scholarly journals Lefschetz fixed point formula on a compact Riemannian manifold with boundary for some boundary conditions

2015 ◽  
Vol 181 (1) ◽  
pp. 43-59
Author(s):  
Rung-Tzung Huang ◽  
Yoonweon Lee
Keyword(s):  
Fixed Point ◽  
Riemannian Manifold ◽  
Boundary Conditions ◽  
Compact Riemannian Manifold ◽  
Lefschetz Fixed Point Formula ◽  
Manifold With Boundary

scholarly journals Boundary layers to a singularly perturbed Klein–Gordon–Maxwell–Proca system on a compact Riemannian manifold with boundary

2017 ◽  
Vol 8 (1) ◽  
pp. 559-582 ◽  
Author(s):  
Mónica Clapp ◽  
Marco Ghimenti ◽  
Anna Maria Micheletti
Keyword(s):  
Riemannian Manifold ◽  
Boundary Conditions ◽  
Compact Riemannian Manifold ◽  
Semiclassical Limit ◽  
Arbitrary Dimension ◽  
Neumann Boundary ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Manifold With Boundary ◽  
Klein Gordon

Abstract We study the semiclassical limit to a singularly perturbed nonlinear Klein–Gordon–Maxwell–Proca system, with Neumann boundary conditions, on a Riemannian manifold {\mathfrak{M}} with boundary. We exhibit examples of manifolds, of arbitrary dimension, on which these systems have a solution which concentrates at a closed submanifold of the boundary of {\mathfrak{M}} , forming a positive layer, as the singular perturbation parameter goes to zero. Our results allow supercritical nonlinearities and apply, in particular, to bounded domains in {\mathbb{R}^{N}} . Similar results are obtained for the more classical electrostatic Klein–Gordon–Maxwell system with appropriate boundary conditions.

scholarly journals Hearing the shape of a compact Riemannian manifold with a finite number of piecewise impedance boundary conditions

1997 ◽  
Vol 20 (2) ◽  
pp. 397-402 ◽  
Author(s):  
E. M. E. Zayed
Keyword(s):  
Riemannian Manifold ◽  
Boundary Conditions ◽  
Finite Number ◽  
Spectral Function ◽  
Compact Riemannian Manifold ◽  
Smooth Boundary ◽  
Beltrami Operator ◽  
Smooth Functions ◽  
Impedance Boundary ◽  
Impedance Boundary Conditions

The spectral functionΘ(t)=∑i=1∞exp(−tλj), where{λj}j=1∞are the eigenvalues of the negative Laplace-Beltrami operator−Δ, is studied for a compact Riemannian manifoldΩof dimension “k” with a smooth boundary∂Ω, where a finite number of piecewise impedance boundary conditions(∂∂ni+γi)u=0on the parts∂Ωi(i=1,…,m)of the boundary∂Ωcan be considered, such that∂Ω=∪i=1m∂Ωi, andγi(i=1,…,m)are assumed to be smooth functions which are not strictly positive.

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.

EXTREMAL FUNCTIONS FOR MOSER–TRUDINGER INEQUALITIES ON 2-DIMENSIONAL COMPACT RIEMANNIAN MANIFOLDS WITH BOUNDARY

2006 ◽  
Vol 17 (03) ◽  
pp. 313-330 ◽  
Author(s):  
YUNYAN YANG
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Compact Riemannian Manifold ◽  
Extremal Functions ◽  
Manifolds With Boundary ◽  
Manifold With Boundary ◽  
Blowing Up

Let (M,g) be a 2-dimensional compact Riemannian manifold with boundary. In this paper, we use the method of blowing up analysis to prove the existence of the extremal functions for some Moser–Trudinger inequalities on (M,g).

scholarly journals Heat asymptotics with spectral boundary conditions. II

2003 ◽  
Vol 133 (2) ◽  
pp. 333-361 ◽  
Author(s):  
Peter Gilkey ◽  
Klaus Kirsten
Keyword(s):  
Riemannian Manifold ◽  
Boundary Conditions ◽  
Compact Riemannian Manifold ◽  
Smooth Boundary ◽  
Dirac Type ◽  
Heat Trace ◽  
Operator Of Dirac Type ◽  
Laplace Type

Let P be an operator of Dirac type on a compact Riemannian manifold with smooth boundary. We impose spectral boundary conditions and study the asymptotics of the heat trace of the associated operator of Laplace type.

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