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We consider a family of spherically symmetric, asymptotically Euclidean manifolds with two trapped sets, one which is unstable and one which is semi-stable. The phase space structure is that of an inflection transmission set. We prove a sharp local smoothing estimate for the linear Schrodinger equation with a loss which depends on how flat the manifold is near each of the trapped sets. The result interpolates between the family of similar estimates in cite{ChWu-lsm}. As a consequence of the techniques of proof, we also show a sharp high energy resolvent estimate with a polynomial loss depending on how flat the manifold is near each of the trapped sets.
We prove a local smoothing result for the Schrodinger equation on a class of surfaces of revolution which have infinitely many trapped geodesics. Our main result is a local smoothing estimate with loss (compared to cite{ChMe-lsm}) depending on the ac
Let $(X,g)$ be a compact manifold with conic singularities. Taking $Delta_g$ to be the Friedrichs extension of the Laplace-Beltrami operator, we examine the singularities of the trace of the half-wave group $e^{- i t sqrt{ smash[b]{Delta_g}}}$ arisin
Consider the transmission eigenvalue problem [ (Delta+k^2mathbf{n}^2) w=0, (Delta+k^2)v=0 mbox{in} Omega;quad w=v, partial_ u w=partial_ u v=0 mbox{on} partialOmega. ] It is shown in [12] that there exists a sequence of eigenfunctions $(w_m, v_
Let $M$ be a complete, simply connected Riemannian manifold with negative curvature. We obtain some Moser-Trudinger inequalities with sharp constants on $M$.
The paper is devoted to a comprehensive study of smoothness of inertial manifolds for abstract semilinear parabolic problems. It is well known that in general we cannot expect more than $C^{1,varepsilon}$-regularity for such manifolds (for some posit