W 2,1 regularity for solutions of the Monge–Ampère equation

In this paper we prove that a strictly convex Alexandrov solution u of the Monge–Ampère equation, with right-hand side bounded away from zero and infinity, is $W^{2,1}_{\mathrm{loc}}$ . This is obtained by showing higher integrability a priori estimates for D ² u, namely D ² u∈Llog ^k L for any k∈?.     

How to recognize convexity of a set from its marginals

We investigate the regularity of the marginals onto hyperplanes for sets of finite perimeter. We prove, in particular, that if a set of finite perimeter has log-concave marginals onto a.e. hyperplane then the set is convex. Our proof relies on measuring the perimeter of a set through a Hilbertian fractional Sobolev norm, a fact that we believe has its own interest.     

A critical Kirchhoff‐type problem driven by a p (·)‐fractional Laplace operator with variable s (·) ‐order

The paper deals with the following Kirchhoff‐type problem $$\left\{\begin{array}{lll}& M\left({?}_{{?}^{2N}}\frac{1}{p(x,y)}\frac{|v(x)?v(y){|}^{p(x,y)}}{|x?y{|}^{N+p(x,y)s(x,y)}}dxdy\right){(?\mathrm{\Delta })}_{p(·)}^{s(·)}v(x)=\mu g(x,v)+|v{|}^{r(x)?2}v& \text{in}\mathrm{\Omega },\\ & v=0& \text{in}{?}^N\backslash \mathrm{\Omega },\end{array}\right.$$ where M models a Kirchhoff coefficient, ${(?\mathrm{\Delta })}_{p(·)}^{s(·)}$ is a variable s(·) ‐order p(·) ‐fractional Laplace operator, with $s(·):{?}^{2N}\to (0,1)$ and $p(·):{?}^{2N}\to (1,\infty )$. Here, $\mathrm{\Omega }?{?}^N$ is a bounded smooth domain with N > p(x, y)s(x, y) for any $(x,y)\in \overline{\mathrm{\Omega }}\times \overline{\mathrm{\Omega }}$, μ is a positive parameter, g is a continuous and subcritical function, while variable exponent r(x) could be close to the critical exponent $p_s^{?}(x)=N\overline{p}(x)/(N?\overline{s}(x)\overline{p}(x))$, given with $\overline{p}(x)=p(x,x)$ and $\overline{s}(x)=s(x,x)$ for $x\in \overline{\mathrm{\Omega }}$. We prove the existence and asymptotic behavior of at least one non‐trivial solution. For this, we exploit a suitable tricky step analysis of the critical mountain pass level, combined with a Brézis and Lieb‐type lemma for fractional Sobolev spaces with variable order and variable exponent.     

Optimization of membrane area and catalyst distribution in a permeative-stage membrane reactor for methane steam reforming

Methane steam reforming to produce high purity hydrogen is conveniently carried out in membrane reactors. The interest on this process has led to several studies about the influence of the key operating variables on the system performances. In the present paper we focus on the optimization of the catalyst and membrane area axial distribution by computer simulation. To this purpose, a numerical model of a permeative-stage membrane reactor (PSMR), a system composed of traditional reactive and membrane permeative stages laid out in alternated series, has been developed. This is based on 1D mass, momentum and energy balances including thermal effects on all the system properties. First, an equi-sized nine-stage PSMR is analyzed at various temperatures and membrane thicknesses, in comparison with a conventional membrane reactor, showing that improved performances, especially in terms of methane conversion, can be obtained. Then, considering the stage lengths as design variables, the optimal system performances are determined by maximizing two objective functions, namely the methane conversion and the hydrogen recovery factor. Different optimal length distributions are found utilizing the two criteria, corresponding to privileged kinetic or permeative aspects. In comparison to the conventional MR, the PSMR leads to a higher conversion and lower recovery factor when the conversion is maximized, whilst the opposite situation occurs when maximizing the recovery factor. Furthermore, the role of the heat exchange area between the furnace and the reactor is studied, considering a PSMR with adiabatic permeative stages. From this case, it is possible to see that the greater advantage of this apparatus (having the same membrane area, catalyst amount and heat-exchange area as the MR) consists in a much higher recovery factor (+21% ca.), which at the end demonstrates to be the most convenient objective function by means of the introduction of an overall performance index, identified as the hydrogen recovery yield. Therefore, the analysis proposed in the present paper shows that the reactive/permeative stage distribution has to be considered an important reactor design parameter, which can be opportunely modified to improve the performances of the process.     

Effects of multiple-stage membrane process designs on the achievable performance of automatic control

There is limited information from literature on the dynamic operability of membrane processes with multiple stages or loops. Such information is useful for assessing the performance achievable by an automatic controller proposed for a process design before the actual controller is implemented. Based on dynamic modeling of an industrial whey ultrafiltration process with an increasing number of stages up to a maximum of 12, the dynamic responses of the flowrate and concentration of the retentate were obtained. Features of the dynamic responses were used to determine the performance, in terms of quality and speed, that can be achieved by automatic controllers. In particular, limitations on the performance are indicated by features of dynamic responses such as effective time delay and inverse responses. Changes in effective time delay and inverse responses with the number of stages in the whey ultrafiltration process demonstrate a trade-off between process performance and control performance. This trade-off should be considered during process and controller design to maximize the economic return from the production of whey protein concentrates.     

Sample covariance shrinkage for high dimensional dependent data

For high dimensional data sets the sample covariance matrix is usually unbiased but noisy if the sample is not large enough. Shrinking the sample covariance towards a constrained, low dimensional estimator can be used to mitigate the sample variability. By doing so, we introduce bias, but reduce variance. In this paper, we give details on feasible optimal shrinkage allowing for time series dependent observations.     

Stationary layered solutions in {\mathbb{R}}^2 for a class of non autonomous Allen-Cahn equations

We consider a class of non autonomous Allen-Cahn equations where is a multiple-well potential and is a periodic, positive, non-constant function. We look for solutions to (0.1) having uniform limits as corresponding to minima of W. We show, via variational methods, that under a nondegeneracy condition on the set of heteroclinic solutions of the associated ordinary differential equation the equation (0.1) has solutions which depends on both the variables x andy. In contrast, when a is constant such nondegeneracy condition is not satisfied and all such solutions are known to depend only on x. Received April 16, 1999 / Accepted October 1, 1999 / Published online June 28, 2000     

Coupled nonlinear Schrödinger systems with potentials

Coupled nonlinear Schrödinger systems describe some physical phenomena such as the propagation in birefringent optical fibers, Kerr-like photorefractive media in optics and Bose-Einstein condensates. In this paper, we study the existence of concentrating solutions of a singularly perturbed coupled nonlinear Schrödinger system, in presence of potentials. We show how the location of the concentration points depends strictly on the potentials.     

Finite element and finite volume‐element simulation of pseudo‐ECGs and cardiac alternans

In this paper, we are interested in the spatio‐temporal dynamics of the transmembrane potential in paced isotropic and anisotropic cardiac tissues. In particular, we observe a specific precursor of cardiac arrhythmias that is the presence of alternans in the action potential duration. The underlying mathematical model consists of a reaction–diffusion system describing the propagation of the electric potential and the nonlinear interaction with ionic gating variables. Either conforming piecewise continuous finite elements or a finite volume‐element scheme are employed for the spatial discretization of all fields, whereas operator splitting strategies of first and second order are used for the time integration. We also describe an efficient mechanism to compute pseudo‐ECG signals, and we analyze restitution curves and alternans patterns for physiological and pathological cardiac rhythms. Copyright © 2014 John Wiley & Sons, Ltd.