ﻻ يوجد ملخص باللغة العربية
This paper presents a highly non-trivial two-fold study of the fractional differential couples - derivatives ($ abla^{0<s<1}_+=(-Delta)^frac{s}{2}$) and gradients ($ abla^{0<s<1}_-= abla (-Delta)^frac{s-1}{2}$) of basic importance in the theory of fractional advection-dispersion equations: one is to discover the sharp Hardy-Rellich ($sp<p<n$) $|$ Adams-Moser ($sp=n$) $|$ Morrey-Sobolev ($sp>n$) inequalities for $ abla^{0<s<1}_pm$; the other is to handle the distributional solutions $u$ of the duality equations $[ abla^{0<s<1}_pm]^ast u=mu$ (a nonnegative Radon measure) and $[ abla^{0<s<1}_pm]^ast u=f$ (a Morrey function).
The sharp trace inequality of Jose Escobar is extended to traces for the fractional Laplacian on R^n and a complete characterization of cases of equality is discussed. The proof proceeds via Fourier transform and uses Liebs sharp form of the Hardy-Littlewood-Sobolev inequality.
We prove matrix and scalar differential Harnack inequalities for linear parabolic equations on Riemannian and Kahler manifolds.
We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions. This is essential for modelling using spatial fractional derivatives. We show well-posednes
In this paper, we establish concentration inequalities both for functionals of the whole solution on an interval [0, T ] of an additive SDE driven by a fractional Brownian motion with Hurst parameter H $in$ (0, 1) and for functionals of discrete-time
In this paper we classify all positive extremal functions to a sharp weighted Sobolev inequality on the upper half space, which involves divergent operators with degeneracy on the boundary. As an application of the results, we can derive a sharp Sobo