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Schrödinger equation on the Heisenberg group

Jacek Zienkiewicz (2004)

Studia Mathematica

Solvability of invariant sublaplacians on spheres and group contractions

Fulvio Ricci, Jérémie Unterberger (2001)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In the first part of this paper we study the local and global solvability and the hypoellipticity of a family of left-invariant sublaplacians $L_{α}$ on the spheres $S^{2 n + 1} ≃ U (n + 1) / U (n)$ . In the second part, we introduce a larger family of left-invariant sublaplacians $L_{α, β}$ on $S^{3} ≃ S U (2)$ and study the corresponding properties by means of a Lie group contraction to the Heisenberg group.

Some Examples of Hypoelliptic Partial Differential Equations.

A. Menikoff (1976)

Mathematische Annalen

Sous-ellipticité et hypoellipticité maximale pour des systèmes différentiels

Henri-Michel Maire (1980)

Journées équations aux dérivées partielles

Stability of mappings with bounded distortion in the Sobolev norm on the John domains of Heisenberg groups.

Isangulova, D.V. (2009)

Sibirskij Matematicheskij Zhurnal

Stochastic calculus and degenerate boundary value problems

Patrick Cattiaux (1992)

Annales de l'institut Fourier

Consider the boundary value problem (L.P): $(h - A) u = f$ in $D$ , $(v - Γ) u = g$ on $\partial D$ where $A$ is written as $A = 1 / 2 \sum_{i = 1}^{m} Y_{i}^{2} + Y_{0}$ , and $Γ$ is a general Venttsel’s condition (including the oblique derivative condition). We prove existence, uniqueness and smoothness of the solution of (L.P) under the Hörmander’s condition on the Lie brackets of the vector fields $Y_{i}$ ( $0 \leq i \leq m$ ), for regular open sets $D$ with a non-characteristic boundary.Our study lies on the stochastic representation of $u$ and uses the stochastic calculus of variations for the $(A, Γ)$ -diffusion process...