Abstract and Applied Analysis

Local Hypoellipticity by Lyapunov Function

E. R. Aragão-Costa

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
Accepted: 20 December 2015
First available in Project Euclid: 10 February 2016

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

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