Geometric approach to nonvariational singular elliptic equations

In this work we develop a systematic geometric approach to study fully nonlinear elliptic equations with singular absorption terms as well as their related free boundary problems. The magnitude of the singularity is measured by a negative parameter $(\gamma -1)$, for $0 < \gamma < 1$, which reflects on lack of smoothness for an existing solution along the singular interface between its positive and zero phases. We establish existence as well sharp regularity properties of solutions. We further prove that minimal solutions are non-degenerate and obtain fine geometric-measure properties of the free boundary $\mathfrak{F} = \partial \{u > 0 \}$. In particular we show sharp Hausdorff estimates which imply local finiteness of the perimeter of the region $\{u > 0 \}$ and $\mathcal{H}^{n-1}$ a.e. weak differentiability property of $\mathfrak{F}$.Comment: Paper from D. Araujo's Ph.D. thesis, distinguished at the 2013 Carlos Gutierrez prize for best thesis, Archive for Rational Mechanics and Analysis 201

Repeated games for eikonal equations, integral curvature flows and non-linear parabolic integro-differential equations

The main purpose of this paper is to approximate several non-local evolution equations by zero-sum repeated games in the spirit of the previous works of Kohn and the second author (2006 and 2009): general fully non-linear parabolic integro-differential equations on the one hand, and the integral curvature flow of an interface (Imbert, 2008) on the other hand. In order to do so, we start by constructing such a game for eikonal equations whose speed has a non-constant sign. This provides a (discrete) deterministic control interpretation of these evolution equations. In all our games, two players choose positions successively, and their final payoff is determined by their positions and additional parameters of choice. Because of the non-locality of the problems approximated, by contrast with local problems, their choices have to "collect" information far from their current position. For integral curvature flows, players choose hypersurfaces in the whole space and positions on these hypersurfaces. For parabolic integro-differential equations, players choose smooth functions on the whole space

On the Path Integral in Imaginary Lobachevsky Space

The path integral on the single-sheeted hyperboloid, i.e.\ in $D$-dimensional imaginary Lobachevsky space, is evaluated. A potential problem which we call ``Kepler-problem'', and the case of a constant magnetic field are also discussed.Comment: 16 pages, LATEX, DESY 93-14

Dynamics of a lattice Universe

We find a solution to Einstein field equations for a regular toroidal lattice of size L with equal masses M at the centre of each cell; this solution is exact at order M/L. Such a solution is convenient to study the dynamics of an assembly of galaxy-like objects. We find that the solution is expanding (or contracting) in exactly the same way as the solution of a Friedman-Lema\^itre-Robertson-Walker Universe with dust having the same average density as our model. This points towards the absence of backreaction in a Universe filled with an infinite number of objects, and this validates the fluid approximation, as far as dynamics is concerned, and at the level of approximation considered in this work.Comment: 14 pages. No figure. Accepted version for Classical and Quantum Gravit

Convergence of nonlocal threshold dynamics approximations to front propagation

In this note we prove that appropriately scaled threshold dynamics-type algorithms corresponding to the fractional Laplacian of order $\alpha \in (0,2)$ converge to moving fronts. When $\alpha \geqq 1$ the resulting interface moves by weighted mean curvature, while for $\alpha <1$ the normal velocity is nonlocal of ``fractional-type.'' The results easily extend to general nonlocal anisotropic threshold dynamics schemes.Comment: 19 page

The heart of a convex body

We investigate some basic properties of the {\it heart} $\heartsuit(\mathcal{K})$ of a convex set $\mathcal{K}.$ It is a subset of $\mathcal{K},$ whose definition is based on mirror reflections of euclidean space, and is a non-local object. The main motivation of our interest for $\heartsuit(\mathcal{K})$ is that this gives an estimate of the location of the hot spot in a convex heat conductor with boundary temperature grounded at zero. Here, we investigate on the relation between $\heartsuit(\mathcal{K})$ and the mirror symmetries of $\mathcal{K};$ we show that $\heartsuit(\mathcal{K})$ contains many (geometrically and phisically) relevant points of $\mathcal{K};$ we prove a simple geometrical lower estimate for the diameter of $\heartsuit(\mathcal{K});$ we also prove an upper estimate for the area of $\heartsuit(\mathcal{K}),$ when $\mathcal{K}$ is a triangle.Comment: 15 pages, 3 figures. appears as "Geometric Properties for Parabolic and Elliptic PDE's", Springer INdAM Series Volume 2, 2013, pp 49-6

