dirichlet Laplacian acting in $L^2(\Omega)$, $|\Omega|$ denotes the Lebesgue measure of $\Omega$, $\mathcal{P}(\Omega)$ denotes the perimeter of $\Omega$, and where $\mathcal{T}$ is in a suitable class set functions. The latter include for example the perimeter of $\Omega$, and the moment of inertia of $\Omega$ with respect to its center of mass.Comment: 15 page

Do public works decrease farmers' soil degradation? Labour income and the use of fertilisers in India's semi-arid tropics

This paper investigates the possibility of using public works to stimulate farmers' fertiliser use in India's SAT. Inadequate replenishment of removed nutrients and organic matter has reduced fertility and increased erosion rates. Fertiliser use, along with other complementary measures, can help reverse this process, which ultimately leads to poverty, hunger, and further environmental degradation. In a high-risk environment like India's SAT, there may be a strong relation between off-farm income and smallholder fertiliser use. Farmers can use the main source of off-farm income, wage income, to manage risk as well as to finance inputs. Consequently, the introduction of public works programmes in areas with high dry-season unemployment may affect fertiliser use. This study confirms the relevance of risk for decisions regarding fertiliser use in two Indian villages. Nevertheless, governments cannot use employment policies to stimulate fertiliser use. Public works even decrease fertiliser use in the survey setting

Optimization problems involving the first Dirichlet eigenvalue and the torsional rigidity

We present some open problems and obtain some partial results for spectral optimization problems involving measure, torsional rigidity and first Dirichlet eigenvalue.Comment: 18 pages, 4 figure

Torsional rigidity for cylinders with a Brownian fracture

We obtain bounds for the expected loss of torsional rigidity of a cylinder $\Omega_L=(-L/2,L/2) \times \Omega\subset \R^3$ of length $L$ due to a Brownian fracture that starts at a random point in $\Omega_L,$ and runs until the first time it exits $\Omega_L$. These bounds are expressed in terms of the geometry of the cross-section $\Omega \subset \R^2$. It is shown that if $\Omega$ is a disc with radius $R$, then in the limit as $L \rightarrow \infty$ the expected loss of torsional rigidity equals $cR^5$ for some $c\in (0,\infty)$. We derive bounds for $c$ in terms of the expected Newtonian capacity of the trace of a Brownian path that starts at the centre of a ball in $\R^3$ with radius $1,$ and runs until the first time it exits this ball.Comment: 18 page

Motivations for local climate adaptation in Dutch municipalities: climate change impacts and the role of local-level government

The local government level is considered to be crucial in preparing society for climate change impact. Yet little is known about why local authorities do or do not take action to adapt their community for climate change impacts. In order to implement effective adaptation policy, the motivations for local climate adaptation need to be examined. This paper explores these motivations in Dutch communities by comparing nine urban and rural cases. To be able to draw general conclusions, cases are selected on „projected risk‟ and „extreme weather event experience‟. Motivations for local climate adaptation appear much more determined by local institutional factors such as a green party aldermen or innovative network membership then projected risk or extreme weather event experience. This could be explained by the empiric data showing diffuse channels of climate change knowledge into the local government level and limited capacity to translate this knowledge into genuine adaptation strategie

Non-Extensive Bose-Einstein Condensation Model

The imperfect Boson gas supplemented with a gentle repulsive interaction is completely solved. In particular it is proved that it has non-extensive Bose-Einstein condensation, i.e., there is condensation without macroscopic occupation of the ground state (k=0) level

Heat content with singular initial temperature and singular specific heat

Let (M,g) be a compact Riemannian manifold without boundary. Let D be a compact subdomain of M with smooth boundary. We examine the heat content asymptotics for the heat flow from D into M where both the initial temperature and the specific heat are permitted to have controlled singularities on the boundary of D. The operator driving the heat process is assumed to be an operator of Laplace typ