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Solvability near the characteristic set for a class of planar vector fields of infinite type. (English) Zbl 1063.35051
Summary: We study the solvability of equations associated with a complex vector field \(L\) in \({\mathbb R}^2\) with \(C^\infty\) or \(C^\omega\) coefficients. We assume that \(L\) is elliptic everywhere except on a simple and closed curve \(\Sigma\). We assume that, on \(\Sigma\), \(L\) is of infinite type and that \(L\wedge\overline{L}\) vanishes to a constant order. The equations considered are of the form \(Lu=pu+f\), with \(f\) satisfying compatibility conditions. We prove, in particular, that when the order of vanishing of \(L\wedge\overline{L}\) is \(>1\), the equation \(Lu=f\) is solvable in the \(C^\infty\) category but not in the \(C^\omega\) category.

MSC:
35F05 Linear first-order PDEs
30G20 Generalizations of Bers and Vekua type (pseudoanalytic, \(p\)-analytic, etc.)
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References:
[1] Perturbation of globally hypoelliptic operators, J. Differential Equations, 114, 513-526, (1994) · Zbl 0815.35009
[2] Remarks about global analytic hypoellipticity, Trans. Amer. Math. Soc., 351, 4113-4126, (1999) · Zbl 0932.35046
[3] Global properties of a class of vector fields in the plane, J. Diff. Equations, 74, 179-199, (1988) · Zbl 0662.58021
[4] Globally hypoelliptic systems of vector fields, J. Funct. Analysis, 114, 267-285, (1993) · Zbl 0777.58041
[5] Global solvability for a class of complex vector fields on the two-torus, Comm. Partial Differential Equations, 29, 785-819, (2004) · Zbl 1065.35088
[6] Semiglobal solvability of a class of planar vector fields of infinite type, Mat. Contemp., 18, 31-42, (2000) · Zbl 0983.35036
[7] On rotationally invariant vector fields in the plane, Manuscripta Math., 89, 355-371, (1996) · Zbl 0858.35021
[8] Global properties of a class of planar vector fields of infinite type, Comm. Partial Differential Equations, 22, 99-142, (1997) · Zbl 0882.35029
[9] A generalized similarity principle for complex vector fields and applications, Trans. Amer. Math. Soc., 353, 1661-1675, (2001) · Zbl 0982.35030
[10] A property of functions and distributions annihilated by locally integrable system of vector fields, Ann. of Math., 113, 341-421, (1981) · Zbl 0491.35036
[11] Normalization of complex-valued planar vector fields which degenerate along a real curve, Adv. Math., 184, 89-118, (2004) · Zbl 1129.35419
[12] Global analytic hypoellipticity for a class of degenerate elliptic operators on the torus, Math. Res. Letters, 1, 501-510, (1994) · Zbl 0836.35036
[13] Homology and cohomology in hypoanalytic structures of the hypersurface type, J. Geo. Analysis, 1, 39-70, (1991) · Zbl 0724.32009
[14] Hypoelliptic vector fields and continued fractions, Proc. Amer. Math. Soc., 31, 115-118, (1972) · Zbl 0229.35024
[15] Global properties in spaces of generalized functions on the torus for second-order differential operators with variable coefficients, Rend. Sem. Mat. Univ. Pol. Torino, 51, 145-172, (1993) · Zbl 0824.35027
[16] The analysis of linear partial differential operators IV, (1984), New York · Zbl 0612.35001
[17] On the similarity principle for planar vector fields: applications to second order PDE, J. Differential Equations, 157, 1-19, (1999) · Zbl 0937.35079
[18] On real analytic planar vector fields near the characteristic set, Contemp. Math., 251, 429-438, (2000) · Zbl 0960.35016
[19] On planar elliptic structures with infinite type degeneracy, J. Funct. Anal., 179, 333-373, (2001) · Zbl 0973.35083
[20] Elliptic planar vector fields with degeneracies · Zbl 1246.35090
[21] Solvability of a first order linear partial differential equation, Comm. Pure Applied Math., 16, 331-351, (1963) · Zbl 0117.06104
[22] Remarks about certain first-order linear PDE in two variables, Comm. Partial Differential Equations, 5, 381-425, (1980) · Zbl 0523.35012
[23] Hypo-analytic structures: local theory, (1992) · Zbl 0787.35003
[24] Generalized analytic functions, (1962) · Zbl 0100.07603
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