On hyperbolic contact problems

2009 ◽  
Vol 43 (1) ◽  
pp. 25-40
Author(s):  
Igor Bock ◽  
Jiří Jarušek
Keyword(s):  
Variational Inequality ◽  
Differential Operators ◽  
Existence Of Solutions ◽  
Contact Problems ◽  
Large Deflections ◽  
Two Dimensional ◽  
Airy Stress Function ◽  
Penalization Method ◽  
Nonlinear Elliptic ◽  
First Case

Abstract We deal with hyperbolic variational inequalities modeling vibrations of two-dimensional structures with an obstacle. We focus on the plates with moderately large deflections. The nonlinear strain-displacements relations imply nonlinear elliptic parts of differential operators in considered problems.We distinguish two types of problems. In the first case only the deflections are considered with accelerations and the plane displacements are expressed using the Airy stress function. In the case of plane accelerations the full von K´arm´an system consisting of two equations and one variational inequality is considered. The existence of solutions is derived using the penalization method.

Topological degree methods for partial differential operators in generalized Sobolev spaces

2021 ◽  
Vol 39 (2) ◽  
pp. 39-61
Author(s):  
Mustapha Ait Hammou ◽  
Elhoussine Azroul ◽  
Badr Lahmi
Keyword(s):  
Topological Degree ◽  
Nonlinear Elliptic Equation ◽  
Differential Operators ◽  
Existence Of Solutions ◽  
Divergence Form ◽  
Generalized Sobolev Spaces ◽  
Nonlinear Elliptic ◽  
Partial Differential Operators ◽  
Topological Degree Methods ◽  
General Divergence

The main aim of this paper is to prove, by using the topological degree methods, the existence of solutions for nonlinear elliptic equation Au = f where Au  is partial dierential operators of general divergence form.

scholarly journals Bra-ket wormholes in gravitationally prepared states

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Yiming Chen ◽  
Victor Gorbenko ◽  
Juan Maldacena
Keyword(s):  
Hyperbolic Space ◽  
Path Integral ◽  
Flat Space ◽  
Low Energy ◽  
Two Dimensional ◽  
De Sitter ◽  
Euclidean Case ◽  
Fine Grained ◽  
Entropy Formula ◽  
First Case

Abstract We consider two dimensional CFT states that are produced by a gravitational path integral.As a first case, we consider a state produced by Euclidean AdS2 evolution followed by flat space evolution. We use the fine grained entropy formula to explore the nature of the state. We find that the naive hyperbolic space geometry leads to a paradox. This is solved if we include a geometry that connects the bra with the ket, a bra-ket wormhole. The semiclassical Lorentzian interpretation leads to CFT state entangled with an expanding and collapsing Friedmann cosmology.As a second case, we consider a state produced by Lorentzian dS2 evolution, again followed by flat space evolution. The most naive geometry also leads to a similar paradox. We explore several possible bra-ket wormholes. The most obvious one leads to a badly divergent temperature. The most promising one also leads to a divergent temperature but by making a projection onto low energy states we find that it has features that look similar to the previous Euclidean case. In particular, the maximum entropy of an interval in the future is set by the de Sitter entropy.

FOURIER ANALYSIS OVER HYPERBOLIC ALGEBRA, PSEUDO-DIFFERENTIAL OPERATORS, AND HYPERBOLIC DEFORMATION OF CLASSICAL MECHANICS

2007 ◽  
Vol 10 (03) ◽  
pp. 421-438 ◽  
Author(s):  
ANDREI KHRENNIKOV
Keyword(s):  
Quantum Mechanics ◽  
Fourier Analysis ◽  
Poisson Bracket ◽  
Commutative Algebra ◽  
Differential Operators ◽  
Classical Mechanics ◽  
Complex Numbers ◽  
Two Dimensional ◽  
Quantum Deformation ◽  
Pseudo Differential Operators

We develop Fourier analysis over hyperbolic algebra (the two-dimensional commutative algebra with the basis e1 = 1, e2 = j, where j2 = 1). We demonstrated that classical mechanics has, besides the well-known quantum deformation over complex numbers, another deformation — so-called hyperbolic quantum mechanics. The classical Poisson bracket can be obtained as the limit h → 0 not only of the ordinary Moyal bracket, but also a hyperbolic analogue of the Moyal bracket.

The magneto-hydrodynamic boundary layer in the two-dimensional steady flow past a semi-infinite flat plate II. The Falkner-Skan problem

1963 ◽  
Vol 273 (1355) ◽  
pp. 509-517 ◽  
Keyword(s):  
Flat Plate ◽  
Iterative Procedure ◽  
Hypergeometric Functions ◽  
Close Agreement ◽  
Adverse Pressure Gradient ◽  
Iterative Solution ◽  
Two Dimensional ◽  
Confluent Hypergeometric Functions ◽  
First Case ◽  
Hydrodynamic Boundary Layer

The problem investigated is the flow of a viscous liquid past a semi-infinite flat plate against an adverse pressure gradient. The method used is an iterative method suggested in a paper by Weyl. To start the iterative procedure a function is chosen which satisfies some of the boundary conditions and by using this function the first iterative solution has been obtained analytically in terms of confluent hypergeometric functions. Two different starting functions have been considered. In the first case it has been found possible to compare the results obtained with the well-known Hartree numerical solution and even at the first iteration close agreement is achieved. In the second case, the first iterative solution behaves correctly at infinity but the agreement with Hartree ’s solution is not as good as it is in the first case.

Load History Effects on Fretting Contacts of Isotropic Materials

2004 ◽  
Vol 126 (2) ◽  
pp. 385-390 ◽  
Author(s):  
P. T. Rajeev ◽  
H. Murthy ◽  
T. N. Farris
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Singular Integral Equations ◽  
Contact Problems ◽  
Two Dimensional ◽  
Load History ◽  
Jet Engine ◽  
Isotropic Materials ◽  
History Effects ◽  
Cauchy Singular Integral

The load history that blade/disk contacts in jet engine attachment hardware are subject to can be very complex. Using finite element method (FEM) to track changes in the contact tractions due to changing loads can be computationally very expensive. For two-dimensional plane-strain contact problems with friction involving similar/dissimilar isotropic materials, the contact tractions can be related to the initial gap function and the slip function using coupled Cauchy singular integral equations (SIEs). The effect of load history on the contact tractions is illustrated by presenting results for an example fretting “mission.” For the case of dissimilar isotropic materials the mission results show the effect of the coupling between the shear traction and the contact pressure.

On the Dirichlet problem for weakly non-linear elliptic partial differential equations

1977 ◽  
Vol 76 (4) ◽  
pp. 283-300 ◽  
Author(s):  
E. N. Dancer
Keyword(s):  
Dirichlet Problem ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Existence Of Solutions ◽  
Elliptic Partial Differential Equations ◽  
Partial Differential ◽  
Weakly Nonlinear ◽  
Partial Derivatives ◽  
Nonlinear Elliptic ◽  
Non Linear

SynopsisWe study the existence of solutions of the Dirichlet problem for weakly nonlinear elliptic partial differential equations. We only consider cases where the nonlinearities do not depend on any partial derivatives. For these cases, we prove the existence of solutions for a wide variety of nonlinearities.

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