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- ItemA logistic equation with nonlocal interactions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Caffarelli, Luis; Dipierro, Serena; Outrata, Jir̆íWe consider here a logistic equation, modeling processes of nonlocal character both in the diffusion and proliferation terms. More precisely, for populations that propagate according to a Levy process and can reach resources in a neighborhood of their position, we compare (and find explicit threshold for survival) the local and nonlocal case. As ambient space, we can consider: bounded domains, periodic environments, transition problems, where the environment consists of a block of infinitesimal diffusion and an adjacent nonlocal one. In each of these cases, we analyze the existence/nonexistence of solutions in terms of the spectral properties of the domain. In particular, we give a detailed description of the fact that nonlocal populations may better adapt to sparse resources and small environments.
- ItemGraph properties for nonlocal minimal surfaces(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Dipierro, Serena; Savin, Ovidiu; Valdinoci, EnricoIn this paper we show that a nonlocal minimal surface which is a graph outside a cylinder is in fact a graph in the whole of the space. As a consequence, in dimension 3, we show that the graph is smooth. The proofs rely on convolution techniques and appropriate integral estimates which show the pointwise validity of an Euler-Lagrange equation related to the nonlocal mean curvature.
- ItemMehler-Heine asymptotics of a class of generalized hypergeometric polynomials(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Bracciali, Cleonice F.; Moreno-Balcázar, Juan JoséWe obtain a Mehler–Heine type formula for a class of generalized hypergeometric polynomials. This type of formula describes the asymptotics of polynomials scale conveniently. As a consequence of this formula, we obtain the asymptotic behavior of the corresponding zeros. We illustrate these results with numerical experiments and some figures.
- ItemAn asymptotic analysis for a generalized Cahn--Hilliard system with fractional operators(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Colli, Pierluigi; Gilardi, Gianni; Sprekels, JürgenIn a recent paper the same authors have proved existence, uniqueness and regularity results for a class of viscous and nonviscous Cahn--Hilliard systems of two operator equations in which nonlinearities of double-well type, like regular or logarithmic potentials, as well as nonsmooth potentials with indicator functions, were admitted. The operators appearing in the system equations are fractional powers in the spectral sense of general linear operators, which are densely defined, unbounded, selfadjoint, and monotone in the Hilbert space of square-integrable functions on a bounded and smooth three-dimensional domain, and have compact resolvents. Here, for the case of the viscous system, we analyze the asymptotic behavior of the solution as the fractional power coefficient of the second operator tends to zero. We prove convergence to a phase relaxation problem at the limit, and we also investigate this limiting problem, in which an additional term containing the projection of the phase variable on the kernel of the second operator appears.
- ItemError control for the approximation of Allen-Cahn and Cahn-Hilliard equations with a logarithmic potential(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Bartels, Sören; Müller, RüdigerA fully computable upper bound for the finite element approximation error of Allen-Cahn and Cahn-Hilliard equations with logarithmic potentials is derived. Numerical experiments show that for the sharp interface limit this bound is robust past topological changes. Modifications of the abstract results to derive quasi-optimal error estimates in different norms for lowest order finite element methods are discussed and lead to weaker conditions on the residuals under which the conditional error estimates hold.
- ItemStrong solutions for the interaction of a rigid body and a viscoelastic fluid(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Götze, KarolineWe study a coupled system of equations describing the movement of a rigid body which is immersed in a viscoelastic fluid. It is shown that under natural assumptions on the data and for general goemetries of the rigid body, excluding contact scenarios, a unique local-in-time strong solution exists.
- ItemEinsatz von 3D-Seismik zur Reduzierung des Fündigkeitsrisikos bei Geothermieprojekten : Endbericht ; Laufzeit des Vorhabens: 01.11.2006 - 30.06.2011(Hannover : Technische Informationsbibliothek (TIB), 2011) Schulz, Rüdiger; Buness, Hermann; Beilecke, Thies; von Hartmann, Hartwig; Musmann, Patrick; Bauer, Stefan; Donath, Andreas; Rüter, Horst[no abstract available]
- ItemDiffraction of stochastic point sets : exactly solvable examples(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Baake, Michael; Birkner, Matthias; Moody, Robert V.Stochastic point sets are considered that display a diffraction spectrum of mixed type, with special emphasis on explicitly computable cases together with a unified approach of reasonable generality. Several pairs of autocorrelation and diffraction measures are discussed that show a duality structure that may be viewed as analogues of the Poisson summation formula for lattice Dirac combs.
- ItemEnergy consistent discontinuous Galerkin methods for a quasi-incompressible diffuse two phase flow model(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Giesselmann, Jan; Pryer, TristanWe design consistent discontinuous Galerkin finite element schemes for the approximation of a quasi-incompressible two phase flow model of AllenCahn/CahnHilliard/NavierStokesKorteweg type which allows for phase transitions.We show that the scheme is mass conservative and monotonically energy dissipative. In this case the dissipation is isolated to discrete equivalents of those effects already causing dissipation on the continuous level, that is, there is no artificial numerical dissipation added into the scheme. In this sense the methods are consistent with the energy dissipation of the continuous PDE system.
- ItemKernmagnetische Ordnung in Metallen mit starker Hyperfeinwechselwirkung : Schlussbericht(Dresden : Forschungszentrum Dresden-Rossendorf, 2000) Pobell, Frank[no abstract available]