Encyclopedia of Thermal Stresses

2014 Edition
| Editors: Richard B. Hetnarski

Linear Partial Differential Equations: Fundamental Solutions, Hypoellipticity, Local Existence

Reference work entry
DOI: https://doi.org/10.1007/978-94-007-2739-7_29
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Overview

This entry contains an introduction to the theory of linear partial differential equations. We introduce the notion of fundamental solution for the case with constant coefficients and sketch the proof of the fact that any nontrivial operator of this type has one. We show some applications of fundamental solutions to the study of existence and regularity of solutions of linear partial differential operators with constant coefficients. Then we construct fundamental solutions for the basic operators of mathematical physics. We conclude with the celebrated Lewy’s counterexample, showing that the local existence theory, which is available for operators with constant coefficients, does not hold if the coefficients are allowed to be variable.

This entry is just a short introduction to a highly developed subject. For more detailed expositions, we refer to [1, 2, 4, 7, 8, 11, 13, 14].

Basic Notations

We shall employ the notations introduced in [5] and [6]. More generally, we shall...

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di BolognaBolognaItaly