## Abstract and Applied Analysis

- Abstr. Appl. Anal.
- Volume 2016 (2016), Article ID 7210540, 8 pages.

### Local Hypoellipticity by Lyapunov Function

#### Abstract

We treat the local hypoellipticity, in the first degree, for a class of abstract differential operators complexes; the ones are given by the following differential operators: ${L}_{j}=\partial /\partial {t}_{j}+(\partial \varphi /\partial {t}_{j})(t,A)A$, $j=\mathrm{1,2},\dots ,n$, where $A:D(A)\subset H\to H$ is a self-adjoint linear operator, positive with $\mathrm{0}\in \rho (A)$, in a Hilbert space $H$, and $\varphi =\varphi (t,A)$ is a series of nonnegative powers of ${A}^{-\mathrm{1}}$ with coefficients in ${C}^{\mathrm{\infty}}(\mathrm{\Omega})$, $\mathrm{\Omega}$ being an open set of ${\mathbb{R}}^{n}$, for any $n\in \mathbb{N}$, different from what happens in the work of Hounie (1979) who studies the problem only in the case $n=\mathrm{1}$. We provide sufficient condition to get the local hypoellipticity for that complex in the elliptic region, using a Lyapunov function and the dynamics properties of solutions of the Cauchy problem $t$′$(s)=-\nabla \mathrm{R}\mathrm{e}\mathrm{}{\varphi}_{\mathrm{0}}(t(s))$, $s\ge \mathrm{0}$, $t(\mathrm{0})={t}_{\mathrm{0}}\in \mathrm{\Omega},{\varphi}_{\mathrm{0}}:\mathrm{\Omega}\to \mathbb{C}$ being the first coefficient of $\varphi (t,A)$. Besides, to get over the problem out of the elliptic region, that is, in the points $t$^{∗} $\in \mathrm{\Omega}$ such that $\nabla \mathrm{R}\mathrm{e}{\varphi}_{\mathrm{0}}(t$^{∗}$)$ = 0, we will use the techniques developed by Bergamasco et al. (1993) for the particular operator $A=\mathrm{1}-\mathrm{\Delta}:{H}^{\mathrm{2}}({\mathbb{R}}^{N})\subset {L}^{\mathrm{2}}({\mathbb{R}}^{N})\to {L}^{\mathrm{2}}({\mathbb{R}}^{N})$.

#### Article information

**Source**

Abstr. Appl. Anal. Volume 2016 (2016), Article ID 7210540, 8 pages.

**Dates**

Received: 7 July 2015

Accepted: 20 December 2015

First available in Project Euclid: 10 February 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.aaa/1455115145

**Digital Object Identifier**

doi:10.1155/2016/7210540

**Mathematical Reviews number (MathSciNet)**

MR3454567

#### Citation

Aragão-Costa, E. R. Local Hypoellipticity by Lyapunov Function. Abstr. Appl. Anal. 2016 (2016), Article ID 7210540, 8 pages. doi:10.1155/2016/7210540. https://projecteuclid.org/euclid.aaa/1455115145