Abstract
We will discuss here problems which lead to integro-differential forms which involve derivatives of first order. Such a problem is called elliptic when the L2-norms of all first derivatives can be estimated by the corresponding form. In [4] it is shown that it suffices to estimate the ‖ ‖ε-norm (with 0<ε≤1) in order to establish smoothness of solutions, discretness of spectrum and other properties such problems have in common with the elliptic case. Here we consider the case when the L2-norms of only some derivatives are bounded.
The following is a special case of an estimate proved by Hörmander in 1. The proof that we give here uses only elementary properties of pseudo-differential operators.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
HöRMANDER, L., “Hypoelliptic second order differential equations” Acta Math. Vol. 119 (1967), 147–171.
KOHN, J.J;, “Boundaries of complex manifolds”. Proc. Conf. Complex Analysis (Minneapolis 1964), 81–94. Springer Verlag, Berlin. 1965.
KOHN, J.J. and ROSSI, H., “On the extension of holomorphic functions from the the boundary of a complex manifold”. Ann. Math. vol. 81 (1965), 451–472.
KOHN, J.J. and NIRENBERG, L., “Non-Coercive boundary value problems”. Conn. P. A. Math. vol. 18 (1965), 443–492.
KOHN, J.J. and NIRENBERG, L., “An algebra of pseudo-differential operators”. Conn. P. A. vol 18 (1965), 269–305.
Editor information
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Kohn, J.J. (2010). Pseudo-Differential Operators and Non-Elliptic Problems. In: Nirenberg, L. (eds) Pseudo-differential Operators. C.I.M.E. Summer Schools, vol 47. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11074-0_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-11074-0_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-11073-3
Online ISBN: 978-3-642-11074-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)