TY  - JOUR
A1  - Alsaedy, Ammar
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - A Hilbert Boundary Value Problem for Generalised Cauchy-Riemann Equations
JF  - Advances in applied Clifford algebras
N2  - We elaborate a boundary Fourier method for studying an analogue of the Hilbert problem for analytic functions within the framework of generalised Cauchy-Riemann equations. The boundary value problem need not satisfy the Shapiro-Lopatinskij condition and so it fails to be Fredholm in Sobolev spaces. We show a solvability condition of the Hilbert problem, which looks like those for ill-posed problems, and construct an explicit formula for approximate solutions.
KW  - Dirac operator
KW  - Clifford algebra
KW  - Riemann-Hilbert problem
KW  - Fredholm operators
Y1  - 2017
U6  - https://doi.org/10.1007/s00006-016-0676-8
SN  - 0188-7009
SN  - 1661-4909
VL  - 27
SP  - 931
EP  - 953
PB  - Springer
CY  - Basel
ER  - 
TY  - JOUR
A1  - Roos, Saskia
T1  - The Dirac operator under collapse to a smooth limit space
JF  - Annals of global analysis and geometry
N2  - Let (M-i, g(i))(i is an element of N) be a sequence of spin manifolds with uniform bounded curvature and diameter that converges to a lower-dimensional Riemannian manifold (B, h) in the Gromov-Hausdorff topology. Then, it happens that the spectrum of the Dirac operator converges to the spectrum of a certain first-order elliptic differential operator D-B on B. We give an explicit description of D-B and characterize the special case where D-B equals the Dirac operator on B.
KW  - Collapse
KW  - Dirac operator
KW  - Spin geometry
Y1  - 2019
U6  - https://doi.org/10.1007/s10455-019-09691-8
SN  - 0232-704X
SN  - 1572-9060
VL  - 57
IS  - 1
SP  - 121
EP  - 151
PB  - Springer
CY  - Dordrecht
ER  - 
TY  - JOUR
A1  - Bandara, Menaka Lashitha
A1  - Rosen, Andreas
T1  - Riesz continuity of the Atiyah-Singer Dirac operator under perturbations of local boundary conditions
JF  - Communications in partial differential equations
N2  - On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that Atiyah-Singer Dirac operator in depends Riesz continuously on perturbations of local boundary conditions The Lipschitz bound for the map depends on Lipschitz smoothness and ellipticity of and bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius away from a compact neighbourhood of the boundary. More generally, we prove perturbation estimates for functional calculi of elliptic operators on manifolds with local boundary conditions.
KW  - Boundary value problems
KW  - Dirac operator
KW  - functional calculus
KW  - real-variable harmonic analysis
KW  - Riesz continuity
KW  - spectral flow
Y1  - 2019
U6  - https://doi.org/10.1080/03605302.2019.1611847
SN  - 0360-5302
SN  - 1532-4133
VL  - 44
IS  - 12
SP  - 1253
EP  - 1284
PB  - Taylor & Francis Group
CY  - Philadelphia
ER  -