@unpublished{AntonioukKiselevStepanenkoetal.2012,
  author    = {Antoniouk, Alexandra Viktorivna and Kiselev, Oleg and Stepanenko, Vitaly and Tarkhanov, Nikolai Nikolaevich},
  title     = {Asymptotic solutions of the Dirichlet problem for the heat equation at a characteristic point},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-61987},
  year      = {2012},
  abstract  = {The Dirichlet problem for the heat equation in a bounded domain is characteristic, for there are boundary points at which the boundary touches a characteristic hyperplane t = c, c being a constant. It was I.G. Petrovskii (1934) who first found necessary and sufficient conditions on the boundary which guarantee that the solution is continuous up to the characteristic point, provided that the Dirichlet data are continuous. This paper initiated standing interest in studying general boundary value problems for parabolic equations in bounded domains. We contribute to the study by constructing a formal solution of the Dirichlet problem for the heat equation in a neighbourhood of a characteristic boundary point and showing its asymptotic character.},
  language  = {en}
}
@unpublished{AlsaedyTarkhanov2012,
  author    = {Alsaedy, Ammar and Tarkhanov, Nikolai Nikolaevich},
  title     = {The method of Fischer-Riesz equations for elliptic boundary value problems},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-61792},
  year      = {2012},
  abstract  = {We develop the method of Fischer-Riesz equations for general boundary value problems elliptic in the sense of Douglis-Nirenberg. To this end we reduce them to a boundary problem for a (possibly overdetermined) first order system whose classical symbol has a left inverse. For such a problem there is a uniquely determined boundary value problem which is adjoint to the given one with respect to the Green formula. On using a well elaborated theory of approximation by solutions of the adjoint problem, we find the Cauchy data of solutions of our problem.},
  language  = {en}
}
@unpublished{DyachenkoTarkhanov2012,
  author    = {Dyachenko, Evgueniya and Tarkhanov, Nikolai Nikolaevich},
  title     = {Degeneration of boundary layer at singular points},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-60135},
  year      = {2012},
  abstract  = {We study the Dirichlet problem in a bounded plane domain for the heat equation with small parameter multiplying the derivative in t. The behaviour of solution at characteristic points of the boundary is of special interest. The behaviour is well understood if a characteristic line is tangent to the boundary with contact degree at least 2. We allow the boundary to not only have contact of degree less than 2 with a characteristic line but also a cuspidal singularity at a characteristic point. We construct an asymptotic solution of the problem near the characteristic point to describe how the boundary layer degenerates.},
  language  = {en}
}
@unpublished{Baer2012,
  author    = {B{\"a}r, Christian},
  title     = {Some properties of solutions to weakly hypoelliptic equations},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-60064},
  year      = {2012},
  abstract  = {A linear differential operator L is called weakly hypoelliptic if any local solution u of Lu = 0 is smooth. We allow for systems, i.e. the coefficients may be matrices, not necessarily of square size. This is a huge class of important operators which covers all elliptic, overdetermined elliptic, subelliptic and parabolic equations. We extend several classical theorems from complex analysis to solutions of any weakly hypoelliptic equation: the Montel theorem providing convergent subsequences, the Vitali theorem ensuring convergence of a given sequence, and Riemann's first removable singularity theorem. In the case of constant coefficients we show that Liouville's theorem holds, any bounded solution must be constant and any L^p solution must vanish.},
  language  = {en}
}
@unpublished{Baer2012,
  author    = {B{\"a}r, Christian},
  title     = {Renormalized integrals and a path integral formula for the heat kernel on a manifold},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-60052},
  year      = {2012},
  abstract  = {We introduce renormalized integrals which generalize conventional measure theoretic integrals. One approximates the integration domain by measure spaces and defines the integral as the limit of integrals over the approximating spaces. This concept is implicitly present in many mathematical contexts such as Cauchy's principal value, the determinant of operators on a Hilbert space and the Fourier transform of an L^p function. We use renormalized integrals to define a path integral on manifolds by approximation via geodesic polygons. The main part of the paper is dedicated to the proof of a path integral formula for the heat kernel of any self-adjoint generalized Laplace operator acting on sections of a vector bundle over a compact Riemannian manifold.},
  language  = {en}
}
@unpublished{PfaeffleStephan2012,
  author    = {Pf{\"a}ffle, Frank and Stephan, Christoph A.},
  title     = {Chiral asymmetry and the spectral action},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-60046},
  year      = {2012},
  abstract  = {We consider orthogonal connections with arbitrary torsion on compact Riemannian manifolds. For the induced Dirac operators, twisted Dirac operators and Dirac operators of Chamseddine-Connes type we compute the spectral action. In addition to the Einstein-Hilbert action and the bosonic part of the Standard Model Lagrangian we find the Holst term from Loop Quantum Gravity, a coupling of the Holst term to the scalar curvature and a prediction for the value of the Barbero-Immirzi parameter.},
  language  = {en}
}
@unpublished{PfaeffleStephan2012,
  author    = {Pf{\"a}ffle, Frank and Stephan, Christoph A.},
  title     = {The Holst action by the spectral action principle},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-60032},
  year      = {2012},
  abstract  = {We investigate the Holst action for closed Riemannian 4-manifolds with orthogonal connections. For connections whose torsion has zero Cartan type component we show that the Holst action can be recovered from the heat asymptotics for the natural Dirac operator acting on left-handed spinor fields.},
  language  = {en}
}
@unpublished{BaerBallmann2012,
  author    = {B{\"a}r, Christian and Ballmann, Werner},
  title     = {Boundary value problems for elliptic differential operators of first order},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-60023},
  year      = {2012},
  abstract  = {We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instance. We provide a selfcontained introduction to (nonlocal) elliptic boundary conditions, boundary regularity of solutions, and index theory. In particular, we simplify and generalize the traditional theory of elliptic boundary value problems for Dirac type operators. We also prove a related decomposition theorem, a general version of Gromov and Lawson's relative index theorem and a generalization of the cobordism theorem.},
  language  = {en}
}
@unpublished{BaerPfaeffle2012,
  author    = {B{\"a}r, Christian and Pf{\"a}ffle, Frank},
  title     = {Wiener measures on Riemannian manifolds and the Feynman-Kac formula},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-59998},
  year      = {2012},
  abstract  = {This is an introduction to Wiener measure and the Feynman-Kac formula on general Riemannian manifolds for Riemannian geometers with little or no background in stochastics. We explain the construction of Wiener measure based on the heat kernel in full detail and we prove the Feynman-Kac formula for Schr{\"o}dinger operators with bounded potentials. We also consider normal Riemannian coverings and show that projecting and lifting of paths are inverse operations which respect the Wiener measure.},
  language  = {en}
}
@unpublished{PfaeffleStephan2012,
  author    = {Pf{\"a}ffle, Frank and Stephan, Christoph A.},
  title     = {On gravity, torsion and the spectral action principle},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-59989},
  year      = {2012},
  abstract  = {We consider compact Riemannian spin manifolds without boundary equipped with orthogonal connections. We investigate the induced Dirac operators and the associated commutative spectral triples. In case of dimension four and totally anti-symmetric torsion we compute the Chamseddine-Connes spectral action, deduce the equations of motions and discuss critical points.},
  language  = {en}
}