Explicit sharp estimates for the Green function of the Laplacian in C domains were completed in 1986 by Zhao [42]. Sharp estimates of the Green function of Lipschitz domains were given in 2000 by Bogdan [6]. Explicit qualitatively sharp estimates for the classical heat kernel in C domains were established in 2002 by Zhang [41]. Qualitatively sharp heat kernel estimates in Lipschitz domains were...

The fractional Laplacian operator (−∆) on a bounded domain Ω can be realized as a Dirichlet-to-Neumann map for a degenerate elliptic equation posed in the semi-infinite cylinder Ω × (0,∞). In fact, the Neumann trace on Ω involves a Muckenhoupt weight that, according to the fractional exponent s, either vanishes (s < 1/2) or blows up (s > 1/2). On the other hand, the normal trace of the solution...

We perform an asymptotic analysis of models of population dynamics with a fractional Laplacian and local or nonlocal reaction terms. The first part of the paper is devoted to the long time/long range rescaling of the fractional Fisher-KPP equation. This rescaling is based on the exponential speed of propagation of the population. In particular we show that the only role of the fractional Laplac...

We prove an asymptotic estimate for the growth of variational eigenvalues of fractional p-Laplacian eigenvalue problems on a smooth bounded domain.

We review some results about nonlocal advection-diffusion equations based on lower bounds for the fractional Laplacian. To Haim, with respect and admiration.

In this paper, we consider a Lewy-Stampacchia-type inequality for the fractional Laplacian on bounded domain in Euclidean space. Using inequality, can show well-posedness of fractional-type anomalous unidirectional diffusion equations. This study is an extension work by Akagi-Kimura (2019) standard Laplacian. However, there exist several difficulties due to nonlocal...

We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem in perforated domains.

We obtain a Struwe type global compactness result for a class of nonlinear nonlocal problems involving the fractional p−Laplacian operator and nonlinearities at critical growth.

Keywords: fractal geometry, fractional derivative, fractional Fourier transform, fractional power of a matrix, self similarity, complex partial differential equation, broadband ultrasound, frequency-dependent attenuation, time domain. 1. Backgrounds The rational behind this model is schematically illustrated below: Fractal geometry (irregular soft tissues) → Fractional Fourier transform (freque...