Regularity of solutions of a nonlinear Hörmander type equation

✍ Scribed by G. Citti

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 462 KB
• Volume: 47
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/s0362-546x(01)00193-6

✦ Synopsis

In this talk we prove a regularity result in the intrinsic directions for the solutions of the Levi equation in (\mathbb{R}^{3}). We denote ((x, y, t)) the points of the space, and represent the equation in the form

[
X^{2} u+Y^{2} u-(X a+Y b) \partial_{t} u=q\left(1+a^{2}+b^{2}\right)^{3 / 2}\left(1+u_{t}^{2}\right)^{1 / 2}
]

where (X) and (Y) are first order differential operators

[
X_{u}=\partial_{x}+a \partial_{t}, \quad Y_{u}=\partial_{y}+b \partial_{t}
]

and the coefficients (a) and (b) nonlinearly depend on the gradient. In the nonlinear situation afforded here the structure of the Lie algebra generated by (X) and (Y) is not known. Then we introduce a new iterative procedure, changing our Lie group at each step: for every (k) we prove that, if (u) belongs to a class (C_{d, k-1}^{k, \alpha}) of Hölder continuous functions of order (k+\alpha) with respect to a distance (d_{k-1}), then (u \in C_{d, k}^{k+1, \alpha}). In this way we prove that (u) is of class (C^{\infty}) in an intrinsic sense, with no assumptions on the curvature of its graph.