ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

10. Partial Differential Equations

1991 MS Classification: 35 Eidelman, Samuil, International Solomon University, Ukraine; Kondratiev, Vladimir, Moscow State University, Russia On Properties of Positive Solutions of Elliptic Equations We consider the qualitative properties of positive solutions of elliptic equations of the form:

∑_{α,β} D^{α}(a_{αβ}(x)D^{β}u) = 0  (1)

D^{α} = \frac{∂^{|α|}}{∂x_{1}^{α_{1}}...∂x_{n}^{α_{n}}} , | α |≤ l, | β |≤ 2m − l

where x ∈ Ω, Ω is a smooth bounded domain of R^{n}, 0 < l < 2m, a_{αβ}(x) are continuos functions. The main result is following: any positive weak solution of (1) belongs to L_{p}(Ω) with arbitrary p, p < \frac{n}{n−1}. We use the methods of the papier [1]. References. 1. V. A. Kondratiev and S. D. Eidelman. Positive solutions of linear partial differential equations, Trudy Mosk. matem. obshch. 31(1974), 85-146.