On the Hölder property of one elliptic equation. (English. Russian original) Zbl 1137.35383
J. Math. Sci., New York 129, No. 1, 3523-3536 (2005); translation from Sovrem. Mat. Prilozh. 10, 8-21 (2003).
The paper deals with an elliptic in the unit disk equation of the divergence form with discontinuity lines coinciding with the two-dimensional coordinate system axes. On the above lines it has the order degeneration in the second and forth quadrants as a power function of the radius of the order \(\alpha\) and with the order \(-\alpha\), \(0<\alpha<2\), in the first and third quadrants. Introducing in the natural way classes of generalized solutions in the corresponding weighted Sobolev space, the Hölder property of the solutions of the equation under consideration is studied.
Reviewer: Georg V. Jaiani (Tbilisi)
Keywords:
degenerate elliptic equations; discontinuous coefficients; generalized solutions; Hölder property of solutions; weighted Sobolev spacesReferences:
| [1] | Yu. A. Alkhutov and V. V. Zhikov, ”On the Holder property of solutions of degenerating elliptic equations,” Dokl. Ross. Akad. Nauk, 378, No.5, 583–588 (2001). · Zbl 1090.35510 |
| [2] | E. Fabes, C. Kenig, and R. Serapioni, ”The local regularity of solutions of degenerate elliptic equations,” Commun. Partial Differ. Equations, 7, No.1, 77–116 (1982). · Zbl 0498.35042 · doi:10.1080/03605308208820218 |
| [3] | L. Hormander, The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis, Grundlehren Math. Wiss., 256, Springer-Verlag, Berlin (1990). |
| [4] | B. Muckenhoupt, ”Weighted norm inequalities for the Hardy-Littlewood maximal function,” Trans. Amer. Math. Soc., 15, 207–226 (1972). · Zbl 0236.26016 · doi:10.1090/S0002-9947-1972-0293384-6 |
| [5] | V. V. Zhikov, ”Weighted Sobolev spaces,” Mat. Sb., 189, No.8, 27–58. · Zbl 0919.46026 |
| [6] | V. V. Zhikov, ”On the problem of passing to the limit in divergent, nonuniformly elliptic equations,” Funkts. Anal. Prilozh., 35, No.1, 23–39 (2001). · Zbl 1085.35054 |
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