6,401 research outputs found
Serrin's Overdetermined Problem and Constant Mean Curvature Surfaces
For all , we find smooth entire epigraphs in , namely smooth
domains of the form , which are not half-spaces and in which a problem of the form
in has a positive, bounded solution with 0
Dirichlet boundary data and constant Neumann boundary data on . This answers negatively for large dimensions a question by Berestycki,
Caffarelli and Nirenberg \cite{bcn2}. In 1971, Serrin \cite{serrin} proved that
a bounded domain where such an overdetermined problem is solvable must be a
ball, in analogy to a famous result by Alexandrov that states that an embedded
compact surface with constant mean curvature (CMC) in Euclidean space must be a
sphere. In lower dimensions we succeed in providing examples for domains whose
boundary is close to large dilations of a given CMC surface where Serrin's
overdetermined problem is solvable.Comment: 59 page
Unified Flavor Symmetry from warped dimensions
We propose a scenario which accommodates all the masses and mixings of the SM
fermions in a model of warped extra-dimensions with all matter fields in the
bulk. In this scenario, the same flavor symmetric structure is imposed on all
the fermions of the Standard Model (SM), including neutrinos. Due to the
exponential sensitivity on bulk fermion masses, a small breaking of the
symmetry can be greatly enhanced and produce seemingly un-symmetric
hierarchical masses and small mixing angles among the charged fermion
zero-modes (SM quarks and charged leptons) and wash-out the obvious effects of
the symmetry. With the Higgs field leaking into the bulk, and Dirac neutrinos
sufficiently localized towards the UV boundary, the neutrino mass hierarchy and
flavor structure will still be largely dominated by the fundamental flavor
structure. The neutrino sector would then reflect the fundamental flavor
structure, whereas the quark sector would probe the effects of the flavor
symmetry breaking sector. As an example, we explore these features in the
context of a family permutation symmetry imposed in both quark and lepton
sectors.Comment: 5 pages, 1 figur
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