Estimates for $p$-Laplace type equation in a limiting case

2014 ◽  
Vol 25 (4) ◽  
pp. 445-448 ◽  
Author(s):  
Fernando Farroni ◽  
Luigi Greco ◽  
Gioconda Moscariello
Keyword(s):  
Type Equation ◽  
Limiting Case ◽  
Laplace Type

scholarly journals GENERAL QUASILINEAR PROBLEMS INVOLVING \(p(x)\)-LAPLACIAN WITH ROBIN BOUNDARY CONDITION

2020 ◽  
Vol 6 (1) ◽  
pp. 30
Author(s):  
Hassan Belaouidel ◽  
Anass Ourraoui ◽  
Najib Tsouli
Keyword(s):  
Mountain Pass Theorem ◽  
Type Equation ◽  
Robin Boundary Condition ◽  
Mountain Pass ◽  
Technical Approach ◽  
Symmetric Mountain Pass Theorem ◽  
Existence And Multiplicity ◽  
Quasilinear Problems ◽  
Robin Boundary ◽  
Laplace Type

This paper deals with the existence and multiplicity of solutions for a class of quasilinear problems involving \(p(x)\)-Laplace type equation, namely $$\left\{\begin{array}{lll}-\mathrm{div}\, (a(| \nabla u|^{p(x)})| \nabla u|^{p(x)-2} \nabla u)= \lambda f(x,u)&\text{in}&\Omega,\\n\cdot a(| \nabla u|^{p(x)})| \nabla u|^{p(x)-2} \nabla u +b(x)|u|^{p(x)-2}u=g(x,u) &\text{on}&\partial\Omega.\end{array}\right.$$ Our technical approach is based on variational methods, especially, the mountain pass theorem and the symmetric mountain pass theorem.

Stability for p-Laplace type equation in a borderline case

2015 ◽  
Vol 116 ◽  
pp. 100-111 ◽  
Author(s):  
Fernando Farroni ◽  
Luigi Greco ◽  
Gioconda Moscariello
Keyword(s):  
Type Equation ◽  
Borderline Case ◽  
Laplace Type

Reconstruction of Space Distribution of Characteristics in Compound Structures by General Ray Method

2014 ◽  
Vol 1616 ◽  
Author(s):  
Alexandre Grebennikov
Keyword(s):  
Spline Approximation ◽  
Type Equation ◽  
Plane Domain ◽  
Space Structure ◽  
Space Distribution ◽  
Electrical Tomography ◽  
Ray Method ◽  
The Family ◽  
Plane Scheme ◽  
Laplace Type

ABSTRACTRecognition of material structures, particularly, identification of electrical properties of materials by Electrical Tomography is very important in different applied problems. In a plane case Electrical Tomography can be mathematically described as a coefficient inverse problem for the Laplace type equation, written in the divergent form. The General Ray (GR) Principle, proposed by the author, reduces the Laplace type equation to the family of ordinary differential equations with respect the traces of the potential function and the permittivity function on the lines, which intersect the plane domain. General Ray Principle was realized as General Ray method and fast algorithm for the plane domains. In presented investigation we apply the plane scheme of GR-method for some space domains to identify distribution of structure characteristics inside it. For this we consider the space domain as assemblage of plane slices. Reconstructing desired distribution in each plane slice we obtain then the space internal distribution of electrical characteristics by 3D spline approximation. We consider here specific variant of the measurement scheme for the 3D Electrical Tomography (ET), based on the variant, proposed by the author for the plane domain. Proposed approach gives, in principle, the possibility to use a large number of electrodes, obtain more values of the input data and reconstruct the desired space structure more perfectly. Computer simulation of this 3D scheme is realized as MATLAB software and justified by numerical experiments on simulated examples.

scholarly journals Derivation of a generalized Fowler–Nordheim equation for nanoscopic field-emitters

2015 ◽  
Vol 471 (2174) ◽  
pp. 20140811 ◽  
Author(s):  
A. Kyritsakis ◽  
J. P. Xanthakis
Keyword(s):  
Field Emission ◽  
First Principles ◽  
Enhancement Factor ◽  
Type Equation ◽  
Experimental Confirmation ◽  
Field Emitters ◽  
Standard Field ◽  
Field Dependent ◽  
Good Conductor ◽  
Limiting Case

In this paper, we derive analytically from first principles a generalized Fowler–Nordheim (FN) type equation that takes into account the curvature of a nanoscopic emitter and is generally applicable to any emitter shape provided that the emitter is a good conductor and no field-dependent changes in emitter geometry occur. The traditional FN equation is shown to be a limiting case of our equation in the limit of emitters of large radii of curvature R. Experimental confirmation of the validity of our equation is given by the data of three different groups. Upon applying our equation to experimental FN plots complying with the above limitations, one may deduce (i) R and (ii) standard field emission parameters—e.g. enhancement factor—with better accuracy than by using the FN equation.

