Uniqueness for the Characteristic Cauchy Problem and Strong Unique Continuation for Higher Order Partial Differential Inequalities

Abstract

Introduction. This paper presents new results on the uniqueness for the Cauchy problem with respect to an everywhere characteristic hypersurface, and applications to strong unique continuation for partial differential inequalities of elliptic or hyperbolic type. The main theorem (Theorem 2) states uniqueness for mth order operators of the form P(t, 0, tat, ao), which are polynomials in tat, whose coefficients are pseudodifferential operators in 0 (depending smoothly on t) on a compact manifold M. Note that for non compact M, uniqueness does not hold in general [4], [9]. Our assumptions and results are parallel to those of Calderon in the classical local uniqueness theorem for the noncharacteristic Cauchy problem ([11], see also Nirenberg [16]), the operator tat replacing at. The proof of Theorem 2 combines pseudodifferential calculus on M with results on differential inequalities in Hilbert space, given in Theorem 1. Convexity and uniqueness results for similar abstract inequalities have been obtained by Agmon and Nirenberg [2], and also by Baiocchi