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Jinggang Tan

Publications and source records attributed to Jinggang Tan.

6 recordsLinked to original sources
arXiv2025

Quantitative unique continuation property for fourth-order Baouendi-Grushin type subelliptic operators with a potential

We investigate the quantitative unique continuation property for solutions to $$Δ^2_{X} u = V u,$$ where $Δ_{X} = Δ_{x} + |x|^{2β} Δ_{y}$ ($0 < β\leq 1$), with $x \in \mathbb{R}^{m}$ and $y \in \mathbb{R}^{n}$, denotes a class of subelliptic operators of Baouendi-Grushin type. The potential $V$ is assumed to be bounded and satisfy $|Z V| \leq K ψ$ for some constant $K>0$, where $Z= \sum_{i=1}^m x_i \partial_{x_i} + (β+1)\sum_{j=1}^n y_j \partial_{y_j}$, $ψ$ is the angle function given by $ψ= \frac{|x|^{2β}}{ρ^{2β}}$, and $$ρ(x,y) = \left(|x|^{2(β+1)} + (β+1)^2 |y|^2\right)^{\frac{1}{2(β+1)}}$$ defines the associated pseudo-gauge. By adapting Almgren's approach, we establish an almost monotonicity formula for the frequency function. As a consequence, we derive a quantitative unique continuation result for solutions to the fourth-order subelliptic equation.

math.AP↗
arXiv2024

Decay estimates and Strichartz inequalities for a class of dispersive equations on H-type groups

Let $\mathcal{L}$ be the sub-Laplacian on H-type groups and $ϕ: \mathbb{R}^+ \to \mathbb{R}$ be a smooth function. The primary objective of the paper is to study the decay estimate for a class of dispersive semigroup given by $e^{itϕ(\mathcal{L})}$. Inspired by earlier work of Guo-Peng-Wang \cite{GPW2008} in the Euclidean space and Song-Yang \cite{SY2023} on the Heisenberg group, we overcome the difficulty arising from the non-homogeneousness of $ϕ$ by frequency localization, which is based on the non-commutative Fourier transform on H-type groups, the properties of the Laguerre functions and Bessel functions, and the stationary phase theorem. Finally, as applications, we derive the new Strichartz inequalities for the solutions of some specific equations, such as the fractional Schrödinger equation, the fourth-order Schrödinger equation, the beam equation and the Klein-Gordon equation, which corresponds to $ϕ(r)=r^α$, $r^2+r,\sqrt{1+r^2},\sqrt{1+r}$, respectively. Moreover, we also prove that the time decay is sharp in these cases.

math.AP↗
arXiv2015

A nonlinear Liouville theorem for fractional equations in the Heisenberg group

We establish a Liouville-type theorem for a subcritical nonlinear problem, involving a fractional power of the sub-Laplacian in the Heisenberg group. To prove our result we will use the local realization of fractional CR covariant operators, which can be constructed as the Dirichlet-to-Neumann operator of a degenerate elliptic equation in the spirit of Caffarelli and Silvestre, as established in \cite{FGMT}. The main tools in our proof are the CR inversion and the moving plane method, applied to the solution of the lifted problem in the half-space $\mathbb H^n\times \mathbbR^+$.

math.AP↗
arXiv2013

An extension problem for the CR fractional Laplacian

We show that the conformally invariant fractional powers of the sub-Laplacian on the Heisenberg group are given in terms of the scattering operator for an extension problem to the Siegel upper halfspace. Remarkably, this extension problem is different from the one studied, among others, by Caffarelli and Silvestre.

math.DG↗
arXiv2009

Positive solutions of nonlinear problems involving the square root of the Laplacian

We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of Gidas-Spruck type. In addition, among other results, we prove a symmetry theorem of Gidas-Ni-Nirenberg type.

math.AP↗