Short talks

This talk concerns Dirichlet and Neumann boundary value problems for second order elliptic divergence form equations with non-symmetric, or more general complex, and measurable coefficients independent of the transversal coordinate in the upper half space. I shall discuss some recent results obtained which shows that, in the case of non-symmetric coefficients, the solutions obtained with the boundary equation method may differ from the solutions obtained with Lax–Milgram’s method. Surprisingly, for sufficiently non-symmetric coefficients, the boundary equation solutions for the Dirichlet problem are given by a signed harmonic measure. The variational solutions in the examples given, have earlier been studied by Kenig et al. I shall also talk on recent joint work with Auscher and McIntosh. This gives perturbation results for well posedness of the boundary value problems for complex coefficients, and uniqueness of the solutions. Badr, Nadine (Université Paris-Sud, France): Real Interpolation of Sobolev spaces on metric-measure spaces. Abstract: We study the real interpolation of Sobolev spaces on metric-measure spaces. We prove that the nonhomogeneous Sobolev spaces W 1 p (resp. homogeneous Ẇ 1 p ) defined on a Riemannian manifold satisfying a doubling property and a Poincaré inequality, form a real interpolation scale on an interval of values of p. We extend this result to other geometric frames. By means of an example, we study the necessity of Poincaré inequality to the interpolation. We study the real interpolation of Sobolev spaces on metric-measure spaces. We prove that the nonhomogeneous Sobolev spaces W 1 p (resp. homogeneous Ẇ 1 p ) defined on a Riemannian manifold satisfying a doubling property and a Poincaré inequality, form a real interpolation scale on an interval of values of p. We extend this result to other geometric frames. By means of an example, we study the necessity of Poincaré inequality to the interpolation. Baklouti, Ali (Faculty of Sciences of Sfax, Tunisia): On Hardy’s Theorem on Solvable Lie groups. Abstract: Let G be a solvable connected and simply connected Lie group. It is well known that Hardy’s Theorem holds when G is nilpotent. In this talk, we show that the result fails to hold for general exponential Lie groups. As such, we get a positive answer whenever the center of the group in question is not trivial. We finally show that the last condition is not necessary when G is merely solvable. This is a joint work with E. Kanuith. Let G be a solvable connected and simply connected Lie group. It is well known that Hardy’s Theorem holds when G is nilpotent. In this talk, we show that the result fails to hold for general exponential Lie groups. As such, we get a positive answer whenever the center of the group in question is not trivial. We finally show that the last condition is not necessary when G is merely solvable. This is a joint work with E. Kanuith. Ben Ali, Besma (University of Paris XI, France): Maximal inequalities and Riesz transform estimates on Lp spaces for Schrödinger operators . Abstract: We give various estimates for the Schrödinger operator We give various estimates for the Schrödinger operator