A counterexample to strong uniqueness for partial differential equations of Schrödinger’s type. (English) Zbl 0806.35023

Alinhac, S.; Baouendi, M. S.

Commun. Partial Differ. Equations 19, No. 9-10, 1727-1733 (1994).

From the introduction: We consider the question of strong uniqueness from the origin for smooth solutions of equations of the type \(\Delta u + a \cdot \nabla u = 0\), or of differential inequalities of the form \(| \Delta u | \leq V | \nabla u |\). We are interested in the singular limiting case where \(a\) or \(V\) behaves like \(C/ | x |\). For any \(C>1\), we construct an example in the plane of a flat complex valued function \(u\) satisfying \(| \Delta u | \leq {C \over | x |} | \nabla u |\) and \(\text{supp} u = \mathbb{R}^ 2\).

MSC:

35J15	Second-order elliptic equations
35A07	Local existence and uniqueness theorems (PDE) (MSC2000)

Keywords:

strong uniqueness; differential inequalities

References:

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