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Register a new userOn the fractional Laplacian of variable order
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We present a novel definition of variable-order fractional Laplacian on Rn based on a natural generalization of the standard Riesz potential. Our definition holds for values of the fractional parameter spanning the entire open set (0, n/2). We then discuss some properties of the fractional Poissons equation involving this operator and we compute the corresponding Green function, for which we provide some instructive examples for specific problems.
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We consider an evolution equation with the Caputo-Dzhrbashyan fractional derivative of order $alpha in (1,2)$ with respect to the time variable, and the second order uniformly elliptic operator with variable coefficients acting in spatial variables. This equation describes the propagation of stress pulses in a viscoelastic medium. Its properties are intermediate between those of parabolic and hyperbolic equations. In this paper, we construct and investigate a fundamental solution of the Cauchy problem, prove existence and uniqueness theorems for such equations.
In the whole space $mathbb R^d$, linear estimates for heat semi-group in Besov spaces are well established, which are estimates of $L^p$-$L^q$ type, maximal regularity, e.t.c. This paper is concerned with such estimates for semi-group generated by the Dirichlet Laplacian of fractional order in terms of the Besov spaces on an arbitrary open set of $mathbb R^d$.
We consider the three-dimensional Laplacian with a magnetic field created by an infinite rectilinear current bearing a constant current. The spectrum of the associated hamiltonian is the positive half-axis as the range of an infinity of band functions all decreasing toward 0. We make a precise asymptotics of the band function near the ground energy and we exhibit a semi-classical behavior. We perturb the hamiltonian by an electric potential. Helped by the analysis of the band functions, we show that for slow decaying potential, an infinite number of negative eigenvalues are created whereas only finite number of eigenvalues appears for fast decaying potential. Our results show different borderline type conditions that in the case where there is no magnetic field.
We investigate diffusion equations with time-fractional derivatives of space-dependent variable order. We examine the well-posedness issue and prove that the space-dependent variable order coefficient is uniquely determined among other coefficients of these equations, by the knowledge of a suitable time-sequence of partial Dirichlet-to-Neumann maps.
We propose two asymptotic expansions of the two interrelated integral-type averages, in the context of the fractional $infty$-Laplacian $Delta_infty^s$ for $sin (frac{1}{2},1)$. This operator has been introduced and first studied in [Bjorland-Caffarelli-Figalli, 2012]. Our expansions are parametrised by the radius of the removed singularity $epsilon$, and allow for the identification of $Delta_infty^sphi(x)$ as the $epsilon^{2s}$-order coefficient of the deviation of the $epsilon$-average from the value $phi(x)$, in the limit $epsilonto 0+$. The averages are well posed for functions $phi$ that are only Borel regular and bounded.
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