Asymptotic behavior of critical points of an energy involving a loop-well potential

We describe the asymptotic behavior of critical points of $\int_{\Omega} [(1/2)|\nabla u|^2+W(u)/\varepsilon^2]$ when $\varepsilon\to 0$. Here, $W$ is a Ginzburg-Landau type potential, vanishing on a simple closed curve $\Gamma$. Unlike the case of the standard Ginzburg-Landau potential $W(u)=(1-|u|^2)^2/4$, studied by Bethuel, Brezis and H\'elein, we do not assume any symmetry on $W$ or $\Gamma$. In order to overcome the difficulties due to the lack of symmetry, we develop new tools which might be of independent interest

Asymptotics of eigenstates of elliptic problems with mixed boundary data on domains tending to infinity

We analyze the asymptotic behavior of eigenvalues and eigenfunctions of an elliptic operator with mixed boundary conditions on cylindrical domains when the length of the cylinder goes to infinity. We identify the correct limiting problem and show in particular, that in general the limiting behavior is very different from the one for the Dirichlet boundary conditions.Comment: Asymptotic Analysis, 201

Albert Renold Memorial Lecture: Molecular Background of Nutritionally Induced Insulin Resistance Leading to Type 2 Diabetes – From Animal Models to Humans

Albert Renold strived to gain insight into the abnormalities of human diabetes by defining the pathophysiology of the disease peculiar to a given animal. He investigated the Israeli desert-derived spiny mice (Acomys cahirinus), which became obese on fat-rich seed diet. After a few months hyperplasia and hypertrophy of β-cells occurred leading to a sudden rupture, insulin loss and ketosis. Spiny mice were low insulin responders, which is probably a characteristic of certain desert animals, protecting against insulin oversecretion when placed on an abundant diet. We have compared the response to overstimulation of several mutant diabetic species and nutritionally induced nonmutant animals when placed on affluent diet. Some endowed with resilient β-cells sustain long-lasting oversecretion, compensating for the insulin resistance, without lapsing into overt diabetes. Some with labile beta cells exhibit apoptosis and lose their capacity of coping with insulin resistance after a relatively short period. The wide spectrum of response to insulin resistance among different diabetes prone species seems to represent the varying response of human beta cells among the populations. In search for the molecular background of insulin resistance resulting from overnutrition we have studied the Israeli desert gerbil Psammomys obesus (sand rat), which progresses through hyperinsulinemia, followed by hyperglycemia and irreversible beta cell loss. Insulin resistance was found to be the outcome of reduced activation of muscle insulin receptor tyrosine kinase by insulin, in association with diminished GLUT4 protein and DNA content and overexpression of PKC isoenzymes, notably of PKCε. This overexpression and translocation to the membrane was discernible even prior to hyperinsulinemia and may reflect the propensity to diabetes in nondiabetic species and represent a marker for preventive action. By promoting the phosphorylation of serine/threonine residues on certain proteins of the insulin signaling pathway, PKCε exerts a negative feedback on insulin action. PKCε was also found to attenuate the activity of PKB and to promote the degradation of insulin receptor, as determined by co-incubation in HEK 293 cells. PKCε overexpression was related to the rise in muscle diacylglycerol and lipid content, which are prevalent on lascivious nutrition especially if fat-rich. Thus, Psammomys illustrates the probable antecedents of the development of worldwide diabetes epidemic in human populations emerging from food scarcity to nutritional affluence, inappriopriate to their metabolic capacity