Horizontal convex envelope in the Heisenberg group and applications to sub-elliptic equations

Regularity Results for Quasilinear Elliptic Equations in the Heisenberg Group

We prove regularity results for solutions to a class of quasilinear elliptic equations in divergence form in the Heisenberg group H. The model case is the nondegenerate p-Laplacean operator

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

In this work, we study the Dirichlet problem for a class of semi-linear subelliptic equations on the Heisenberg group with a singular potential. The singularity is controlled by Hardy’s inequality, and the nonlinearity is controlled by Sobolev’s inequality. We prove the existence of a nontrivial solution for a homogenous Dirichlet problem.

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

Convex Functions on the Heisenberg Group

Convex functions in Euclidean space can be characterized as universal viscosity subsolutions of all homogeneous fully nonlinear second order elliptic partial differential equations. This is the starting point we have chosen for a theory of convex functions on the Heisenberg group.

Vertical versus Horizontal Poincaré Inequalities on the Heisenberg Group

Let H = 〈a, b | a[a, b] = [a, b]a ∧ b[a, b] = [a, b]b〉 be the discrete Heisenberg group, equipped with the left-invariant word metric dW (·, ·) associated to the generating set {a, b, a−1, b−1}. Letting Bn = {x ∈ H : dW (x, eH) 6 n} denote the corresponding closed ball of radius n ∈ N, and writing c = [a, b] = aba−1b−1, we prove that if (X, ‖·‖X) is a Banach space whose modulus of uniform conve...