Ngô Quốc Anh

Talks

  1. Positive solutions for a class of semilinear elliptic systems via the dual variational method [2006]. In this talk, we consider the existence of non-trivial solutions for semilinear elliptic systems with n-equations on a bounded domain of \mathbb{R}^N, with zero Dirichlet boundary conditions

    -\Delta u + Au = f(u),
    u\left| {_{\partial \Omega } } \right. = 0,

    where

    u = \left( {u_1 ,u_2 ,..,u_n } \right), f\left( u \right) = \left( {f_1 \left( {u_1 } \right),f_2 \left( {u_2 } \right),..,f_n \left( {u_n } \right)} \right), {f_k \left( {u_k } \right)} (k=1,2,..,n)

    are nonlinear functions for u_k defined in \Omega and A = \left( {a_{ij} } \right)_{n \times n} is a matrix of real entries satisfying a_{ij} = a_{ji} for all i \ne j. Here, we use the dual variational method.

  2. An application of dual variational method to semilinear elliptic systems on a bounded domain [Feb, 2007].
  3. Morse theory and several applications to partial differential equations [May, 2007].
  4. Scalar curvatures of manifolds with negative conformal invariant [Oct, 2009].
  5. The constraint equations for the Einstein-scalar field system on manifolds [Jan, 2010]. While much is understood about constant mean curvature solutions of the constraint equation for the vacuum Einstein, Einstein–Maxwell and Einstein-Yang-Mills fields, much less is known about the solutions of  constant mean curvature equation for the Einstein-scalar field. It is well known that the conformal method can be applied in the Einstein-scalar case. The difficulty is that  the Lichnerowicz equation includes terms which are not seen in other cases, and which need more refined analysis. In this talk, we show how to overcome some of these difficulties for a compact manifold. Our approach allows us to include the vacuum constraint equations with an arbitrary cosmological constant, the Yamabe equation and even (all cases of) the prescribed scalar curvature problem as special cases. We also discuss a conjecture on Einstein-scalar field constraints on asymptotically hyperboloidal case and other related questions.
  6. The vacuum Einstein constraint equations with freely specified mean curvature [Oct, 2010]. The conformal method of solving the Einstein constraint equations is remarkably effective when the mean curvature is constant, and is remarkably recalcitrant when it is not. In this talk, we will discuss some progresses towards our understanding of the non-CMC case.
  7. Existence results for the Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds in the null case [Oct 14; Oct 21, 2011]. While the conformal method can be effectively applied for solving the Einstein constraint equations in most cases, it should be pointed out that there are several cases for which either partial result or no result was achieved, especially when gravity is coupled to field sources. Based on a division recently obtained in <Class. Quantum Grav. 24 (2007), pp. 808-828>, one can observe that there are two cases corresponding to the non-positive Yamabe-scalar field conformal invariant with sign-changing nonlinearities, for which no result was achieved. This is basically due to the fact that the Einstein-scalar field Lichnerowicz equation includes terms which are not seen in other cases, which need more refined analysis. In this talk, we show how to overcome those difficulties for compact manifolds by developing a new approach that suits for our analysis. During the talk, some non-existence, existence, and multiplicity results are also presented.

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