9,107 research outputs found

    Multiple boundary peak solutions for some singularly perturbed Neumann problems

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    We consider the problem \left \{ \begin{array}{rcl} \varepsilon^2 \Delta u - u + f(u) = 0 & \mbox{ in }& \ \Omega\\ u > 0 \ \mbox{ in} \ \Omega, \ \frac{\partial u}{\partial \nu} = 0 & \mbox{ on }& \ \partial\Omega, \end{array} \right. where \Omega is a bounded smooth domain in R^N, \varepsilon>isasmallparameterandfisasuperlinear,subcriticalnonlinearity.Itisknownthatthisequationpossessesboundaryspikesolutionssuchthatthespikeconcentrates,asεapproacheszero,atacriticalpointofthemeancurvaturefunctionH(P),PΩ.ItisalsoknownthatthisequationhasmultipleboundaryspikesolutionsatmultiplenondegeneratecriticalpointsofH(P)ormultiplelocalmaximumpointsofH(P).Inthispaper,weprovethatforanyfixedpositiveinteger is a small parameter and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses boundary spike solutions such that the spike concentrates, as \varepsilon approaches zero, at a critical point of the mean curvature function H(P), P \in \partial \Omega . It is also known that this equation has multiple boundary spike solutions at multiple nondegenerate critical points of H(P) or multiple local maximum points of H(P). In this paper, we prove that for any fixed positive integer Kthereexistboundary there exist boundary K-peaksolutionsatalocalminimumpointof solutions at a local minimum point of H(P).Thisimpliesthatforanysmoothandboundeddomaintherealwaysexistboundary. This implies that for any smooth and bounded domain there always exist boundary K-peak$ solutions. We first use the Liapunov-Schmidt method to reduce the problem to finite dimensions. Then we use a maximizing procedure to obtain multiple boundary spikes

    Generalised CP and A4A_4 Family Symmetry

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    We perform a comprehensive study of family symmetry models based on A4A_4 combined with the generalised CP symmetry HCPH_{\rm{CP}}. We investigate the lepton mixing parameters which can be obtained from the original symmetry A4HCPA_4\rtimes H_{\rm{CP}} breaking to different remnant symmetries in the neutrino and charged lepton sectors. We find that only one case is phenomenologically viable, namely GCPνZ2S×HCPνG^{\nu}_{\rm{CP}}\cong Z^{S}_2\times H^{\nu}_{\rm{CP}} in the neutrino sector and GCPlZ3THCPlG^{l}_{\rm{CP}}\cong Z^{T}_3\rtimes H^{l}_{\rm{CP}} in the charged lepton sector, leading to the prediction of no CP violation, namely δCP\delta_{CP} and the Majorana phases α21\alpha_{21} and α31\alpha_{31} are all equal to either zero or π\pi. We then propose an effective supersymmetric model based on the symmetry A4HCPA_4\rtimes H_{\rm{CP}} in which trimaximal lepton mixing is predicted together with either zero CP violation or δCP±π/2\delta_{CP}\simeq\pm \pi/2 with non-trivial Majorana phases. An ultraviolet completion of the effective model yields a neutrino mass matrix which depends on only three real parameters. As a result of this, all three CP phases and the absolute neutrino mass scale are determined, the atmospheric mixing angle is maximal, and the Dirac CP can either be preserved with δCP=0,π\delta_{CP}=0,\pi or maximally broken with δCP=±π/2\delta_{CP}=\pm \pi /2 and sharp predictions for the Majorana phases and neutrinoless double beta decay.Comment: 38 pages, 3 figure

    Generalized μτ\mu-\tau reflection symmetry and leptonic CP violation

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    We propose a generalized μτ\mu-\tau reflection symmetry to constrain the lepton flavor mixing parameters. We obtain a new correlation between the atmospheric mixing angle θ23\theta_{23} and the "Dirac" CP violation phase δCP\delta_{\rm CP}. Only in a specific limit our proposed CP transformation reduces to standard μτ\mu-\tau reflection, for which θ23\theta_{23} and δCP\delta_{CP} are both maximal. The "Majorana" phases are predicted to lie at their CP-conserving values with important implications for the neutrinoless double beta decay rates. We also study the phenomenological implications of our scheme for present and future neutrino oscillation experiments including T2K, NOν\nuA and DUNE.Comment: 14 pages, 9 figures, latex, Final version to appear in Physics Letters

    Electron-positron pair creation in a vacuum by an electromagnetic field in 3+1 and lower dimensions

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    We calculate the probability of electron-positron pair creation in vacuum in 3+1 dimensions by an external electromagnetic field composed of a constant uniform electric field and a constant uniform magnetic field, both of arbitrary magnitudes and directions. The same problem is also studied in 2+1 and 1+1 dimensions in appropriate external fields and similar results are obtained.Comment: REVTeX, 10 pages, no figure, a brief note and some more references added in the proo

    Scattering by a contact potential in three and lower dimensions

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    We consider the scattering of nonrelativistic particles in three dimensions by a contact potential Ω2δ(r)/2μrα\Omega\hbar^2\delta(r)/ 2\mu r^\alpha which is defined as the a0a\to 0 limit of Ω2δ(ra)/2μrα\Omega\hbar^2\delta(r-a)/2\mu r^\alpha. It is surprising that it gives a nonvanishing cross section when α=1\alpha=1 and Ω=1\Omega=-1. When the contact potential is approached by a spherical square well potential instead of the above spherical shell one, one obtains basically the same result except that the parameter Ω\Omega that gives a nonvanishing cross section is different. Similar problems in two and one dimensions are studied and results of the same nature are obtained.Comment: REVTeX, 9 pages, no figur

    Phase-field-crystal modeling of the (2x1)-(1x1) phase-transitions of Si(001) and Ge(001) surfaces

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    We propose a two-dimensional phase-field-crystal model for the (2×\times1)-(1×\times1) phase transitions of Si(001) and Ge(001) surfaces. The dimerization in the 2×\times1 phase is described with a phase-field-crystal variable which is determined by solving an evolution equation derived from the free energy. Simulated periodic arrays of dimerization variable is consistent with scanning-tunnelling-microscopy images of the two dimerized surfaces. Calculated temperature dependence of the dimerization parameter indicates that normal dimers and broken ones coexist between the temperatures describing the charactristic temperature width of the phase-transition, TLT_L and THT_H, and a first-order phase transition takes place at a temperature between them. The dimerization over the whole temperature is determined. These results are in agreement with experiment. This phase-field-crystal approach is applicable to phase-transitions of other reconstructed surface phases, especially semiconductor n×n\times1 reconstructed surface phases.Comment: 10 pages with 4 figures include
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