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Register a new userSharp local smoothing for manifolds with smooth inflection transmission
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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.
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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 accumulation rate of the critical points of the profile curve. The proof uses an h-dependent version of semiclassical propagation of singularities, and a result on gluing an h-dependent number of cutoff resolvent estimates.
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 $e^{- i t sqrt{ smash[b]{Delta_g}}}$ arising from strictly diffractive closed geodesics. Under a generic nonconjugacy assumption, we compute the principal amplitude of these singularities in terms of invariants associated to the geodesic and data from the cone point. This generalizes the classical theorem of Duistermaat-Guillemin on smooth manifolds and a theorem of Hillairet on flat surfaces with cone points.
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 $(w_m, v_m)_{minmathbb{N}}$ associated with $k_mrightarrow infty$ such that either ${w_m}_{minmathbb{N}}$ or ${v_m}_{minmathbb{N}}$ are surface-localized, depending on $mathbf{n}>1$ or $0<mathbf{n}<1$. In this paper, we discover a new type of surface-localized transmission eigenmodes by constructing a sequence of transmission eigenfunctions $(w_m, v_m)_{minmathbb{N}}$ associated with $k_mrightarrow infty$ such that both ${w_m}_{minmathbb{N}}$ and ${v_m}_{minmathbb{N}}$ are surface-localized, no matter $mathbf{n}>1$ or $0<mathbf{n}<1$. Though our study is confined within the radial geometry, the construction is subtle and technical.
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 positive, but small $varepsilon$). Nevertheless, as shown in the paper, under the natural assumptions, the obstacles to the existence of a $C^n$-smooth inertial manifold (where $ninmathbb N$ is any given number) can be removed by increasing the dimension and by modifying properly the nonlinearity outside of the global attractor (or even outside the $C^{1,varepsilon}$-smooth IM of a minimal dimension). The proof is strongly based on the Whitney extension theorem.
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