Related papers: A partial solution of the isoperimetric problem fo…

In this paper, we consider the Dirichlet problem for a class of Hessian quotient equations on Riemannian manifolds. Under the assumption of an admissible subsolution, we solve the existence and the uniquness for the Dirichlet problem in a…

Symmetries of finite Heisenberg groups represent an important tool for the study of deeper structure of finite-dimensional quantum mechanics. This short contribution presents extension of previous investigations to composite quantum systems…

In this short exposition we provide a simplified proof of Buser's result for Cheeger's isoperimetric constant.

We prove asymptotically isometric, coarsely geodesic metrics on a toral relatively hyperbolic group are coarsely equal. The theorem applies to all lattices in SO(n,1). This partly verifies a conjecture by Margulis. In the case of hyperbolic…

We show that there exists an algorithm to decide any single equation in the Heisenberg group in finite time. The method works for all two-step nilpotent groups with rank-one commutator, which includes the higher Heisenberg groups. We also…

We prove the isodiametric inequality in the spherical and in the hyperbolic space

We prove a monotonicity result at specific points for the Horizontal Perimeter for a class of surfaces in the Heisenberg group.

Algebraic derivation of modified Heisenberg commutation rules for restricted Landau problem is given.

An improved a.e. lower bound is given for Hausdorff dimension under vertical projections in the first Heisenberg group.

We consider the isoperimetric problem in planar sectors with density $r^{p}$, and with density $a>1$ inside the unit disk and $1$ outside. We characterize solutions as a function of sector angle. We also solve the isoperimetric problem in…

In this paper, we consider the Neumann problem for parabolic Hessian quotient equations. We show that the $k$-admissible solution of the parabolic Hessian quotient equation exists for all time and converges to the smooth solution of…

We study the horizontal mean curvature flow in the Heisenberg group by using the level-set method. We prove the uniqueness, existence and stability of axisymmetric viscosity solutions of the level-set equation. An explicit solution is given…

We show that metric bisectors with respect to the Kor\'anyi metric in the Heisenberg group are spinal spheres and vice versa. We also calculate explicitly their horizontal mean curvature.

An isoperimetric upper bound on the resistance is given. As a corollary we resolve two problems, regarding mean commute time on finite graphs and resistance on percolation clusters. Further conjectures are presented.

We study the following nonlocal diffusion equation in the Heisenberg group $\mathbb{H}_n$, \[ u_t(z,s,t)=J\ast u(z,s,t)-u(z,s,t), \] where $\ast$ denote convolution product and $J$ satisfies appropriated hypothesis. For the Cauchy problem…

We consider two families of Lorentzian problems on the Heisenberg group and their asymptotic behaviour as the parameter of a family tends to a limit.

We prove that the Folland's fundamental solution for the sub-Laplacian on Heisenberg groups can also be derived form the resolvent kernel of this sub-Laplacian. This provides us with a new integral representation for this fundamental…

In this paper we propose a conjecture concerning partial sums of an arbitrary finite subset of an abelian group, that naturally arises investigating simple Heffter systems. Then, we show its connection with related open problems and we…

We derive two equivalent definitions of the viscosity solutions to the homogeneous sub-p- Laplace parabolic equations on the Heisenberg group, and characterize the viscosity solutions in terms of an asymptotic mean value formula. Moreover,…

In this note, we establish the existence of a positive solution and its stability to the following problem $$\Delta_{\mathbb{H}^n}^2u=a(\xi)u-f(\xi,u)\text{ in }\Omega, \,\,\, u|_{\partial\Omega} = 0 =\left.\Delta_{\mathbb{H}^n}…