A strong uniqueness theorem for planar vector fields.

*(English)*Zbl 1043.35051From 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.

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 |

##### References:

[1] | [BT] M. S. Baouendi and F. Treves,A property of the functions and distributions annihilated by a locally integrable system of complex vector fields, Ann. of Math.,113 (1981), 387-421. · Zbl 0491.35036 · doi:10.2307/2006990 |

[2] | [BH1] S. Berhanu and J. Hounie,An F. and M. Riesz theorem for planar vector fields, Math. Ann.,320, (2001), 463-485. · Zbl 0984.35045 · doi:10.1007/PL00004482 |

[3] | [BH2] S. Berhanu and J. HounieOn boundary properties of solutions of complex vector fields, to appear in Jour of Functional Analysis. |

[4] | [C] P. Cohen,The non-uniqueness of the Cauchy problem, O.N.R. Tech. Report,93 (1960), Stanford. · Zbl 0151.38101 |

[5] | [Co] P. Cordaro,Approximate solutions in locally integrable structures, Fields Institute Communications volume: Differential Equations and Dynamical Systems (in honor to Waldyr Muniz Oliva), to appear |

[6] | [Du] P. Duren,Theory of H p spaces, Academic Press, 1970. · Zbl 0215.20203 |

[7] | [F] P. Fatou,S?ries trigonom?triques e s?ries de Taylor, Acta Math.,30 (1906), 335-400. · JFM 37.0283.01 · doi:10.1007/BF02418579 |

[8] | [HM] J. Hounie and J. MalaguttiOn the convergence of the Baouendi-Treves approximation formula, Comm. P.D.E.,23 (1998), 1305-1347. · Zbl 0911.35031 · doi:10.1080/03605309808821386 |

[9] | [HT] J. Hounie and J. Tavares,On removable singularities of locally solvable differential operators, Invent. Math.,126 (1996), 589-623. · Zbl 0869.58051 · doi:10.1007/s002220050110 |

[10] | [Hor] L. H?rmander,The Analysis of linear partial differential operators I, Springer-Verlag, 1990. |

[11] | [J] B. J?ricke,Deformation of CR manifolds, minimal points and CR manifolds with the microlocal analytic extension property, J. Geom. Anal., (1996), 555-611. · Zbl 0917.32007 |

[12] | [RR] F. Riesz and M. Riesz,?ber die Randwerte einer analytischen Funktion Quatri?me Congr?s de Math. Scand. Stockholm, (1916), 27-44. |

[13] | [T1] F. Treves,Hypo-analytic structures, local theory, Princeton University Press, 1992. · Zbl 0787.35003 |

[14] | [T2] F. Treves,Approximation and representation of solutions in locally integrable structures with boundary, Aspects of Math. and Applications, (1986), 781-816. |

[15] | [T3] F. Treves,Approximation and representation of functions and distributions annihilated by a system of complex vector fields, Centre de Math?matiques. ?cole Polytechnique, Palaiseau, France, 1981. |

[16] | [Z] C. ZuilyUniqueness and non-uniqueness in the Cauchy Problem, Birkh?user, Boston-Basel-Stuttgart, 1983. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.