TheC *-algebra of a singular elliptic problem on a noncompact Riemannian manifold

1977 ◽  
Vol 153 (2) ◽  
pp. 101-116 ◽  
Author(s):  
Heinz O. Cordes ◽  
Robert C. McOwen
Keyword(s):  
Riemannian Manifold ◽  
Elliptic Problem ◽  
Noncompact Riemannian Manifold

scholarly journals On the Existence of Nodal Solutions for a Nonlinear Elliptic Problem on Symmetric Riemannian Manifolds

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
Anna Maria Micheletti ◽  
Angela Pistoia
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Elliptic Problem ◽  
Multiplicity Result ◽  
Nonlinear Elliptic Problem ◽  
Nodal Solutions ◽  
Nonlinear Elliptic ◽  
Sign Changing Solutions

Given thatis a smooth compact and symmetric Riemannian -manifold, , we prove a multiplicity result for antisymmetric sign changing solutions of the problem in . Here if and if .

Multiplicity and nondegeneracy of positive solutions to quasilinear equations on compact Riemannian manifolds

2015 ◽  
Vol 17 (02) ◽  
pp. 1450029 ◽  
Author(s):  
Silvia Cingolani ◽  
Giuseppina Vannella ◽  
Daniela Visetti
Keyword(s):  
Riemannian Manifold ◽  
Positive Solutions ◽  
Riemannian Manifolds ◽  
Elliptic Problem ◽  
Quasilinear Equations ◽  
Metric Tensor ◽  
Poincaré Polynomial ◽  
Quasilinear Elliptic Problem ◽  
Quasilinear Problem ◽  
Quasilinear Elliptic

We consider a compact, connected, orientable, boundaryless Riemannian manifold (M, g) of class C∞ where g denotes the metric tensor. Let n = dim M ≥ 3. Using Morse techniques, we prove the existence of [Formula: see text] nonconstant solutions u ∈ H1,p(M) to the quasilinear problem [Formula: see text] for ε > 0 small enough, where 2 ≤ p < n, p < q < p*, p* = np/(n - p) and [Formula: see text] is the p-laplacian associated to g of u (note that Δ2,g = Δg) and [Formula: see text] denotes the Poincaré polynomial of M. We also establish results of genericity of nondegenerate solutions for the quasilinear elliptic problem (Pε).

A coarse Mayer–Vietoris principle

1993 ◽  
Vol 114 (1) ◽  
pp. 85-97 ◽  
Author(s):  
Nigel Higson ◽  
John Roe ◽  
Guoliang Yu
Keyword(s):  
Riemannian Manifold ◽  
Metric Space ◽  
Metric Spaces ◽  
Compact Operators ◽  
Cohomology Theory ◽  
Coarse Geometry ◽  
Locally Compact ◽  
Finite Propagation ◽  
Noncompact Riemannian Manifold

In [1], [4], and [6] the authors have studied index problems associated with the ‘coarse geometry’ of a metric space, which typically might be a complete noncompact Riemannian manifold or a group equipped with a word metric. The second author has introduced a cohomology theory, coarse cohomology, which is functorial on the category of metric spaces and coarse maps, and which can be computed in many examples. Associated to such a metric space there is also a C*-algebra generated by locally compact operators with finite propagation. In this note we will show that for suitable decompositions of a metric space there are Mayer–Vietoris sequences both in coarse cohomology and in the K-theory of the C*-algebra. As an application we shall calculate the K-theory of the C*-algebra associated to a metric cone. The result is consistent with the calculation of the coarse cohomology of the cone, and with a ‘coarse’ version of the Baum–Connes conjecture.

scholarly journals Hölder Compactification for Some Manifolds with Pinched Negative Curvature Near Infinity

2008 ◽  
Vol 60 (6) ◽  
pp. 1201-1218 ◽  
Author(s):  
Eric Bahuaud ◽  
Tracey Marsh
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Negative Curvature ◽  
Exponential Map ◽  
Noncompact Riemannian Manifold ◽  
Compact Submanifold ◽  
Curvature Pinching ◽  
Negative Sectional Curvature

AbstractWe consider a complete noncompact Riemannian manifold M and give conditions on a compact submanifold K ⊂ M so that the outward normal exponential map off the boundary of K is a diffeomorphism onto M\K. We use this to compactify M and show that pinched negative sectional curvature outside K implies M has a compactification with a well-defined Hölder structure independent of K. The Hölder constant depends on the ratio of the curvature pinching. This extends and generalizes a 1985 result of Anderson and Schoen.

scholarly journals Existence of Infinitely Many Distinct Solutions to the Quasirelativistic Hartree-Fock Equations

2009 ◽  
Vol 2009 ◽  
pp. 1-20 ◽  
Author(s):  
M. Enstedt ◽  
M. Melgaard
Keyword(s):  
Riemannian Manifold ◽  
Kinetic Energy ◽  
Ground State ◽  
Multiple Solutions ◽  
Density Operator ◽  
Coulomb Systems ◽  
Total Charge ◽  
Hartree Fock ◽  
Noncompact Riemannian Manifold ◽  
Semilinear Elliptic

We establish existence of infinitely many distinct solutions to the semilinear elliptic Hartree-Fock equations forN-electron Coulomb systems with quasirelativistic kinetic energy−α−2Δxn+α−4−α−2for thenthelectron. Moreover, we prove existence of a ground state. The results are valid under the hypotheses that the total chargeZtotofKnuclei is greater thanN−1and thatZtotis smaller than a critical chargeZc. The proofs are based on a new application of the Fang-Ghoussoub critical point approach to multiple solutions on a noncompact Riemannian manifold, in combination with density operator techniques.

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