Boundary problems on a manifold with edge

2017 ◽  
Vol 10 (02) ◽  
pp. 1750087 ◽  
Author(s):  
S. Khalil ◽  
B.-W. Schulze
Keyword(s):  
Boundary Value Problems ◽  
Compact Manifold ◽  
Smooth Boundary ◽  
Boundary Value ◽  
Boundary Problems ◽  
Manifold With Boundary ◽  
Edge Based

We establish a calculus of boundary value problems (BVPs) on a manifold [Formula: see text] with boundary and edge, based on Boutet de Monvel’s theory of BVPs in the case of a smooth boundary and on the edge calculus, where in the present case the model cone has a base which is a compact manifold with boundary. The corresponding calculus with boundary and edge is a unification of both structures and controls different operator-valued symbolic structures, in order to obtain ellipticity and parametrices.

A generalization of the asymptotic behavior of Palais-Smale sequences on a manifold with boundary

2018 ◽  
Vol 29 (10) ◽  
pp. 1850069
Author(s):  
Hong Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Boundary Value Problem ◽  
Mean Curvature ◽  
Compact Manifold ◽  
Boundary Value ◽  
Fundamental Solutions ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
Manifold With Boundary ◽  
Prescribed Mean Curvature Equation

In this paper, we study the asymptotic behavior of Palais-Smale sequences associated with the prescribed mean curvature equation on a compact manifold with boundary. We prove that every such sequence converges to a solution of the associated equation plus finitely many “bubbles” obtained by rescaling fundamental solutions of the corresponding Euclidean boundary value problem.

Deformation quantization and boundary value problems

2016 ◽  
Vol 13 (02) ◽  
pp. 1650007
Author(s):  
Nikolai Tarkhanov
Keyword(s):  
Harmonic Oscillator ◽  
Boundary Value Problems ◽  
Symplectic Manifold ◽  
Deformation Quantization ◽  
Boundary Value ◽  
Index Theorem ◽  
Manifold With Boundary ◽  
Quantum Observables ◽  
Compact Symplectic Manifold

We describe a natural construction of deformation quantization on a compact symplectic manifold with boundary. On the algebra of quantum observables a trace functional is defined which as usual annihilates the commutators. This gives rise to an index as the trace of the unity element. We formulate the index theorem as a conjecture and examine it by the classical harmonic oscillator.

scholarly journals Stability of mathematical models of the main problems of the anisotropic theory of elasticity

2020 ◽  
Vol 30 (1) ◽  
pp. 112-124
Author(s):  
A.V. Yudenkov ◽  
A.M. Volodchenkov
Keyword(s):  
Boundary Value Problems ◽  
Analytic Functions ◽  
Boundary Value ◽  
Research Result ◽  
Theory Of Elasticity ◽  
Boundary Problems ◽  
Main Research ◽  
Contour Shape ◽  
Variable Function ◽  
Anisotropic Bodies

The boundary problems of the complex-variable function theory are effectively used while investigating equilibrium of homogeneous elastic mediums. The most complicated systems of the boundary value problems correspond to the case when an elastic body exhibits anisotropic properties. Anisotropy of the medium results in the drift of boundary conditions of the function that in general disrupts analyticity of the functions of interest. The paper studies systems of the boundary value problems with drift for analytic vectors corresponding to the primal elastic problems (first, second and mixed problems). Systems of analytic vectors with drift are reduced to equivalent systems of Hilbert boundary value problems for analytic functions with weak singularity integrators. The obtained general solution of the primal boundary value problems for the anisotropic theory of elasticity allows us to check the above problems for stability with respect to perturbations of boundary value conditions and contour shape. The research is relevant as there is necessity to apply approximate numerical methods to the boundary value problems with drift. The main research result comes to be a proof of stability of the systems of the vector boundary value problems with drift for analytic functions on the Hölder space corresponding to the primal problems of the elastic theory for anisotropic bodies in the case of change in the boundary value conditions and contour shape.

scholarly journals Boundary Value Problems for the Lorentzian Dirac Operator

2018 ◽  
pp. 3-18
Author(s):  
Christian Bär ◽  
Sebastian Hannes
Keyword(s):  
Riemannian Manifold ◽  
Dirac Operator ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Classical Case ◽  
Spin Manifold ◽  
Globally Hyperbolic ◽  
Manifold With Boundary ◽  
Smooth Space

On a compact globally hyperbolic Lorentzian spin manifold with smooth space-like Cauchy boundary, the (hyperbolic) Dirac operator is known to be Fredholm when Atiyah–Patodi–Singer boundary conditions are imposed. This chapter explores to what extent these boundary conditions can be replaced by more general ones and how the index then changes. There are some differences to the classical case of the elliptic Dirac operator on a Riemannian manifold with boundary.

scholarly journals On non-elliptic boundary problems

1982 ◽  
Vol 86 ◽  
pp. 1-38
Author(s):  
Yoshio Kato
Keyword(s):  
Differential Equation ◽  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Second Order ◽  
Elliptic Differential Equation ◽  
Boundary Problems

The purpose of this paper is to study the boundary value problems for the second order elliptic differential equation

ON THE SOLUTION SETS OF FOUR-POINT BOUNDARY VALUE PROBLEMS FOR NONCONVEX DIFFERENTIAL INCLUSIONS

2011 ◽  
Vol 08 (01) ◽  
pp. 23-37 ◽  
Author(s):  
ADEL MAHMOUD GOMAA
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Differential Inclusions ◽  
Boundary Value ◽  
Point Boundary ◽  
Boundary Problems ◽  
Solution Sets ◽  
Nonempty Compact ◽  
Nonconvex Differential Inclusions

We consider the multivalued problem [Formula: see text] under four boundary conditions u(0) = x0, u(η) = u(θ) = u(T) where 0 < η < θ < T and for F is a multifunctions from [0, T] × ℝn × ℝn to the nonempty compact subsets of ℝn not necessary convex. We give a lemma which is useful in the study of four boundary problems for the differential equations and the differential inclusions. Further we have results that improve earlier theorems.

scholarly journals On Homotopic Classification of Elliptic Problems with Contractions and K-Groups of Corresponding C∗-Algebras

2018 ◽  
Vol 64 (1) ◽  
pp. 164-179
Author(s):  
A Yu Savin
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Elliptic Problems ◽  
Nonlocal Problems ◽  
Boundary Problems ◽  
C Algebras ◽  
Codimension One ◽  
Homotopic Classification

We consider calculation of a group of stable homotopic classes for pseudodifferential elliptic boundary problems. We study this problem in terms of topological K-groups of some spaces in the following cases: for boundary-value problems on manifolds with boundaries, for conjugation problems with conditions on a closed submanifold of codimension one, and for nonlocal problems with contractions.

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