The Cauchy problem for Douglis-Nirenberg elliptic systems of partial differential equations

Author:
Richard J. Kramer

Journal:
Trans. Amer. Math. Soc. **182** (1973), 211-225

MSC:
Primary 35J55

DOI:
https://doi.org/10.1090/S0002-9947-1973-0333439-1

MathSciNet review:
0333439

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Abstract: Several partial answers are given to the question: Suppose *U* is a solution of the Douglis-Nirenberg elliptic system where *F* is analytic and *L* has analytic coefficients. If in some appropriate sense on a hyperplane (or any analytic hypersurface) must *U* vanish identically?

One answer follows from introducing a so-called formal Cauchy problem for Douglis-Nirenberg elliptic systems and establishing existence and uniqueness theorems.

A second Cauchy problem, in some sense a more natural one, is discussed for an important subclass of the Douglis-Nirenberg elliptic systems. The results in this case give a second partial answer to the original question.

The methods of proof employed are largely algebraic. The systems are reduced to systems to which the Cauchy-Kowalewski theorem applies.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1973-0333439-1

Keywords:
Cauchy problem,
formal Cauchy data,
Douglis-Nirenberg elliptic systems,
Cauchy-Kowalewski theorem

Article copyright:
© Copyright 1973
American Mathematical Society