Doubly periodic Riemann boundary value problem for non-rectifiable curves. (English) Zbl 1405.30041
Summary: The known results on the doubly periodic Riemann boundary value problem concern the case of piecewise-smooth contours. In the present paper, we study it for non-rectifiable curves in terms of so called Marcinkiewicz exponents.
MSC:
| 30E25 | Boundary value problems in the complex plane |
| 28A75 | Length, area, volume, other geometric measure theory |
Keywords:
Riemann boundary value problem; doubly periodic functions; non-rectifiable curve; Marcinkiewicz exponentsReferences:
| [1] | F. D. Gakhov, Boundary Value Problems (Nauka, Moscow, 1977) [In Russian]. · Zbl 0449.30030 |
| [2] | N. I. Muskhelishvili, Singular Integral Equations (Nauka, Moscow, 1962). · Zbl 0103.07502 |
| [3] | L. I. Chibrikova, Doctoral Dissertation (Minsk, 1962). |
| [4] | L. I. Chibrikova, The Main Boundary Value Problems for Analytic Functions (Izd. Kazan Univ., Kazan, 1977) (In Russian). |
| [5] | Lu, J., No article title, J. Wufan Univ., 3, 1 (1979) |
| [6] | Lu, J., No article title, Chin. Ann. Math., 1, 289 (1980) · Zbl 0476.30038 |
| [7] | Cai, H-t; Lu, J-k, Mathematical Theory in Periodic Plane Elasticity (2000), Amsterdam · Zbl 0959.74002 |
| [8] | Garifyanov, F. N., No article title, Mat. Zametki, 59, 932 (1996) · doi:10.4213/mzm1794 |
| [9] | Wang, Y.; Wang, Y., No article title, Math. Nachr., 287, 1886 (2014) · Zbl 1303.30041 · doi:10.1002/mana.201300124 |
| [10] | Kats, B. A., No article title, Dokl. Akad. Nauk USSR, 267, 789 (1982) |
| [11] | Kats, B. A., No article title, Complex Variables and Elliptic Equations, 59, 1053 (2014) · Zbl 1314.30063 · doi:10.1080/17476933.2013.809574 |
| [12] | D. B. Katz, In Abstracts of ISAAC 9th Congress, Krakow, 106 (2013). |
| [13] | Katz, D. B., No article title, Trudy Matematicheskogo Tsentra im. Lobachevskogo, 46, 235 (2013) |
| [14] | Katz, D. B., No article title, Izv. Vuz. Mat., 3, 68 (2014) |
| [15] | E. M. Stein, Singular Integrals and Differential Properties of Functions (Princeton Univ. Press, Princeton, 1970). · Zbl 0207.13501 |
| [16] | P. Painleve, Sur les lignes singulieres des fonctions analytiques (Paris, 1887). · JFM 19.0393.01 |
| [17] | Dolzhenko, E. P., No article title, Uspekhi Mat. Nauk, 18, 135 (1963) · Zbl 0204.39001 |
| [18] | Kolmogorov, A. N.; Tikhomirov, VM, No article title, Uspekhi Mat. Nauk, 14, 3 (1959) · Zbl 0090.33503 |
| [19] | C. Tricot, Curves and Fractal Dimension (Springer, Heidelberg, 1995) · Zbl 0847.28004 · doi:10.1007/978-1-4612-4170-6 |
| [20] | K. J. Falconer, Fractal Geometry, 3rd ed. (Wiley, Chichester, 2014) · Zbl 1285.28011 |
| [21] | Kats, B. A., No article title, Complex Anal. Operator Theory, 6, 1147 (2012) · Zbl 1275.30004 · doi:10.1007/s11785-010-0111-4 |
| [22] | Abreu-Blaya, R.; Bory-Reyes, J.; Kats, B. A., No article title, J.Math. Anal. Appl., 380, 177 (2011) · Zbl 1218.30104 · doi:10.1016/j.jmaa.2011.02.068 |
| [23] | I. N. Vekua, Generalized Analytical Functions (Nauka, Moscow, 1980) [In Russian]. |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
