Abstract. We prove Strichartz estimates with a loss of derivatives for the Schrödinger equation on polygonal domains with either Dirichlet or Neumann homogeneous boundary conditions. Using a standard doubling procedure, estimates on the polygon follow from those on Euclidean surfaces with conical singularities. We develop a Littlewood-Paley squarefunction estimate with respect to the spectrum of the Laplacian on these spaces. This allows us to reduce matters to proving estimates at each frequency scale. The problem can be localized in space provided the time intervals are sufficiently small. Strichartz estimates then follow from a recent result of the second author regarding the Schrödinger equation on the Euclidean cone.
Abstract. Let (X, g) be a compact manifold with conic singularities. Taking ∆g to be the Friedrichs extension of the Laplace-Beltrami operator, we examine the singularities of the trace of the half-wave group e −it √ ∆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. IntroductionIn this paper, we consider the trace of the half-wave group U(t) def = e −it √ ∆g on a compact manifold with conic singularities (X, g). Our main result is a description of the singularities of this trace at the lengths of closed geodesics undergoing diffractive interaction with the cone points. Under the generic assumption that the cone points of X are pairwise nonconjugate along the geodesic flow, the resulting singularity at such a length t = L has the oscillatory integral representationwhere the amplitude a is to leading orderas |ξ| → ∞ and the index j is cyclic in {1, . . . , k}. Here, n is the dimension of X and k the number of diffractions along the geodesic, and χ is a smooth function supported in [1, ∞) and equal to 1 on [2, ∞). L 0 is the primitive length, in case the geodesic is an iterate of a shorter closed geodesic. The product is over the diffractions undergone by the geodesic, with D αj a quantity determined by the functional calculus of the Laplacian on the link of the j-th cone point Y αj , the factor Θ − 1 2 (Y αj → Y αj+1 ) is (at least on a formal level) the determinant of the differential of the flow between the j-th and (j + 1)-st cone points, and m γj is the Morse index of the geodesic segment γ j from the j-th to (j + 1)-st cone points. All of these factors are described in more detail below.To give this result some context, we recall the known results for the Laplace- is a well-defined distribution on R t . Moreover, it satisfies the "Poisson relation": it is singular only on the length spectrum of (X, g),L is the length of a closed geodesic in (X, g)} .(This was shown independently by Chazarain [Cha74]; see also [CdV73].) Subject to a nondegeneracy hypothesis, the singularity at the length t = ±L of a closed geodesic has a specific leading form encoding the geometry of that geodesic-the formula involves the linearized Poincaré map and the Morse index of the geodesic. The proofs of these statements center around the identificationwhere∆g is (half of) the propagator for solutions to the wave equation on X; in particular, U(t) is a Fourier integral operator.In this paper, we prove an analogue of the Duistermaat-Guillemin trace formula on compact manifolds with conic singularities (or "conic manifolds"), generalizing results of Hillairet [Hil05] from the case of flat surfaces with conic singularities. We again consider the trace Tr U(t), a spectral invariant equal to ∞ j=0 e −itλj . The Poiss...
We consider the solution operator for the wave equation on the flat Euclidean cone over the circle of radius ρ > 0, the manifold R + × R 2πρZ equipped with the metric g(r, θ) = dr 2 + r 2 dθ 2 . Using explicit representations of the solution operator in regions related to flat wave propagation and diffraction by the cone point, we prove dispersive estimates and hence scale invariant Strichartz estimates for the wave equation on flat cones. We then show that this yields corresponding inequalities on wedge domains, polygons, and Euclidean surfaces with conic singularities. This in turn yields well-posedness results for the nonlinear wave equation on such manifolds. Morawetz estimates on the cone are also treated.
We investigate the singularities of the trace of the half-wave group, Tr e −it √ ∆ , on Euclidean surfaces with conical singularities (X, g). We compute the leading-order singularity associated to periodic orbits with successive degenerate diffractions. This result extends the previous work of the third author [Hil05] and the two-dimensional case of the work of the first author and Wunsch [FW] as well as the seminal result of Duistermaat and Guillemin [DG75] in the smooth setting. As an intermediate step, we identify the wave propagators on X as singular Fourier integral operators associated to intersecting Lagrangian submanifolds, originally developed by Melrose and Uhlmann [MU79].
The problem relates to the transient vibration of a symmetrical, continuous, simply supported two-span beam which is traversed by a constant force moving with constant velocity. The beam is of slender proportions, flexure alone being considered. Damping is zero, and there is no mass associated with the moving force. Exact theoretical solutions for bending stress have been derived in general form. They consist of three infinite series, each related to one of three time eras as follows: (a) Where force is crossing first span; (b) is crossing second span; (c) has left the beam. Each term of a series is related to a natural mode of vibration. Quantitative theoretical studies show the variation in individual terms of the series, and also in summations of the first five terms, as the traversing velocity is varied. A mechanical model with electrical recording of stress was employed to obtain a more complete quantitative solution than was feasible analytically. The agreement between theory and experiment was reasonably good. Large magnifications of stress (of the order of 2.5) were found in the neighborhood of resonance with the fundamental mode.
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