Semi-linear Sub-elliptic Equations on the Heisenberg Group with a Singular Potential

ar X iv : 0 70 8 . 36 55 v 1 [ m at h . A P ] 2 7 A ug 2 00 7 GRADIENT REGULARITY FOR ELLIPTIC EQUATIONS IN THE HEISENBERG GROUP

Abstract. We give dimension-free regularity conditions for a class of possibly degenerate sub-elliptic equations in the Heisenberg group exhibiting super-quadratic growth in the horizontal gradient; this solves an issue raised in [40], where only dimension dependent bounds for the growth exponent are given. We also obtain explicit a priori local regularity estimates, and cover the case of the h...

GGMRES: A GMRES--type algorithm for solving singular linear equations with index one

In this paper, an algorithm based on the Drazin generalized conjugate residual (DGMRES) algorithm is proposed for computing the group-inverse solution of singular linear equations with index one. Numerical experiments show that the resulting group-inverse solution is reasonably accurate and its computation time is significantly less than that of group-inverse solution obtained by the DGMRES alg...

Existence of Solution of Sub-elliptic Equations on the Heisenberg Group with Critical Growth and Double Singularities

Adaptive Methods for Semi-linear Elliptic Equations with Critical Exponents and Interior Singularities Adaptive Methods for Semi-linear Elliptic Equations with Critical Exponents and Interior Singularities

We consider the eeectiveness of adaptive nite-element methods for nding the nite element solutions of the parametrised semi-linear elliptic equation u+u+u 5 = 0; u > 0, where u 2 C 2 ((); for a domain IR 3 and u = 0 on the boundary of : This equation is important in analysis and it is known that there is a value 0 > 0 such that no solutions exist for < 0 and a singularity forms as ! 0. Furtherm...

Explicit Equations of Some Elliptic Modular Surfaces

We present explicit equations of semi-stable elliptic surfaces (i.e., having only type In singular fibers) which are associated to the torsion-free genus zero congruence subgroups of the modular group as classified by A. Sebbar.