Freeze fracture imaging and computer simulation of liposome structure: Membrane geometry and defects

1983 ◽  
Vol 41 ◽  
pp. 646-647
Author(s):  
Joseph A. Zasadzinski
Keyword(s):  
Liquid Crystalline ◽  
Freeze Fracture ◽  
Continuum Theory ◽  
Closed Surfaces ◽  
Straight Line ◽  
Low Weight ◽  
Concentric Spheres ◽  
Membrane Geometry ◽  
Dupin Cyclides ◽  
Limiting Case

At low weight fractions, many surfactant and biological amphiphiles form dispersions of lamellar liquid crystalline liposomes in water. Amphiphile molecules tend to align themselves in parallel bilayers which are free to bend. Bilayers must form closed surfaces to separate hydrophobic and hydrophilic domains completely. Continuum theory of liquid crystals requires that the constant spacing of bilayer surfaces be maintained except at singularities of no more than line extent. Maxwell demonstrated that only two types of closed surfaces can satisfy this constraint: concentric spheres and Dupin cyclides. Dupin cyclides (Figure 1) are parallel closed surfaces which have a conjugate ellipse (r1) and hyperbola (r2) as singularities in the bilayer spacing. Any straight line drawn from a point on the ellipse to a point on the hyperbola is normal to every surface it intersects (broken lines in Figure 1). A simple example, and limiting case, is a family of concentric tori (Figure 1b).To distinguish between the allowable arrangements, freeze fracture TEM micrographs of representative biological (L-α phosphotidylcholine: L-α PC) and surfactant (sodium heptylnonyl benzenesulfonate: SHBS)liposomes are compared to mathematically derived sections of Dupin cyclides and concentric spheres.

Suppression of γ’ Precipitation in a Laser-Melted Nickel-Base Alloy

1977 ◽  
Vol 35 ◽  
pp. 270-271
Author(s):  
J. M. Walsh ◽  
J. C. Whittles ◽  
B. H. Kear ◽  
E. M. Breinan
Keyword(s):  
Cooling Rate ◽  
Base Alloy ◽  
Material Surface ◽  
Intimate Contact ◽  
Absorbed Power ◽  
Perfect Contact ◽  
Nickel Base ◽  
Rapidly Solidified ◽  
Limiting Case ◽  
The Heat Transfer Coefficient

Conventionally cast γ’ precipitation hardened nickel-base superalloys possess well-defined dendritic structures and normally exhibit pronounced segregation. Splat quenched, or rapidly solidified alloys, on the other hand, show little or no evidence for phase decomposition and markedly reduced segregation. In what follows, it is shown that comparable results have been obtained in superalloys processed by the LASERGLAZE™ method.In laser glazing, a sharply focused laser beam is traversed across the material surface at a rate that induces surface localized melting, while avoiding significant surface vaporization. Under these conditions, computations of the average cooling rate can be made with confidence, since intimate contact between the melt and the self-substrate ensures that the heat transfer coefficient is reproducibly constant (h=∞ for perfect contact) in contrast to the variable h characteristic of splat quenching. Results of such computations for pure nickel are presented in Fig. 1, which shows that there is a maximum cooling rate for a given absorbed power density, corresponding to the limiting case in which melt depth approaches zero.

Cartesian Fingerprints in Beckett's Imagination Dead Imagine

2012 ◽  
Vol 21 (2) ◽  
pp. 223-243
Author(s):  
Irit Degani-Raz
Keyword(s):  
The Self ◽  
Levels Of Abstraction ◽  
The Media ◽  
The Mind ◽  
Limiting Case

The idea that Beckett investigates in his works the limits of the media he uses has been widely discussed. In this article I examine the fiction Imagination Dead Imagine as a limiting case in Beckett's exploration of limits at large and the limits of the media he uses in particular. Imagination Dead Imagine is shown to be the self-reflexive act of an artist who imaginatively explores the limits of that ultimate medium – the artist's imagination itself. My central aim is to show that various types of structural homologies (at several levels of abstraction) can be discerned between this poetic exploration of the limits of imagination and Cartesian thought. The homologies indicated here transcend what might be termed as ‘Cartesian typical topics’ (such as the mind-body dualism, the cogito, rationalism versus empiricism, etc.). The most important homologies that are indicated here are those existing between the role of imagination in Descartes' thought - an issue that until only a few decades ago was quite neglected, even by Cartesian scholars - and Beckett's perception of imagination. I suggest the use of these homologies as a tool for tracing possible sources of inspiration for Beckett's Imagination Dead Imagine.

