Homogenization of Variational Inequalities for the p-Laplace Operator in Perforated Media Along Manifolds

We address homogenization problems of variational inequalities for the p-Laplace operator in a domain of Rn (n ? 3, p ? [2, n)) periodically perforated by balls of radius O(??) where ? > 1 and ? is the size of the period. The perforations are distributed along a (n ? 1)-dimensional manifold ? , and we impose constraints for solutions and their fluxes (associated with the p-Laplacian) on the boundary of the perforations. These constraints imply that the solution is positive and that the flux is bounded from above by a negative, nonlinear monotonic function of the solution multiplied by a parameter ? ?? , ? ? R and ? is a small parameter that we shall make to go to zero. We analyze different relations between the parameters p, n, ?, ? and ?, and obtain homogenized problems which are completely new in the literature even for the case p = 2.This work has been partially supported by the Spanish grant MINECO:MTM2013-44883-P

Non existence of critical scales in the homogenization of the problem with p-Laplace diffusion and nonlinear reaction in the boundary of periodically distributed particles in n-dimensional domains when p>n

In previous works, the homogenization of the problem with p-Laplace diffusion and nonlinear reaction in the boundary of periodically distributed particles in n-dimensional domains has been studied in the cases where p≤ n. The main trait of the cases p≤ n is the existence of a critical size of the particles, for which the nonlinearity arising of the limit problem does not coincide with the non linear term of the microscopic reaction. The main result of this paper proves that in the case p> n there exists no critical size

Non existence of critical scales in the homogenization of the problem with p-Laplace diffusion and nonlinear reaction in the boundary of periodically distributed particles in n-dimensional domains when p>n

In previous works, the homogenization of the problem with p-Laplace diffusion and nonlinear reaction in the boundary of periodically distributed particles in n-dimensional domains has been studied in the cases where p≤ n. The main trait of the cases p≤ n is the existence of a critical size of the particles, for which the nonlinearity arising of the limit problem does not coincide with the non linear term of the microscopic reaction. The main result of this paper proves that in the case p> n there exists no critical size

On the asymptotic limit of the effectiveness of reaction-diffusion equations in periodically structured media

This paper addresses an investigation of the asymptotic behaviour as ε→0 of the solution to the boundary value problem associated with the p-Laplace operator in an ε-periodically structured domain with a nonlinear Robin-type condition specified on the boundary of the periodic subdomains. This kind of domains include the so called perforated media as well as the case of isolated particles distributed in a periodic way. This second case arises quite often in Chemical Engineering. Here we consider a non-critical size of the particles. The objective of this paper is twofold. First we study the homogenization of solutions in the case of a continuous nonlinear reaction term on the boundary of the periodic structure. Then, we move to studying the homogenization of the effectiveness factor of the reactor, which is of relevance in Chemical Engineering

Instrumental tactile diagnostics in robot-assisted surgery

Rozalia F Solodova,1,2 Vladimir V Galatenko,1,2 Eldar R Nakashidze,3 Igor L Andreytsev,3 Alexey V Galatenko,1 Dmitriy K Senchik,2 Vladimir M Staroverov,1 Vladimir E Podolskii,1,2 Mikhail E Sokolov,1,2 Victor A Sadovnichy1,2 1Faculty of Mechanics and Mathematics, 2Institute of Mathematical Studies of Complex Systems, Lomonosov Moscow State University, 31st Surgery Department, Clinical Hospital 31, Moscow, Russia Background: Robotic surgery has gained wide acceptance due to minimizing trauma in patients. However, the lack of tactile feedback is an essential limiting factor for the further expansion. In robotic surgery, feedback related to touch is currently kinesthetic, and it is mainly aimed at the minimization of force applied to tissues and organs. Design and implementation of diagnostic tactile feedback is still an open problem. We hypothesized that a sufficient tactile feedback in robot-assisted surgery can be provided by utilization of Medical Tactile Endosurgical Complex (MTEC), which is a novel specialized tool that is already commercially available in the Russian Federation. MTEC allows registration of tactile images by a mechanoreceptor, real-time visualization of these images, and reproduction of images via a tactile display. Materials and methods: Nine elective surgeries were performed with da Vinci™ robotic system. An assistant performed tactile examination through an additional port under the guidance of a surgeon during revision of tissues. The operating surgeon sensed registered tactile data using a tactile display, and the assistant inspected the visualization of tactile data. First, surgeries where lesion boundaries were visually detectable were performed. The goal was to promote cooperation between the surgeon and the assistant and to train them in perception of the tactile feedback. Then, instrumental tactile diagnostics was utilized in case of visually undetectable boundaries. Results: In robot-assisted surgeries where lesion boundaries were not visually detectable, instrumental tactile diagnostics performed using MTEC provided valid identification and localization of lesions. The results of instrumental tactile diagnostics were concordant with the results of intraoperative ultrasound examination. However, in certain cases, for example, thoracoscopy, ultrasound examination is inapplicable, while MTEC-based tactile diagnostics can be efficiently utilized. Conclusion: The study proved that MTEC can be efficiently used in robot-assisted surgery allowing correct localization of visually undetectable lesions and visually undetectable boundaries of pathological changes of tissues. Keywords: tactile feedback, instrumental palpation, Medical Tactile Endosurgical Complex, tactile lesion localizatio