zbMATH — the first resource for mathematics

On first and second order planar elliptic equations with degeneracies. (English) Zbl 1250.35114
Mem. Am. Math. Soc. 1019, iii, 77 p. (2012).
In this monograph, the author studies the properties of solutions of first and second order equations in the plane. These equations are generated by a complex vector field \(X\) that is elliptic everywhere except along a simple closed curve. The technique used by the author consists to give a thorough study of the operator \(\mathcal{L}\) defined by a unique vector field conjugated to \(X\). The main properties of solutions to \(\mathcal{L}u=0\) or the nonhomogeneous equation are established in a neighborhood of the degeneracy curve through integral and series representations. An application to a class of second order elliptic operators with punctual singularity is given.

35J70 Degenerate elliptic equations
35F05 Linear first-order PDEs
35G20 Nonlinear higher-order PDEs
Full Text: DOI arXiv
[1] Heinrich Begehr and Dao-Qing Dai, On continuous solutions of a generalized Cauchy-Riemann system with more than one singularity, J. Differential Equations 196 (2004), no. 1, 67-90. · Zbl 1109.30033 · doi:10.1016/j.jde.2003.07.013
[2] Heinrich G. W. Begehr, Complex analytic methods for partial differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1994. An introductory text. · Zbl 0840.35001
[3] Shiferaw Berhanu, Paulo D. Cordaro, and Jorge Hounie, An introduction to involutive structures, New Mathematical Monographs, vol. 6, Cambridge University Press, Cambridge, 2008. · Zbl 1151.35011
[4] Paulo D. Cordaro and Xianghong Gong, Normalization of complex-valued planar vector fields which degenerate along a real curve, Adv. Math. 184 (2004), no. 1, 89-118. · Zbl 1129.35419 · doi:10.1016/S0001-8708(03)00139-7
[5] Antonio Gilioli and François Trèves, An example in the solvability theory of linear PDE’s, Amer. J. Math. 96 (1974), 367-385. · Zbl 0308.35022
[6] M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Springer-Verlag, New York-Heidelberg, 1973. Graduate Texts in Mathematics, Vol. 14. · Zbl 0294.58004
[7] Lars Hörmander, Propagation of singularities and semiglobal existence theorems for (pseudo)differential operators of principal type, Ann. of Math. (2) 108 (1978), no. 3, 569-609. · Zbl 0396.35087 · doi:10.2307/1971189
[8] Abdelhamid Meziani, On planar elliptic structures with infinite type degeneracy, J. Funct. Anal. 179 (2001), no. 2, 333-373. · Zbl 0973.35083 · doi:10.1006/jfan.2000.3695
[9] Abdelhamid Meziani, Elliptic planar vector fields with degeneracies, Trans. Amer. Math. Soc. 357 (2005), no. 10, 4225-4248 (electronic). · Zbl 1246.35090 · doi:10.1090/S0002-9947-04-03658-X
[10] Abdelhamid Meziani, Representation of solutions of a singular Cauchy-Riemann equation in the plane, Complex Var. Elliptic Equ. 53 (2008), no. 12, 1111-1130. · Zbl 1162.30029 · doi:10.1080/17476930802509239
[11] Abdelhamid Meziani, Properties of solutions of a planar second-order elliptic equation with a singularity, Complex Var. Elliptic Equ. 54 (2009), no. 7, 677-688. · Zbl 1178.35123 · doi:10.1080/17476930902998928
[12] François Trèves, Hypo-analytic structures, Princeton Mathematical Series, vol. 40, Princeton University Press, Princeton, NJ, 1992. Local theory. · Zbl 0565.35079
[13] F. Treves: On planar vector fields with complex linear coefficients, Preprint · Zbl 1194.35101
[14] A. Tungatarov, On the theory of the generalized Cauchy-Riemann system with a singular point, Sibirsk. Mat. Zh. 34 (1993), no. 4, 207-216, v, xi (Russian, with English and Russian summaries); English transl., Siberian Math. J. 34 (1993), no. 4, 776-785. · Zbl 0824.35096 · doi:10.1007/BF00975183
[15] A. Tungatarov, Continuous solutions of the generalized Cauchy-Riemann system with a finite number of singular points, Mat. Zametki 56 (1994), no. 1, 105-115, 157 (Russian, with Russian summary); English transl., Math. Notes 56 (1994), no. 1-2, 722-729 (1995). · Zbl 0836.30030 · doi:10.1007/BF02110563
[16] Z. D. Usmanov, Generalized Cauchy-Riemann systems with a singular point, Complex Variables Theory Appl. 26 (1994), no. 1-2, 41-52. · Zbl 0990.30501
[17] Z. D. Usmanov, Generalized Cauchy-Riemann systems with a singular point, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 85, Longman, Harlow, 1997. · Zbl 0873.35002
[18] I. N. Vekua, Generalized analytic functions, Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass., 1962. · Zbl 0127.03505
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.