location:  Publications → journals
Search results

Search: All articles in the CJM digital archive with keyword fundamental solutions

 Expand all        Collapse all Results 1 - 2 of 2

1. CJM 2010 (vol 62 pp. 1116)

Jin, Yongyang; Zhang, Genkai
 Degenerate p-Laplacian Operators and Hardy Type Inequalities on H-Type Groups Let \$\mathbb G\$ be a step-two nilpotent group of H-type with Lie algebra \$\mathfrak G=V\oplus \mathfrak t\$. We define a class of vector fields \$X=\{X_j\}\$ on \$\mathbb G\$ depending on a real parameter \$k\ge 1\$, and we consider the corresponding \$p\$-Laplacian operator \$L_{p,k} u= \operatorname{div}_X (|\nabla_{X} u|^{p-2} \nabla_X u)\$. For \$k=1\$ the vector fields \$X=\{X_j\}\$ are the left invariant vector fields corresponding to an orthonormal basis of \$V\$; for \$\mathbb G\$ being the Heisenberg group the vector fields are the Greiner fields. In this paper we obtain the fundamental solution for the operator \$L_{p,k}\$ and as an application, we get a Hardy type inequality associated with \$X\$. Keywords:fundamental solutions, degenerate Laplacians, Hardy inequality, H-type groupsCategories:35H30, 26D10, 22E25

2. CJM 1999 (vol 51 pp. 673)

Barlow, Martin T.; Bass, Richard F.
 Brownian Motion and Harmonic Analysis on Sierpinski Carpets We consider a class of fractal subsets of \$\R^d\$ formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion \$X\$ and determine its basic properties; and extend some classical Sobolev and Poincar\'e inequalities to this setting. Keywords:Sierpinski carpet, fractal, Hausdorff dimension, spectral dimension, Brownian motion, heat equation, harmonic functions, potentials, reflecting Brownian motion, coupling, Harnack inequality, transition densities, fundamental solutionsCategories:60J60, 60B05, 60J35