2,512 research outputs found

    Dynamics of the heat semigroup in Jacobi analysis

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    Let Δ\Delta be the Jacobi Laplacian. We study the chaotic and hypercyclic behaviour of the strongly continuous semigroups of operators generated by perturbations of Δ\Delta with a multiple of the identity on LpL^p spaces

    A Bayesian numerical homogenization method for elliptic multiscale inverse problems

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    A new strategy based on numerical homogenization and Bayesian techniques for solving multiscale inverse problems is introduced. We consider a class of elliptic problems which vary at a microscopic scale, and we aim at recovering the highly oscillatory tensor from measurements of the fine scale solution at the boundary, using a coarse model based on numerical homogenization and model order reduction. We provide a rigorous Bayesian formulation of the problem, taking into account different possibilities for the choice of the prior measure. We prove well-posedness of the effective posterior measure and, by means of G-convergence, we establish a link between the effective posterior and the fine scale model. Several numerical experiments illustrate the efficiency of the proposed scheme and confirm the theoretical findings

    Paley--Wiener Theorems for the U(n)--spherical transform on the Heisenberg group

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    We prove several Paley--Wiener-type theorems related to the spherical transform on the Gelfand pair (Hn⋊U(n),U(n))\big(H_n\rtimes U(n),U(n)\big), where HnH_n is the 2n+12n+1-dimensional Heisenberg group. Adopting the standard realization of the Gelfand spectrum as the Heisenberg fan in R2{\mathbb R}^2, we prove that spherical transforms of U(n) U(n)--invariant functions and distributions with compact support in HnH_n admit unique entire extensions to C2{\mathbb C}^2, and we find real-variable characterizations of such transforms. Next, we characterize the inverse spherical transforms of compactly supported functions and distributions on the fan, giving analogous characterizations

    A weighted anisotropic Sobolev type inequality and its applications to Hardy inequalities

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    In this paper we focus our attention on an embedding result for a weighted Sobolev space that involves as weight the distance function from the boundary taken with respect to a general smooth gauge function FF. Starting from this type of inequalities we prove some refined Hardy-type inequalities

    Optimal Szeg\"o-Weinberger type inequalities

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    Denote with ÎŒ1(Ω;eh(∣x∣))\mu_{1}(\Omega;e^{h\left(|x|\right)}) the first nontrivial eigenvalue of the Neumann problem \begin{equation*} \left\{\begin{array}{lll} -\text{div}\left(e^{h\left(|x|\right)}\nabla u\right) =\mu e^{h\left(|x|\right)}u & \text{in} & \Omega & & \frac{\partial u}{\partial \nu}=0 & \text{on} & \partial \Omega , \end{array} \right. \end{equation*} where Ω\Omega is a bounded and Lipschitz domain in RN\mathbb{R}^{N}. Under suitable assumption on hh we prove that the ball centered at the origin is the unique set maximizing ÎŒ1(Ω;eh(∣x∣))\mu_{1}(\Omega;e^{h\left(|x|\right)}) among all Lipschitz bounded domains Ω\Omega of RN\mathbb{R}^{N} of prescribed eh(∣x∣)dxe^{h\left(|x|\right)}dx-measure and symmetric about the origin. Moreover, an example in the model case h(∣x∣)=∣x∣2,h\left(|x|\right) =|x|^{2}, shows that, in general, the assumption on the symmetry of the domain cannot be dropped. In the one-dimensional case, i.e. when Ω\Omega reduces to an interval (a,b),(a,b), we consider a wide class of weights (including both Gaussian and anti-Gaussian). We then describe the behavior of the eigenvalue as the interval (a,b)(a,b) slides along the xx-axis keeping fixed its weighted length
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