Lavrentiev Phenomenon in Microstructure Theory

A variational problem arising as a model in martensitic phase transformation including surface energy is studied. It explains the complex, multi-dimensional pattern of twin branching which is often observed in a martensitic phase near the austenite interface. We prove that a Lavrentiev phenomenon can occur if the domain is a rectangle. We show that this phenomenon disappears under arbitrarily small shears of the domain. We also prove that other perturbations of the problem lead to an extinction of the Lavrentiev phenomenon

The Complex X-ray Spectrum of the Sefyert 1.5 Source NGC 6860

The X-ray spectrum of the Seyfert 1.5 source NGC 6860 is among the most complex of the sources detected in the Swift Burst Alert Telescope all-sky survey. A short XMM-Newton follow-up observation of the source revealed a flat spectrum both above and below 2 keV. To uncover the complexity of the source, in this paper we analyze both a 40 ks Suzaku and a 100 ks XMM-Newton observation of NGC 6860. While the spectral state of the source changed between the newer observations presented here and the earlier short XMM-Newton spectrum - showing a higher flux and steeper power law component - the spectrum of NGC 6860 is still complex with clearly detected warm absorption signatures. We find that a two component warm ionized absorber is present in the soft spectrum, with column densities of about 10^20 and 10^21 cm$^-2, ionization parameters of xi = 180 and 45 ergs cm s^-1, and outflow velocities for each component in the range of 0-300 km s^-1. Additionally, in the hard spectrum we find a broad (approx 11000 km s^-1) Fe K-alpha emission line, redshifted by approx 2800 km s^-1.Comment: 35 pages, 9 figures, Accepted to Ap

Uncovering Local Absorbed Active Galactic Nuclei with Swift and Suzaku

Detection of absorbed active galactic nuclei and their properties remains an elusive and important problem in understanding the evolution and activation of black holes. With the very hard X-ray survey conducted by Swift's Burst Alert Telescope - the first all-sky survey in 30 years - we are beginning to uncover the characteristics of obscured AGN. The synergy between Suzaku and Swift has been crucial in pinning down the X-ray properties of newly detected heavily obscured but bright hard X-ray sources. We review the X-ray and optical spectroscopic properties of obscured AGN in the local Universe, as detected in the Swift survey. We discuss the relative distribution of absorbed/unabsorbed sources, including "hidden" and Compton thick AGN populations. Among the results from the survey, we find that absorbed AGN are less luminous than unabsorbed sources. Optical spectra reveal that sources with emission line ratios indicative of LINERs/H II galaxies/composites are the least luminous objects in the sample, while optical absorbed and unabsorbed Seyferts have the same luminosity distributions. Thus, the least luminous sources are likely accreting in a different mode than the Seyferts.Comment: 8 pages, 5 figures, To appear in the conference proceedings for "Exploring the X-ray Universe: Suzaku and Beyond", the July 2011 Suzaku Science Conferenc

Symmetric and Asymmetric Multiple Clusters In a Reaction-Diffusion System

We consider the Gierer-Meinhardt system in the interval (-1,1) with Neumann boundary conditions for small diffusion constant of the activator and finite diffusion constant of the inhibitor. A cluster is a combination of several spikes concentrating at the same point. In this paper, we rigorously show the existence of symmetric and asymmetric multiple clusters. This result is new for systems and seems not to occur for single equations. We reduce the problem to the computation of two matrices which depend on the coefficient of the inhibitor as well as the number of different clusters and the number of spikes within each cluster

On the stationary Cahn-Hilliard equation: Interior spike solutions

We study solutions of the stationary Cahn-Hilliard equation in a bounded smooth domain which have a spike in the interior. We show that a large class of interior points (the "nondegenerate peak" points) have the following property: there exist such solutions whose spike lies close to a given nondegenerate peak point. Our construction uses among others the methods of viscosity solution, weak convergence of measures and Liapunov-Schmidt reduction

Existence, classification and stability analysis of multiple-peaked solutions for the gierer-meinhardt system in R^1

