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A strong uniqueness theorem for planar vector fields. (English) Zbl 1043.35051
From the introduction: Consider a complex, smooth vector field \(L= \partial/ \partial y+a(x,y)\partial/ \partial x\) defined in a neighborhood of the origin in \(\mathbb{R}^2\). We are interested in the following uniqueness question: if a function \(u(x,y)\) defined in a neighborhood of the origin satisfies \[ Lu=0\text{ for }y>0 \text{ and } u(x,0)=0,\tag{1} \] can we conclude that \(u(x,y)\) vanishes identically in a neighborhood of the origin?
In this article we investigate a stronger uniqueness property for locally integrable vector fields, replacing the condition that \(u(x,0)\) vanish identically by the weaker hypothesis that the integral of \(\ln| u(x,0)|\) be equal to \(-\infty\), and consider one-sided solutions, i.e., \(u(x,y)\) is only assumed to satisfy the equation on one side of the initial curve \(\{y=0\}\), conditions that are classically known to guarantee uniqueness for the Cauchy-Riemann operator. After an appropriate local change of variables that preserves the initial curve \(\{y=0\}\), any elliptic vector field can be transformed into a multiple of the Cauchy-Riemann operator and this shows that elliptic vector fields share this strong uniqueness property. However, this condition is not enough to guarantee uniqueness for the vector field \(\partial_y\) so an additional hypothesis has to be made on \(L\) if it is to possess the strong uniqueness property under scrutiny. It turns out that a much weaker assumption than ellipticity is enough to ensure that \(L\) will share with the Cauchy-Riemann operator this strong uniqueness property for bounded solutions. All we need to assume is that the integral curve of \(X=\text{Re}\,L\) through the origin contains a sequence of points on which \(L\) is elliptic and the sequence converges to the origin (see Theorem 1.2 for the precise statement). It is also shown that this geometric condition is necessary for the validity of the strong uniqueness property. The work [B. Jöricke, J. Geom. Anal. 6, 555–611 (1996; Zbl 0917.32007)] and the recent article P. Cordaro [Fields Inst. Commun. 31, 97–112 (2002; Zbl 1003.35007)] contain results on this kind of uniqueness property.
Finally a theorem on pointwise convergence to weak boundary values is proved.

MSC:
35F15 Boundary value problems for linear first-order PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
30E25 Boundary value problems in the complex plane
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