Uniqueness for the Cauchy Problem

• Lars Hörmander
Part of the Grundlehren der mathematischen Wissenschaften book series (CLASSICS)

Summary

In this chapter we resume the study of uniqueness theorems for differential operators with non-analytic coefficients started in Section 17.2. Section 28.1 is devoted to Calderón’s uniqueness theorem which in its original form states that Theorem 17.2.1 remains valid when there are real characteristics, too. The proofs here start from scratch and rely on factorization in first order pseudo-differential operators. A careful study of these factors leads to more general forms of the Calderón uniqueness theorem.

As in Section 17.2 the basic tool is Carleman estimates. A systematic discussion of such estimates is given in Section 28.2 particularly for operators of real principal type or more generally for operators satisfying a stronger form of condition (P) (principally normal operators). The resulting uniqueness theorems which involve convexity conditions on the initial surface are presented in Section 28.3. More precise results in the second order case are discussed in Section 28.4.

Notes

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