Some homogenization and corrector results for nonlinear monotone operators

This paper deals with the limit behaviour of the solutions of quasi-linear equations of the form \ \ds -\limfunc{div}\left(a\left(x, x/{\varepsilon _h},Du_h\right)\right)=f_h on $\Omega$ with Dirichlet boundary conditions. The sequence $(\varepsilon _h)$ tends to $0$ and the map $a(x,y,\xi )$ is periodic in $y$, monotone in $\xi$ and satisfies suitable continuity conditions. It is proved that $u_h\rightarrow u$ weakly in $H_0^{1,2}(\Omega )$, where $u$ is the solution of a homogenized problem \ -\limfunc{div}(b(x,Du))=f on $\Omega$. We also prove some corrector results, i.e. we find $(P_h)$ such that $Du_h-P_h(Du)\rightarrow 0$ in $L^2(\Omega ,R^n)$

Correctors for some nonlinear monotone operators

In this paper we study homogenization of quasi-linear partial differential equations of the form -\mbox{div}\left( a\left( x,x/\varepsilon _h,Du_h\right) \right) =f_h on $\Omega$ with Dirichlet boundary conditions. Here the sequence $\left( \varepsilon _h\right)$ tends to $0$ as $h\rightarrow \infty$ and the map $a\left( x,y,\xi \right)$ is periodic in $y,$ monotone in $\xi$ and satisfies suitable continuity conditions. We prove that $u_h\rightarrow u$ weakly in $W_0^{1,p}\left( \Omega \right)$ as $h\rightarrow \infty ,$ where $u$ is the solution of a homogenized problem of the form -\mbox{div}\left( b\left( x,Du\right) \right) =f on $\Omega .$ We also derive an explicit expression for the homogenized operator $b$ and prove some corrector results, i.e. we find $\left( P_h\right)$ such that $Du_h-P_h\left( Du\right) \rightarrow 0$ in $L^p\left( \Omega, \mathbf{R}^n\right)$

An intermediate value theorem in ordered Banach spaces

We consider a monotone increasing operator in an ordered Banach space having $u_-$ and $u_+$ as a strong super- and subsolution, respectively. In contrast with the well studied case $u_+ < u_-$, we suppose that $u_- < u_+$. Under the assumption that the order cone is normal and minihedral, we prove the existence of a fixed point located in the ordered interval $[u_-,u_+].

Numerical Bifurcation Analysis of Conformal Formulations of the Einstein Constraints

The Einstein constraint equations have been the subject of study for more than fifty years. The introduction of the conformal method in the 1970's as a parameterization of initial data for the Einstein equations led to increased interest in the development of a complete solution theory for the constraints, with the theory for constant mean curvature (CMC) spatial slices and closed manifolds completely developed by 1995. The first general non-CMC existence result was establish by Holst et al. in 2008, with extensions to rough data by Holst et al. in 2009, and to vacuum spacetimes by Maxwell in 2009. The non-CMC theory remains mostly open; moreover, recent work of Maxwell on specific symmetry models sheds light on fundamental non-uniqueness problems with the conformal method as a parameterization in non-CMC settings. In parallel with these mathematical developments, computational physicists have uncovered surprising behavior in numerical solutions to the extended conformal thin sandwich formulation of the Einstein constraints. In particular, numerical evidence suggests the existence of multiple solutions with a quadratic fold, and a recent analysis of a simplified model supports this conclusion. In this article, we examine this apparent bifurcation phenomena in a methodical way, using modern techniques in bifurcation theory and in numerical homotopy methods. We first review the evidence for the presence of bifurcation in the Hamiltonian constraint in the time-symmetric case. We give a brief introduction to the mathematical framework for analyzing bifurcation phenomena, and then develop the main ideas behind the construction of numerical homotopy, or path-following, methods in the analysis of bifurcation phenomena. We then apply the continuation software package AUTO to this problem, and verify the presence of the fold with homotopy-based numerical methods.Comment: 13 pages, 4 figures. Final revision for publication, added material on physical implication

On approximate solutions of semilinear evolution equations

A general framework is presented to discuss the approximate solutions of an evolution equation in a Banach space, with a linear part generating a semigroup and a sufficiently smooth nonlinear part. A theorem is presented, allowing to infer from an approximate solution the existence of an exact solution. According to this theorem, the interval of existence of the exact solution and the distance of the latter from the approximate solution can be evaluated solving a one-dimensional "control" integral equation, where the unknown gives a bound on the previous distance as a function of time. For example, the control equation can be applied to the approximation methods based on the reduction of the evolution equation to finite-dimensional manifolds: among them, the Galerkin method is discussed in detail. To illustrate this framework, the nonlinear heat equation is considered. In this case the control equation is used to evaluate the error of the Galerkin approximation; depending on the initial datum, this approach either grants global existence of the solution or gives fairly accurate bounds on the blow up time.Comment: 33 pages, 10 figures. To appear in Rev. Math. Phys. (Shortened version; the proof of Prop. 3.4. has been simplified

