141 research outputs found

    Calculus of Variations (hybrid meeting)

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    Calculus of Variations touches several interrelated areas. In this workshop we covered several topics, such as minimal submanifolds, mean curvature and related flows, free boundary problems, variational models of interacting dislocations, defects in physical systems, phase transitions, etc

    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

    Topological and Variational Methods for Differential Equations

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    These notes contain the extended abstracts of the talks presented at the workshop. The range of topics includes nonlinear SchroĢˆdinger equations, singularly perturbed equations, symmetry and nodal properties of solutions, long-time dynamics for parabolic equations, Morse theory

    Homoclinic snaking in bounded domains

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    Homoclinic snaking is a term used to describe the back and forth oscillation of a branch of time-independent spatially localized states in a bistable, spatially reversible system as the localized structure grows in length by repeatedly adding rolls on either side. On the real line this process continues forever. In finite domains snaking terminates once the domain is filled but the details of how this occurs depend critically on the choice of boundary conditions. With periodic boundary conditions the snaking branches terminate on a branch of spatially periodic states. However, with non-Neumann boundary conditions they turn continuously into a large amplitude filling state that replaces the periodic state. This behavior, shown here in detail for the Swift-Hohenberg equation, explains the phenomenon of ā€œsnaking without bistabilityā€, recently observed in simulations of binary fluid convection by Mercader, Batiste, Alonso and Knobloch (preprint)

    The ground state of a Grossā€“Pitaevskii energy with general potential in the Thomasā€“Fermi limit

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    We study the ground state which minimizes a Grossā€“Pitaevskii energy with general non-radial trapping potential, under the unit mass constraint, in the Thomasā€“Fermi limit where a small parameter tends to 0. This ground state plays an important role in the mathematical treatment of recent experiments on the phenomenon of Boseā€“Einstein condensation, and in the study of various types of solutions of nonhomogeneous defocusing nonlinear Schrodinger equations. Many of these applications require delicate estimates for the behavior of the ground state near the boundary of the condensate, as the singular parameter tends to zero, in the vicinity of which the ground state has irregular behavior in the form of a steep corner layer. In particular, the role of this layer is important in order to detect the presence of vortices in the small density region of the condensate, understand the superļ¬‚uid ļ¬‚ow around an obstacle, and also has a leading order contribution in the energy. In contrast to previous approaches, we utilize a perturbation argument to go beyond the classical Thomasā€“Fermi approximation and accurately approximate the layer by the Hastingsā€“McLeod solution of the Painleveā€“II equation. This settles an open problem, answered very recently only for the special case of the model harmonic potential. In fact, we even improve upon previous results that relied heavily on the radial symmetry of the potential trap. Moreover, we show that the ground state has the maximal regularity available, namely it remains uniformly bounded in the 1/2-Holder norm, which is the exact Holder regularity of the singular limit proļ¬le, as the singular parameter tends to zero. Our study is highly motivated by an interesting open problem posed recently by Aftalion, Jerrard, and Royo-Letelier, and an open question of Gallo and Pelinovsky, concerning the removal of the radial symmetry assumption from the potential trap

    Steady states and dynamics of an autocatalytic chemical reaction model with decay

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    The dynamics and steady state solutions of an autocatalytic chemical reaction model with decay in the catalyst are considered. Nonexistence and existence of nontrivial steady state solutions are shown by using energy estimates, upper-lower solution method, and bifurcation theory. The effects of decay order, decay rate and diffusion rates to the dynamical behavior are discussed. (c) 2012 Published by Elsevier Inc
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