The build of nonlinear quantum mechancs and variations of feature of microscopic particles as well as their experimental affirmation

2014 ◽  
Vol 5 (3) ◽  
pp. 871-981 ◽  
Author(s):  
Pang Xiao Feng
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Lagrange Equation ◽  
Minimum Principle ◽  
Superposition Principle ◽  
Type Equation ◽  
Experimental Results ◽  
Traditional Concept ◽  
Nonlinear Quantum ◽  
Nonlinear Quantum Mechanics

We establish the nonlinear quantum mechanics due to difficulties and problems of original quantum mechanics, in which microscopic particles have only a wave feature, not corpuscle feature, which are completely not consistent with experimental results and traditional concept of particle. In this theory the microscopic particles are no longer a wave, but localized and have a wave-corpuscle duality, which are represented by the following facts, the solutions of dynamic equation describing the particles have a wave-corpuscle duality, namely it consists of a mass center with constant size and carrier wave, is localized and stable and has a determinant mass, momentum and energy, which obey also generally conservation laws of motion, their motions meet both the Hamilton equation, Euler-Lagrange equation and Newton-type equation, their collision satisfies also the classical rule of collision of macroscopic particles, the uncertainty of their position and momentum is denoted by the minimum principle of uncertainty. Meanwhile the microscopic particles in this theory can both propagate in solitary wave with certain frequency and amplitude and generate reflection and transmission at the interfaces, thus they have also a wave feature, which but are different from linear and KdV solitary wave’s. Therefore the nonlinear quantum mechanics changes thoroughly the natures of microscopic particles due to the nonlinear interactions. In this investigation we gave systematically and completely the distinctions and variations between linear and nonlinear quantum mechanics, including the significances and representations of wave function and mechanical quantities, superposition principle of wave function, property of microscopic particle, eigenvalue problem, uncertainty relation and the methods solving the dynamic equations, from which we found nonlinear quantum mechanics is fully new and different from linear quantum mechanics. Finally, we verify further the correctness of properties of microscopic particles described by nonlinear quantum mechanics using the experimental results of light soliton in fiber and water soliton, which are described by same nonlinear Schrödinger equation. Thus we affirm that nonlinear quantum mechanics is correct and useful, it can be used to study the real properties of microscopic particles in physical systems.

Mathematical modelling of salt purification by recrystallization. Countercurrent arrangement

1986 ◽  
Vol 51 (11) ◽  
pp. 2481-2488
Author(s):  
Benitto Mayrhofer ◽  
Jana Mayrhoferová ◽  
Lubomír Neužil ◽  
Jaroslav Nývlt
Keyword(s):  
Steady State ◽  
Simple Model ◽  
Mathematical Modelling ◽  
Distribution Coefficient ◽  
Mother Liquor ◽  
Equilibrium Temperature ◽  
Operating Conditions ◽  
Limiting Case

The paper presents a simple model of recrystallization with countercurrent flows of the solution and the crystals being purified. The model assumes steady-state operating conditions, an equilibrium between the outlet streams of each stage, and the same equilibrium temperature and distribution coefficient for all stages. With these assumptions, the model provides the basis for analyzing the variation in the degree of purity as a function of the number of recrystallization stages. The analysis is facilitated by the use of a diagram constructed for the limiting case of perfect removal of the mother liquor from the crystals between the stages.

Five Modes of Scepticism

2019 ◽  
Author(s):  
Stefan Sienkiewicz
Keyword(s):  
Sextus Empiricus ◽  
Theoretical Assumptions ◽  
Limiting Case ◽  
Do So

This book offers an account of the functioning of the five Agrippan modes of scepticism as presented in the works of Sextus Empiricus. These five modes (of disagreement, hypothesis, infinite regression, reciprocity, and relativity) are analysed, individually, in the book’s first five chapters, and, collectively, in its sixth. Two perspectives on these modes are distinguished from one another—a dogmatic perspective which considers how a dogmatic philosopher might come to suspend judgement on the basis of these modes and a sceptical perspective which considers how a sceptic might come to do so. It is argued that the standard way in which these modes have been understood has been from a dogmatic perspective. The book opens up an alternative sceptical perspective on the modes according to which mode of disagreement (or one version of it) is equivalent to the sceptic’s method of equipollence, and the modes of hypothesis, infinite regression, and reciprocity are different instances of that method (with the mode of hypothesis being a limiting case of the method). It is also argued that the mode of relativity is inconsistent with the mode of disagreement and should be discarded when considering how the modes work together in a combined sceptical strategy. The final chapter offers an account of four different ways in which the modes might be combined together and concludes that each of these ways turns on a number of theoretical assumptions which the sceptic is not in a position to make.

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