Multi-site breathers in Klein-Gordon lattices: stability, resonances, and bifurcations

We prove the most general theorem about spectral stability of multi-site breathers in the discrete Klein-Gordon equation with a small coupling constant. In the anti-continuum limit, multi-site breathers represent excited oscillations at different sites of the lattice separated by a number of "holes" (sites at rest). The theorem describes how the stability or instability of a multi-site breather depends on the phase difference and distance between the excited oscillators. Previously, only multi-site breathers with adjacent excited sites were considered within the first-order perturbation theory. We show that the stability of multi-site breathers with one-site holes change for large-amplitude oscillations in soft nonlinear potentials. We also discover and study a symmetry-breaking (pitchfork) bifurcation of one-site and multi-site breathers in soft quartic potentials near the points of 1:3 resonance.Comment: 34 pages, 12 figure

Detailed Spectroscopic and Photometric Analysis of DQ White Dwarfs

We present an analysis of spectroscopic and photometric data for cool DQ white dwarfs based on improved model atmosphere calculations. In particular, we revise the atmospheric parameters of the trigonometric parallax sample of Bergeron et al.(2001), and discuss the astrophysical implications on the temperature scale and mean mass, as well as the chemical evolution of these stars. We also analyze 40 new DQ stars discovered in the first data release of the Sloan Digital Sky Survey.Comment: 6 pages,3 figures, 14th European Workshop on White Dwarfs, ASP Conference Series, in pres

Ellipsometric measurements by use of photon pairs generated by spontaneous parametric down-conversion

We present a novel interferometric technique for performing ellipsometric measurements. This technique relies on the use of a non-classical optical source, namely, polarization-entangled twin photons generated by spontaneous parametric down-conversion from a nonlinear crystal, in conjunction with a coincidence-detection scheme. Ellipsometric measurements acquired with this scheme are absolute; i.e., they do not require source and detector calibration.Comment: 10 pages, accepted for publication in Optics Letter

Adaptive plasticity in the healthy reading network investigated through combined neurostimulation and neuroimaging

The reading network in the human brain comprises several regions, including the left inferior frontal cortex (IFC), ventral occipito-temporal cortex (vOTC) and dorsal temporo-parietal cortex (TPC). The left TPC is crucial for phonological decoding, i.e., for learning and retaining sound-letter mappings. Here, we tested the causal contribution of this area for reading with repetitive transcranial magnetic stimulation (rTMS) and explored the response of the reading network using functional magnetic resonance imaging (fMRI). 28 healthy adult readers overtly read simple and complex words and pseudowords during fMRI after effective or sham TMS over the left TPC. Behaviorally, effective stimulation slowed pseudoword reading. A multivariate pattern analysis showed a shift in activity patterns in the left IFC for pseudoword reading after effective relative to sham TMS. Furthermore, active TMS led to increased effective connectivity from the left vOTC to the left TPC, specifically for pseudoword processing. The observed changes in task-related activity and connectivity suggest compensatory reorganization in the reading network following TMS-induced disruption of the left TPC. Our findings provide first evidence for a causal role of the left TPC for overt pseudoword reading and emphasize the relevance of functional interactions in the healthy reading network for successful pseudoword processing

Optimal Constraint Projection for Hyperbolic Evolution Systems

Techniques are developed for projecting the solutions of symmetric hyperbolic evolution systems onto the constraint submanifold (the constraint-satisfying subset of the dynamical field space). These optimal projections map a field configuration to the ``nearest'' configuration in the constraint submanifold, where distances between configurations are measured with the natural metric on the space of dynamical fields. The construction and use of these projections is illustrated for a new representation of the scalar field equation that exhibits both bulk and boundary generated constraint violations. Numerical simulations on a black-hole background show that bulk constraint violations cannot be controlled by constraint-preserving boundary conditions alone, but are effectively controlled by constraint projection. Simulations also show that constraint violations entering through boundaries cannot be controlled by constraint projection alone, but are controlled by constraint-preserving boundary conditions. Numerical solutions to the pathological scalar field system are shown to converge to solutions of a standard representation of the scalar field equation when constraint projection and constraint-preserving boundary conditions are used together.Comment: final version with minor changes; 16 pages, 14 figure