Surface Energies Arising in Microscopic Modeling of Martensitic Transformations

In this paper we construct and analyze a two-well Hamiltonian on a 2D atomic lattice. The two wells of the Hamiltonian are prescribed by two rank-one connected martensitic twins, respectively. By constraining the deformed configurations to special 1D atomic chains with position-dependent elongation vectors for the vertical direction, we show that the structure of ground states under appropriate boundary conditions is close to the macroscopically expected twinned configurations with additional boundary layers localized near the twinning interfaces. In addition, we proceed to a continuum limit, show asymptotic piecewise rigidity of minimizing sequences and rigorously derive the corresponding limiting form of the surface energy

Rigidity estimates for isometric and conformal maps from Sn−1\mathbb{S}^{n-1} to Rn\mathbb{R}^n

We investigate both linear and nonlinear stability aspects of rigid motions (resp. M\"obius transformations) of $\mathbb{S}^{n-1}$ among Sobolev maps from $\mathbb{S}^{n-1}$ into $\mathbb{R}^n$. Unlike similar in flavour results for maps defined on domains of $\mathbb{R}^n$ and mapping into $\mathbb{R}^n$, not only an isometric (resp. conformal) deficit is necessary in this more flexible setting, but also a deficit measuring the distortion of $\mathbb{S}^{n-1}$ under the maps in consideration. The latter is defined as an associated isoperimetric type of deficit. We mostly focus on the case $n=3$, where we also explain why the estimates are optimal in their corresponding settings. In the isometric case the estimate holds true also when $n=2$ and generalizes in dimensions $n\geq 4$ as well, if one requires apriori boundedness in a certain higher Sobolev norm. We also obtain linear stability estimates for both cases in all dimensions. These can be regarded as Korn-type inequalities for the combination of the quadratic form associated with the isometric (resp. conformal) deficit on $\mathbb{S}^{n-1}$ and the isoperimetric one.Comment: 53 page

Alzheimer's disease-like alterations in peripheral cells from presenilin-1 transgenic mice

Many cases of early-onset inherited Alzheimer's disease (AD) are caused by mutations in the presenilin-1 (PS1) gene. Expression of PS1 mutations in cell culture systems and in primary neurons from transgenic mice increases their vulnerability to cell death. Interestingly, enhanced vulnerability to cell death has also been demonstrated for peripheral lymphocytes from AD patients. We now report that lymphocytes from PS1 mutant transgenic mice show a similar hypersensitivity to cell death as do peripheral cells from AD patients and several cell culture systems expressing PS1 mutations. The cell death-enhancing action of mutant PS1 was associated with increased production of reactive oxygen species and altered calcium regulation, but not with changes of mitochondrial cytochrome c. Our study further emphasizes the pathogenic role of mutant PS1 and may provide the fundamental basis for new efforts to close the gap between studies using neuronal cell lines transfected with mutant PS1, neurons from transgenic animals, and peripheral cells from AD patients. Copyright 2001 Academic Press