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A class of planar vector fields with homogeneous singular points: solvability and boundary value problems. (English) Zbl 1397.35066
Summary: This paper deals with the solvability of planar complex vector fields with homogeneous degeneracies. Hölder continuous solutions are obtained via a Cauchy type integral operator associated to the vector field. An associated boundary value problem of Riemann-Hilbert type is also considered.

MSC:
35F15 Boundary value problems for linear first-order PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35F05 Linear first-order PDEs
35C15 Integral representations of solutions to PDEs
35J70 Degenerate elliptic equations
35Q15 Riemann-Hilbert problems in context of PDEs
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References:
[1] Berhanu, S.; Cordaro, P.; Hounie, J., An introduction to involutive structures, New Math. Mono., vol. 6, (2008), Cambridge University Press Cambridge · Zbl 1151.35011
[2] Begehr, H., Complex analytic methods for partial differential equations. an introductory text, (1994), World Scientific Publishing, NJ · Zbl 0840.35001
[3] Campana, C.; Meziani, A., Boundary value problems for a class of planar complex vector fields, J. Differential Equations, 261, 10, 5609-5636, (2016) · Zbl 1352.35086
[4] Campana, C.; Dattori da Silva, P. L.; Meziani, A., Properties of solutions of a class of hypocomplex vector fields, Contemporary Mathematics, vol. 681, 29-50, (2017), Amer. Math. Soc. · Zbl 1362.35088
[5] Campana, C.; Dattori da Silva, P. L.; Meziani, A., Riemann-Hilbert problem for a class of hypocomplex vector fields, Complex Var. Elliptic Equ., 62, 10, 1413-1424, (2017) · Zbl 1375.35293
[6] Krantz, S. G., Geometric function theory: explorations in complex analysis, (2006), Springer Science & Business Media · Zbl 1086.30001
[7] Meziani, A., On planar elliptic structures with infinite type degeneracy, J. Funct. Anal., 179, 333-373, (2001) · Zbl 0973.35083
[8] Meziani, A., Solvability of planar complex vector fields with homogeneous singularities, Complex Var. Elliptic Equ., 62, 10, 1447-1473, (2017) · Zbl 1375.35295
[9] Nirenberg, L.; Treves, F., Solvability of a first-order linear differential equation, Comm. Pure Appl. Math., 16, 331-351, (1963) · Zbl 0117.06104
[10] Treves, F., Hypo-analytic structures: local theory, Princeton Mathematical Series, vol. 40, (1992), Princeton Univ. Press NJ · Zbl 0787.35003
[11] Vekua, I. N., Generalized analytic functions, (1962), Pergamon Press Oxford · Zbl 0127.03505
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