On Nonlinear Biharmonic Problems on the Heisenberg Group (original) (raw)

Picone’s identity for biharmonic operators on Heisenberg group and its applications

Nonlinear Differential Equations and Applications NoDEA, 2016

In this paper, we establish a nonlinear analogue of Picone's identity for biharmonic operators on Heisenberg group. As an applications of Picone's identity, we obtain Hardy-Rellich type inequality, Morse index, Caccioppoli inequality, Picone inequality for biharmonic operators on Heisenberg group.

A short survey on biharmonic maps between Riemannian manifolds

2006

While the above variational problems are natural generalizations of harmonic maps and minimal immersions, biharmonic Riemannian immersions do not recover Willmore immersions, even when the ambient space is R n . Therefore, the two generalizations give rise to different variational problems.

Neumann boundary value problem in domains of the Heisenberg Group mathbbHn\mathbb H_nmathbbHn

Existence and uniqueness of the solution of the Neumann problem for the Kohn-Laplacian on the Kor\'anyi ball of the Heisenberg group mathbbHn\mathbb{H}_nmathbbHn are discussed. Explicit representations of Green's type function (Neumann function) for the half space and Kor\'anyi ball in mathbbHn\mathbb{H}_nmathbbHn for circular functions have been obtained. These functions are then used on above regions in mathbbHn\mathbb{H}_nmathbbHn to solve the inhomogeneous Neumann boundary value problem for circular data.

GEOMETRIC MECHANICS ON THE HEISENBERG GROUP

2000

We give detailed discussion of subRiemannian geometry which arised from the sub-LaplacianH on the Heisen- berg group. In particular, we calculate the subRiemannian dis- tances along the geodesics. We also find the complex action func- tion and the volume element on the group. Using this action func- tion and the volume element, we obtain the fundamental solution and the heat

DIV–CURL TYPE THEOREM, H-CONVERGENCE AND STOKES FORMULA IN THE HEISENBERG GROUP

Communications in Contemporary Mathematics, 2006

The original purpose of the present paper was somehow different from the final outcome the reader will find below. In fact, the research was meant to attack the notion of Hconvergence in the Heisenberg group and to produce optimal estimates for the so-called effective coefficients in homogenization theory. The notion of H-convergence goes back to Murat and Tartar in the 70's, and fits in the literature between G-convergence for differential equations and Γ-convergence for variational functionals. In fact, H-convergence provides a particular notion of matrix convergence, which is appropriate for instance to study some homogenization problems ([33], [6]).

Generalizations of a Laplacian-Type Equation in the Heisenberg Group and a Class of Grushin-Type Spaces

arXiv (Cornell University), 2011

In [2], Beals, Gaveau and Greiner find the fundamental solution to a 2-Laplace-type equation in a class of sub-Riemannian spaces. This solution is related to the well-known fundamental solution to the p-Laplace equation in Grushin-type spaces [5] and the Heisenberg group [6]. We extend the 2-Laplace-type equation to a p-Laplacetype equation. We show that the obvious generalization does not have desired properties, but rather, our generalization preserves some natural properties.

Some results and examples of the biharmonic maps with potential

Arab Journal of Mathematical Sciences, 2018

In this paper, we will study the class of biharmonic maps with potential, in the particular case represented by conformal maps between equidimensional manifolds. Some examples are constructed in particular cases (Euclidean space and sphere).