Topology and Homoclinic Trajectories of Discrete Dynamical Systems

We show that nontrivial homoclinic trajectories of a family of discrete, nonautonomous, asymptotically hyperbolic systems parametrized by a circle bifurcate from a stationary solution if the asymptotic stable bundles Es(+{\infty}) and Es(-{\infty}) of the linearization at the stationary branch are twisted in different ways.Comment: 19 pages, canceled the appendix (Properties of the index bundle) in order to avoid any text overlap with arXiv:1005.207

Sex, But Not Spontaneous Cardiovagal Baroreflex Sensitivity, Predicts Tolerance To Simulated Hemorrhage

Some, but not all studies, suggest that spontaneous cardiovagal baroreflex sensitivity (cBRS; i.e., autonomic control of heart rate) is lower in females. However, it is unknown whether cBRS values are associated with hemorrhagic tolerance, which has repeatedly been demonstrated to be lower in females. PURPOSE: Therefore, the purpose of this study was to test the hypothesis that resting spontaneous cBRS is lower in females and that cBRS is associated with differences in hemorrhagic tolerance between the sexes. METHODS: 25 females (age: 26 ± 6 years) and 27 males (age: 30 ± 5 years) completed a progressive lower-body negative pressure (LBNP – a simulation of hemorrhage) protocol starting at -40 mmHg, which was reduced by 10 mmHg every 3 minutes until presyncope. Presyncope was defined by the subject feeling faint and/or nauseous; a rapid decline in blood pressure (BP) \u3c systolic BP of 80 mmHg; and/or a relative bradycardia accompanied by narrowing of pulse pressure. LBNP tolerance was quantified as cumulative stress index (CSI; mmHg*min). Heart rate (HR) and beat-to-beat BP (finometer) were measured continuously. Spontaneous cBRS was analyzed using the sequence method (i.e., ≥ 3 consecutive cardiac cycles of concordant changes in R-R interval and systolic BP, r2 ≥ 0.8 for such sequences). Data were compared between sexes using a Mann-Whitney U test. A least squares multiple linear regression was used to compare the effect of sex and cBRS on CSI. Data are presented as median ± IQR. RESULTS: Resting BP and HR were not different between the sexes (p \u3e 0.36 for both). Resting cBRS was not different between females and males (21 ± 16 vs. 22 ± 11 ms/mmHg, respectively, p = 0.73). As expected, females had a lower tolerance to LBNP (Females: 385 ± 322, Males: 918 ± 418 mmHg*min, p \u3c 0.0001). Multiple linear regression analysis revealed a significant effect of sex (β = 408, p= 0.04), but not resting cBRS (β = 2.4, p = 0.69) or sex*cBRS (i.e., interaction; β = 1.32, p = 0.87), on CSI. When data from both sexes were combined, there was no correlation between resting cBRS and CSI (r = 0.05, p = 0.71). CONCLUSION: Our cohort did not exhibit sex-related differences in resting cBRS. As expected, females had a lower tolerance to simulated hemorrhage. Importantly, we demonstrated that resting cBRS does not explain the observed sex differences in hemorrhagic tolerance

Weak versus D-solutions to linear hyperbolic first order systems with constant coefficients

We establish a consistency result by comparing two independent notions of generalized solutions to a large class of linear hyperbolic first-order PDE systems with constant coefficients, showing that they eventually coincide. The first is the usual notion of weak solutions defined via duality. The second is the new notion of D-solutions which we recently introduced and arose in connection to the vectorial calculus of variations in L∞ and fully nonlinear elliptic systems. This new approach is a duality-free alternative to distributions and is based on the probabilistic representation of limits of difference quotients