Volume 39, Issue 2

We prove global and local versions of the so-called div{\operatorname{div}}-curl{\operatorname{curl}}-lemma, a crucial result in the homogenization theory of partial differential equations, for mixed boundary conditions on bounded weak Lipschitz domains in 3D with weak Lipschitz interfaces. We will generalize our results using an abstract Hilbert space setting, which shows corresponding results to hold in arbitrary dimensions as well as for various differential operators. The crucial tools and the core of our arguments are Hilbert complexes and related compact embeddings.

We give another proof of Poisson’s integral formula for harmonic functions in a ball or a half space by using heat kernels with Green’s formula. We wish to emphasize that this method works well even for a half space, which is an unbounded domain; the functions involved are integrable, since the heat kernel decays rapidly. This method needs no trick such as the subordination identity, which is indispensable when applying the Fourier transform method for a half space.

The main aim of this paper is to investigate the solution of generalized fractional kinetic equations involving the Rnm{{}_{m}{R}_{n}} function by using the Laplace transform. Also, we use the differential property of Rnm{{}_{m}{R}_{n}} to solve kinetic equations. The obtained results are expressed in terms of generalized K -functions. The graphical interpretations of the solutions are also presented in this study.