We consider the Gierer-Meinhardt system in R^1. where the exponents (p, q, r, s) satisfy 1< \frac{ qr}{(s+1)( p-1)} < \infty, 1 <p < +\infty, and where \ep<<1, 0<D<\infty, \tau\geq 0, D and \tau are constants which are independent of \ep. We give a rigorous and unified approach to show that the existence and stability of N-peaked steady-states can be reduced to computing two matrices in terms of the coefficients D, N, p, q, r, s. Moreover, it is shown that N-peaked steady-states are generated by exactly two types of peaks, provided their mutual distance is bounded away from zero

Symmetry of Nodal Solutions for Singularly Perturbed Elliptic Problems on a Ball

In [40], it was shown that the following singularly perturbed Dirichlet problem \ep^2 \Delta u - u+ |u|^{p-1} u=0, \ \mbox{in} \ \Om,\] \[ u=0 \ \mbox{on} \ \partial \Om has a nodal solution u_\ep which has the least energy among all nodal solutions. Moreover, it is shown that u_\ep has exactly one local maximum point P_1^\ep with a positive value and one local minimum point P_2^\ep with a negative value and, as \ep \to 0, \varphi (P_1^\ep, P_2^\ep) \to \max_{ (P_1, P_2) \in \Om \times \Om } \varphi (P_1, P_2), where \varphi (P_1, P_2)= \min (\frac{|P_1-P_2}{2}, d(P_1, \partial \Om), d(P_2, \partial \Om)). The following question naturally arises: where is the {\bf nodal surface} \{ u_\ep (x)=0 \}? In this paper, we give an answer in the case of the unit ball \Om=B_1 (0). In particular, we show that for \epsilon sufficiently small, P_1^\ep, P_2^\ep and the origin must lie on a line. Without loss of generality, we may assume that this line is the x_1-axis. Then u_\ep must be even in x_j, j=2, ..., N, and odd in x_1. As a consequence, we show that \{ u_\ep (x)=0 \} = \{ x \in B_1 (0) | x_1=0 \}. Our proof is divided into two steps: first, by using the method of moving planes, we show that P_1^\ep, P_2^\ep and the origin must lie on the x_1-axis and u_\ep must be even in x_j, j=2, ..., N. Then, using the Liapunov-Schmidt reduction method, we prove the uniqueness of u_\ep (which implies the odd symmetry of u_\ep in x_1). Similar results are also proved for the problem with Neumann boundary conditions

Existence and stability of multiple spot solutions for the gray-scott model in R^2

We study the Gray-Scott model in a bounded two dimensional domain and establish the existence and stability of {\bf symmetric} and {\bf asymmetric} multiple spotty patterns. The Green's function and its derivatives together with two nonlocal eigenvalue problems both play a major role in the analysis. For symmetric spots, we establish a threshold behavior for stability: If a certain inequality for the parameters holds then we get stability, otherwise we get instability of multiple spot solutions. For asymmetric spots, we show that they can be stable within a narrow parameter range

Young measures in a nonlocal phase transition problem

A nonlocal variational problem modelling phase transitions is studied in the framework of Young measures. The existence of global minimisers among functions with internal layers on an infinite tube is proved by combining a weak convergence result for Young measures and the principle of concentration-compactness. The regularity of such global minimisers is discussed, and the nonlocal variational problem is also considered on asymptotic tubes

Multi-interior-spike solutions for the Cahn-Hilliard equation with arbitrarily many peaks

We study the Cahn-Hilliard equation in a bounded smooth domain without any symmetry assumptions. We prove that for any fixed positive integer K there exist interior $K$--spike solutions whose peaks have maximal possible distance from the boundary and from one another. This implies that for any bounded and smooth domain there exist interior K-peak solutions. The central ingredient of our analysis is the novel derivation and exploitation of a reduction of the energy to finite dimensions (Lemma 5.5) with variables which are closely related to the location of the peaks. We do not assume nondegeneracy of the points of maximal distance to the boundary but can do with a global condition instead which in many cases is weaker