scholarly journals Three lectures on global boundary conditions and the theory of self-adjoint extensions of the covariant Laplace-Beltrami and Dirac operators on Riemannian manifolds with boundary

2012 ◽  
Author(s):  
A. Ibort
Keyword(s):  
Boundary Conditions ◽  
Riemannian Manifolds ◽  
Dirac Operators ◽  
Manifolds With Boundary

Multiple Solutions for a Class of Neumann Elliptic Problems on Compact Riemannian Manifolds with Boundary

2010 ◽  
Vol 53 (4) ◽  
pp. 674-683 ◽  
Author(s):  
Alexandru Kristály ◽  
Nikolaos S. Papageorgiou ◽  
Csaba Varga
Keyword(s):  
Boundary Condition ◽  
Riemannian Manifolds ◽  
Neumann Boundary Condition ◽  
Multiple Solutions ◽  
Compact Riemannian Manifold ◽  
Neumann Boundary ◽  
Manifolds With Boundary ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Manifold With Boundary

AbstractWe study a semilinear elliptic problem on a compact Riemannian manifold with boundary, subject to an inhomogeneous Neumann boundary condition. Under various hypotheses on the nonlinear terms, depending on their behaviour in the origin and infinity, we prove multiplicity of solutions by using variational arguments.

Twisted dirac operators and Kastler-Kalau-Walze theorems for six-dimensional manifolds with boundary

2020 ◽  
Vol 17 (14) ◽  
pp. 2050211
Author(s):  
Sining Wei ◽  
Yong Wang
Keyword(s):  
Vector Bundle ◽  
Dirac Operators ◽  
Manifolds With Boundary ◽  
Unitary Connection ◽  
Twisted Dirac Operators

In this paper, we establish two kinds of Kastler-Kalau-Walze type theorems for Dirac operators and signature operators twisted by a vector bundle with a non-unitary connection on six-dimensional manifolds with boundary.

scholarly journals ON NONUNIQUENESS FOR THE ANISOTROPIC CALDERÓN PROBLEM WITH PARTIAL DATA

2020 ◽  
Vol 8 ◽  
Author(s):  
THIERRY DAUDÉ ◽  
NIKY KAMRAN ◽  
FRANÇOIS NICOLEAU
Keyword(s):  
Infinite Number ◽  
Riemannian Manifolds ◽  
Riemannian Metrics ◽  
Unique Continuation ◽  
Manifolds With Boundary ◽  
Conformal Class ◽  
Hölder Continuous ◽  
Continuation Principle ◽  
Partial Data ◽  
Calderón Problem

We show that there is nonuniqueness for the Calderón problem with partial data for Riemannian metrics with Hölder continuous coefficients in dimension greater than or equal to three. We provide simple counterexamples in the case of cylindrical Riemannian manifolds with boundary having two ends. The coefficients of these metrics are smooth in the interior of the manifold and are only Hölder continuous of order $\unicode[STIX]{x1D70C}<1$ at the end where the measurements are made. More precisely, we construct a toroidal ring $(M,g)$ and we show that there exist in the conformal class of $g$ an infinite number of Riemannian metrics $\tilde{g}=c^{4}g$ such that their corresponding partial Dirichlet-to-Neumann maps at one end coincide. The corresponding smooth conformal factors are harmonic with respect to the metric $g$ and do not satisfy the unique continuation principle.

scholarly journals On Yamabe-type problems on Riemannian manifolds with boundary

2016 ◽  
Vol 284 (1) ◽  
pp. 79-102 ◽  
Author(s):  
Marco Ghimenti ◽  
Anna Micheletti ◽  
Angela Pistoia
Keyword(s):  
Riemannian Manifolds ◽  
Manifolds With Boundary

Harmonic Tensors on Riemannian Manifolds with Boundary

1952 ◽  
Vol 56 (1) ◽  
pp. 128 ◽  
Author(s):  
G. F. D. Duff ◽  
D. C. Spencer
Keyword(s):  
Riemannian Manifolds ◽  
Manifolds With Boundary

scholarly journals Higher spectral flow for Dirac operators with local boundary conditions

2016 ◽  
Vol 27 (08) ◽  
pp. 1650068
Author(s):  
Jianqing Yu
Keyword(s):  
Boundary Conditions ◽  
Elliptic Boundary ◽  
Dirac Operators ◽  
Spectral Flow ◽  
Local Boundary ◽  
Manifold With Boundary ◽  
Index Bundle ◽  
Adjoint Extension ◽  
Smooth Map ◽  
Higher Spectral Flow

We consider a one parameter family [Formula: see text] of families of fiberwise twisted Dirac type operators on a fibration with the typical fiber an even dimensional compact manifold with boundary, which verifies [Formula: see text] with [Formula: see text] being a smooth map from the fibration to a unitary group [Formula: see text]. For each [Formula: see text], we impose on [Formula: see text] a certain fixed local elliptic boundary condition [Formula: see text] and get a self-adjoint extension [Formula: see text]. Under the assumption that [Formula: see text] has vanishing [Formula: see text]-index bundle, we establish a formula for the higher spectral flow of [Formula: see text], [Formula: see text]. Our result generalizes a recent result of [A. Gorokhovsky and M. Lesch, On the spectral flow for Dirac operators with local boundary conditions, Int. Math. Res. Not. IMRN (2015) 8036–8051.] to the families case.

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