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We study the energy-critical focusing non-linear wave equation, with data in the energy space, in dimensions 3, 4 and 5. We prove that for Cauchy data of energy smaller than the one of the static solution$W$which gives the best constant in the Sobolev embedding, the following alternative holds. If the initial data has smaller norm in the homogeneous Sobolev space$H$^{1}than the one of$W$, then we have global well-posedness and scattering. If the norm is larger than the one of$W$, then we have break-down in finite time.

This paper deals with a free boundary porblem connected with the concept “quadrature surface”. Let Ω⊂$R$^{$n$}be a bounded domain with a$C$^{2}boundary and μ a measure compactly supported in Ω. Then we say ∂Ω is a quadrature surface with respect to μ if the following overdetermined Cauchy problem has a solution. $$\Delta u = - \mu in \Omega ,u = 0 and \frac{{\partial u}}{{\partial v}} = - 1 on \partial \Omega .$$

In the relative trace formula approach to the arithmetic Gan–Gross–Prasad conjecture, we formulate a local conjecture (arithmetic transfer) in the case of an exotic smooth formal moduli space of p-divisible groups, associated to a unitary group relative to a ramified quadratic extension of a p-adic field. We prove our conjecture in the case of a unitary group in three variables.

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