ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

10. Partial Differential Equations

1991 MS Classification: not given Mikhailov, Leonid Grigorevich, Tadjikistan Singular Differential Equations \underline{1. Singular Analysis.} The x = 0 is singular point (s. p.) of y(x), 0 < x ≤ ρ, if for x → 0 , y(x) is unbounded and nonintegrable. If lim(x → 0) x·y′ = P, then lim x → 0(y(x)/lux) = P. If for α > 0 be lim(x → 0) x^{1+α}y′(x) = P_{1,α}, then lim(x → 0) x^{α}y(x) = P_{α} and P_{1,α} = (−α) · P_{α}. There are some analogously results for y^{(m)}(x) and for differentials operators.

2. A new class of n-dimensional singular integral equations (with kernels homogeneous of order (-n)). \underline{3. The property of degeneracy.} If y_{(x)} ∈ IC[0 > 1] is solution of −x′ · y′ = f(x, y), 0 < x ≤ 1, where f(x, y) ∈ IC, y_{0} = y(0), then necessarily f(0, y_{0}) = 0. Analogously for: z̄ · δz̄w = f(z, w) (z = x + iy, 2 · δz̄ = δ_{x} + i · δ_{y}, then f(0, w_{0}) = 0, for −|x|^{2} · Δu = f(x, u) (x = x_{1}, …, x_{n}, |x|^{2} = ∑ x_{k}^{2}) and for equations with singular linears: (r − q) · δ_{z} − w = f(z, w) and for |(r − q)^{2} · Δw = f(z, w) : if w(z) ∈ C is solution, then f(z, w)_{|z|=q} = 0.

4. The total differential systems with s. p.

r^{n} · U_{x} = a(x, y) , r^{n} · U′_{y} = b(x, y) , r^{2} = x^{2} + y^{2} ,

is received a representation formula

U(x, y) = c + q_{n−1} · φ + P_{n−1} · ln r + ∑ r̄^{[(n−1)−k]} · γ_{k}(r, φ) + V (r, φ)

from that follows a wide generalization of the residue-theorem.

5. For −x · y′ = f(x, y), 0 < x ≤ q, is given a method for construct solution of Problem [y]_{x=q} = y, then the value y(0) is determined and may be only zero of f(0, y) or +∞.