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Register a new userHigher order fractional Leibniz rule
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The fractional Leibniz rule is generalized by the Coifman-Meyer estimate. It is shown that the arbitrary redistribution of fractional derivatives for higher order with the corresponding correction terms.
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Taylor series is a useful mathematical tool when describing and constructing a function. With the series representation, some properties of fractional calculus can be revealed clearly. This paper investigates two typical applications: Lebiniz rule and Laplace transform. It is analytically shown that the commonly used Leibniz rule cannot be applied for Caputo derivative. Similarly, the well-known Laplace transform of Riemann-Liouville derivative is doubtful for n-th continuously differentiable function. By the aid of this series representation, the exact formula of Caputo Leibniz rule and the explanation of Riemann-Liouville Laplace transform are presented. Finally, three illustrative examples are revisited to confirm the obtained results.
We study the bilinear estimates in the Sobolev spaces with the Dirichlet and the Neumann boundary condition. The optimal regularity is revealed to get such estimates in the half space case, which is related to not only smoothness of functions and but also boundary behavior. The crucial point for the proof is how to handle boundary values of functions and their derivatives.
In this paper we explore the theory of fractional powers of non-negative (and not necessarily self-adjoint) operators and its amazing relationship with the Chebyshev polynomials of the second kind to obtain results of existence, regularity and behavior asymptotic of solutions for linear abstract evolution equations of $n$-th order in time, where $ngeqslant3$. We also prove generalizations of classical results on structural damping for linear systems of differential equations.
In this paper we study a variational Neumann problem for the higher order $s$-fractional Laplacian, with $s>1$. In the process, we introduce some non-local Neumann boundary conditions that appear in a natural way from a Gauss-like integration formula.
Spectral measurements of boundary localized in-gap modes are commonly used to identify topological insulators via the bulk-boundary correspondence. This can be extended to high-order topological insulators for which the most striking feature is in-gap modes at boundaries of higher co-dimension, e.g. the corners of a 2D material. Unfortunately, this spectroscopic approach is not always viable since the energies of the topological modes are not protected and they can often overlap the bulk bands, leading to potential misidentification. Since the topology of a material is a collective product of all its eigenmodes, any conclusive indicator of topology must instead be a feature of its bulk band structure, and should not rely on specific eigen-energies. For many topological crystalline insulators the key topological feature is fractional charge density arising from the filled bulk bands, but measurements of charge distributions have not been accessible to date. In this work, we experimentally measure boundary-localized fractional charge density of two distinct 2D rotationally-symmetric metamaterials, finding 1/4 and 1/3 fractionalization. We then introduce a new topological indicator based on collective phenomenology that allows unambiguous identification of higher-order topology, even in the absence of in-gap states. Finally, we demonstrate the higher-order bulk-boundary correspondence associated with this fractional feature by using boundary deformations to spectrally isolate localized corner modes where they were previously unobservable.
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