16 research outputs found

    On selection criteria for problems with moving inhomogeneities

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    We study mechanical problems with multiple solutions and introduce a thermodynamic framework to formulate two different selection criteria in terms of macroscopic energy productions and fluxes. Studying simple examples for lattice motion we then compare the implications for both resting and moving inhomogeneities.Comment: revised version contains new introduction, numerical simulations of Riemann problems, and a more detailed discussion of the causality principle; 18 pages, several figure

    Kinetic relations for a lattice model of phase transitions

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    Microscopic Hamiltonian Systems and their Effective Description

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    Uniform estimates for positive solutions of semilinear elliptic equations and related Liouville and one-dimensional symmetry results

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    We consider a semilinear elliptic equation with Dirichlet boundary conditions in a smooth, possibly unbounded, domain. Under suitable assumptions, we deduce a condition on the size of the domain that implies the existence of a positive solution satisfying a uniform pointwise estimate. Here, uniform means that the estimate is independent of the domain. The main advantage of our approach is that it allows us to remove a restrictive monotonicity assumption that was imposed in the recent paper. In addition, we can remove a non-degeneracy condition that was assumed in the latter reference. Furthermore, we can generalize an old result, concerning semilinear elliptic nonlinear eigenvalue problems. Moreover, we study the boundary layer of global minimizers of the corresponding singular perturbation problem. For the above applications, our approach is based on a refinement of a result, concerning the behavior of global minimizers of the associated energy over large balls, subject to Dirichlet conditions. Combining this refinement with global bifurcation theory and the sliding method, we can prove uniform estimates for solutions away from their nodal set. Combining our approach with a-priori estimates that we obtain by blow-up, a doubling lemma, and known Liouville type theorems, we can give a new proof of a known Liouville type theorem without using boundary blow-up solutions. We can also provide an alternative proof, and a useful extension, of a Liouville theorem, involving the presence of an obstacle. Making use of the latter extension, we consider the singular perturbation problem with mixed boundary conditions. Moreover, we prove some new one-dimensional symmetry and rigidity properties of certain entire solutions to Allen-Cahn type equations, as well as in half spaces, convex cylindrical domains. In particular, we provide a new proof of Gibbons' conjecture in two dimensions.Comment: Corrected the subsection on Gibbon's conjecture: As it is, our Gibbons' conjecture proof works only in two dimension

    Universal properties of self-organized localized structures

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    Ziel dieser Arbeit ist es, einen Beitrag zur AufklĂ€rung der universellen teilchenartigen Eigenschaften von selbstorganisierten lokalisierten Strukturen in unterschiedlichen rĂ€umlich ausgedehnten Systemen zu leisten. Untersucht werden zunĂ€chst Mechanismen der Bildung, Dynamik und Wechselwirkung in Prototyp-Modellsystemen wie Reaktions-Diffusions-, Ginzburg-Landau- und Swift-Hohenberg-Gleichungen. Mittels adiabatischer Eliminationsmethoden kann die Feldbeschreibung in vielen FĂ€llen auf gewöhnliche Differentialgleichungen von einheitlicher Form reduziert werden. So kann auch das Verhalten großer Ensembles lokalisierter Strukturen analytisch und numerisch analysiert werden. Hier ergeben sich zahlreiche neue PhĂ€nomene ohne klassisches Analogon. Zuletzt ist die Dynamik von biologischen teilchenartigen Strukturen in Form von Dictyostelium-Zellen von Interesse, die experimentell in mikrofluidischen Aufbauten erforscht wird. This work tries to contribute to explaining the universal particle-like behavior of self-organized localized structures in different spatially extended systems. First, mechanisms of generation, dynamics and interaction in prototype systems like reaction-diffusion, Ginzburg-Landau or Swift-Hohenberg equations are investigated. Using methods of adiabatic elimination, in many cases the field description can be reduced to ordinary differential equations of uniform structure. In this way, one may also explore the behavior of large particle ensembles analytically and numerically. Here, many new phenomena without classical analogue arise. Finally, an experimental analysis of biological particle-like structures in the form of Dictyostelium cells is carried out using microfluidic devices
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