@misc{BandaraRosen2019,
  author    = {Bandara, Menaka Lashitha and Ros{\´e}n, Andreas},
  title     = {Riesz continuity of the Atiyah-Singer Dirac operator under perturbations of local boundary conditions},
  series = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe},
  journal   = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe},
  number    = {758},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-43407},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-434078},
  pages     = {1253 -- 1284},
  year      = {2019},
  abstract  = {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.},
  language  = {en}
}
@phdthesis{Gehring2023,
  author    = {Gehring, Penelope},
  title     = {Non-local boundary conditions for the spin Dirac operator on spacetimes with timelike boundary},
  doi       = {10.25932/publishup-57775},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-577755},
  school      = {Universit{\"a}t Potsdam},
  pages     = {100},
  year      = {2023},
  abstract  = {Non-local boundary conditions - for example the Atiyah-Patodi-Singer (APS) conditions - for Dirac operators on Riemannian manifolds are rather well-understood, while not much is known for such operators on Lorentzian manifolds. Recently, B{\"a}r and Strohmaier [15] and Drago, Große, and Murro [27] introduced APS-like conditions for the spin Dirac operator on Lorentzian manifolds with spacelike and timelike boundary, respectively. While B{\"a}r and Strohmaier [15] showed the Fredholmness of the Dirac operator with these boundary conditions, Drago, Große, and Murro [27] proved the well-posedness of the corresponding initial boundary value problem under certain geometric assumptions. In this thesis, we will follow the footsteps of the latter authors and discuss whether the APS-like conditions for Dirac operators on Lorentzian manifolds with timelike boundary can be replaced by more general conditions such that the associated initial boundary value problems are still wellposed. We consider boundary conditions that are local in time and non-local in the spatial directions. More precisely, we use the spacetime foliation arising from the Cauchy temporal function and split the Dirac operator along this foliation. This gives rise to a family of elliptic operators each acting on spinors of the spin bundle over the corresponding timeslice. The theory of elliptic operators then ensures that we can find families of non-local boundary conditions with respect to this family of operators. Proceeding, we use such a family of boundary conditions to define a Lorentzian boundary condition on the whole timelike boundary. By analyzing the properties of the Lorentzian boundary conditions, we then find sufficient conditions on the family of non-local boundary conditions that lead to the well-posedness of the corresponding Cauchy problems. The well-posedness itself will then be proven by using classical tools including energy estimates and approximation by solutions of the regularized problems. Moreover, we use this theory to construct explicit boundary conditions for the Lorentzian Dirac operator. More precisely, we will discuss two examples of boundary conditions - the analogue of the Atiyah-Patodi-Singer and the chirality conditions, respectively, in our setting. For doing this, we will have a closer look at the theory of non-local boundary conditions for elliptic operators and analyze the requirements on the family of non-local boundary conditions for these specific examples.},
  language  = {en}
}
@unpublished{PolkovnikovTarkhanov2017,
  author    = {Polkovnikov, Alexander and Tarkhanov, Nikolai Nikolaevich},
  title     = {A Riemann-Hilbert problem for the Moisil-Teodorescu system},
  volume    = {6},
  number    = {3},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-397036},
  pages     = {31},
  year      = {2017},
  abstract  = {In a bounded domain with smooth boundary in R^3 we consider the stationary Maxwell equations for a function u with values in R^3 subject to a nonhomogeneous condition (u,v)_x = u_0 on the boundary, where v is a given vector field and u_0 a function on the boundary. We specify this problem within the framework of the Riemann-Hilbert boundary value problems for the Moisil-Teodorescu system. This latter is proved to satisfy the Shapiro-Lopaniskij condition if an only if the vector v is at no point tangent to the boundary. The Riemann-Hilbert problem for the Moisil-Teodorescu system fails to possess an adjoint boundary value problem with respect to the Green formula, which satisfies the Shapiro-Lopatinskij condition. We develop the construction of Green formula to get a proper concept of adjoint boundary value problem.},
  language  = {en}
}
@unpublished{AlsaedyTarkhanov2016,
  author    = {Alsaedy, Ammar and Tarkhanov, Nikolai Nikolaevich},
  title     = {A Hilbert boundary value problem for generalised Cauchy-Riemann equations},
  volume    = {5},
  number    = {1},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-86109},
  pages     = {21},
  year      = {2016},
  abstract  = {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.},
  language  = {en}
}