We study the sub-Finsler prescribed mean curvature equation, associated to a strictly convex body K0⊆R2n, for t-graphs on a bounded domain Ω in the Heisenberg group Hn. When the prescribed datum H is ...constant and strictly smaller than the Finsler mean curvature of ∂Ω, we prove the existence of a Lipschitz solution to the Dirichlet problem for the sub-Finsler CMC equation by means of a Finsler approximation scheme.
We study the obstacle problem related to a wide class of nonlinear integro-differential operators, whose model is the fractional subLaplacian in the Heisenberg group. We prove both the existence and ...uniqueness of the solution, and that solutions inherit regularity properties of the obstacle such as boundedness, continuity and Hölder continuity up to the boundary. We also prove some independent properties of weak supersolutions to the class of problems we are dealing with. Armed with the aforementioned results, we finally investigate the Perron–Wiener–Brelot generalized solution by proving its existence for very general boundary data.
We study the magnetic trajectories in the generalized Heisenberg group H(n,1) of dimension (2n+1) endowed with its quasi-Sasakian structure. We prove that the trajectories are Frenet curves of ...maximum order 5 and we completely classify them.
In this paper, we consider a class of the critical Kirchhoff–Poisson systems in the Heisenberg group. Under suitable assumptions on the Kirchhoff function and on the nonlinear terms, the existence of ...multiple solutions is obtained by using the symmetric mountain pass theorem. Moreover, our results are new even in the Euclidean case.
This paper is devoted to the derivation of L2 - L2 decay estimates for the solution of the homogeneous linear damped wave equation on the Heisenberg group Hn, for its time derivative and for its ...horizontal gradient. Moreover, we consider the improvement of these estimates when further L1(Hn) regularity is required for the Cauchy data. Our approach will rely strongly on the group Fourier transform of Hn and on the properties of the Hermite functions that form a maximal orthonormal system for L2(Rn) of eigenfunctions of the harmonic oscillator.
In Euclidean 3-space, it is well known that the Sine-Gordon equation was considered in the nineteenth century in the course of investigations of surfaces of constant Gaussian curvature K=−1. Such a ...surface can be constructed from a solution to the Sine-Gordon equation, and vice versa. With this as motivation, employing the fundamental theorem of surfaces in the Heisenberg group H1, we show in this paper that the existence of a constant p-mean curvature surface (without singular points) is equivalent to the existence of a solution to a nonlinear second-order ODE (1.2), which is a kind of Liénard equations. Therefore, we turn to investigate this equation. It is a surprise that we give a complete set of solutions to (1.2) (or (1.5)), and hence use the types of the solution to divide constant p-mean curvature surfaces into several classes. As a result, after a kind of normalization, we obtain a representation of constant p-mean curvature surfaces and classify further all constant p-mean curvature surfaces. In Section 9, we provide an approach to construct p-minimal surfaces. It turns out that, in some sense, generic p-minimal surfaces can be constructed via this approach. Finally, as a derivation, we recover the Bernstein-type theorem which was first shown in 14 (or see 19,20).
In this paper, we establish the existence of extremals for two kinds of Stein–Weiss inequalities on the Heisenberg group. More precisely, we prove the existence of extremals for the Stein–Weiss ...inequalities with full weights in Theorem 1.1 and the Stein–Weiss inequalities with horizontal weights in Theorem 1.4. Different from the proof of the analogous inequality in Euclidean spaces given by Lieb 26 using Riesz rearrangement inequality which is not available on the Heisenberg group, we employ the concentration compactness principle to obtain the existence of the maximizers on the Heisenberg group. Our result is also new even in the Euclidean case because we don't assume that the exponents of the double weights in the Stein–Weiss inequality (1.1) are both nonnegative (see Theorem 1.3 and more generally Theorem 1.5). Therefore, we extend Lieb's celebrated result of the existence of extremal functions of the Stein–Weiss inequality in the Euclidean space to the case where the exponents are not necessarily both nonnegative (see Theorem 1.3). Furthermore, since the absence of translation invariance of the Stein–Weiss inequalities, additional difficulty presents and one cannot simply follow the same line of Lions' idea to obtain our desired result. Our methods can also be used to obtain the existence of optimizers for several other weighted integral inequalities (Theorem 1.5).
We find fundamental solutions to p-Laplace equations with drift terms in the Heisenberg group and Grushin-type planes. These solutions are natural generalizations of the fundamental solutions ...discovered by Beals, Gaveau, and Greiner for the Laplace equation with drift term. Our results are independent of the results of Bieske and Childers, in that Bieske and Childers consider a generalization that focuses on the p-Laplace-type equation while we primarily concentrate on a generalization of the drift term.
For more information see https://ejde.math.txstate.edu/Volumes/2021/99/abstr.html
In this paper, we deal with a class of Kirchhoff-type critical elliptic equations involving the Formula: see text-sub-Laplacians operators on the Heisenberg group of the form M(∥DHu∥pp + ∥u∥ p,Vp)−Δ ...H,pu + V (ξ)|u|p−2u = λf(ξ,u) + |u|p∗−2u,ξ ∈ ℍn,u ∈HWV1,p(ℍn), where Formula: see text is the Formula: see text-sub-Laplacian, Formula: see text, Formula: see text is the horizontal Sobolev space on Formula: see text. And Formula: see text is the homogeneous dimension of Formula: see text, Formula: see text is a real parameter, Formula: see text is the critical Sobolev exponent on the Heisenberg group. Under some proper assumptions on the Kirchhoff function Formula: see text, the potential function Formula: see text and Formula: see text, together with the mountain pass theorem and the concentration-compactness principles for classical Sobolev spaces on the Heisenberg group, we prove the existence and multiplicity of nontrivial solutions for the above problem in non-degenerate and degenerate cases on the Heisenberg group. The results of this paper extend or complete recent papers and are new in several directions for the non-degenerate and degenerate critical Kirchhoff equations involving the Formula: see text-Laplacian type operators on the Heisenberg group.
In this note we focus our attention on a stochastic heat equation defined on the Heisenberg group Hn of order n. This equation is written as ∂tu=12Δu+uW˙α, where Δ is the hypoelliptic Laplacian on Hn ...and {W˙α;α>0} is a family of Gaussian space-time noises which are white in time and have a covariance structure generated by (−Δ)−α in space. Our aim is threefold: (i) Give a proper description of the noise Wα; (ii) Prove that one can solve the stochastic heat equation in the Itô sense as soon as α>n2; (iii) Give some basic moment estimates for the solution u(t,x).
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