ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

10. Partial Differential Equations

1991 MS Classification: 35, 31 Horiuchi, Toshio, Ibaraki University, Japan Removable singularities for quasilinear degenerate elliptic equations with absorption terms Let N ≥ 1 and p > 1. Let F be a compact set and Ω be a bounded open set of ℝ^{N} satisfying F ⊂ Ω ⊂ ℝ^{N}. Define a generalized p−harmonic operator by

L_{p}u = −div(A(x)|∇u|^{p−2}∇u).

Here A(x) ∈ C^{1}(Ω′) is positive in Ω \ F and vanishes in Int F. Assume that u ∈ C^{0}(Ω′) ∩ C^{2}(Ω \ F) satisfies in the distribution sense

L_{p}u + B(x)Q(u) = f, in Ω \ F

for some nonnegative function B(x) and a nonlinear term Q(t). Then we shall show under some additional conditions that

lim sup_{x→∂F} |u(x)| < +∞. Removability of singularities

We also study the Dirichlet boundary problem for this equation and establish the unique existence of the bounded solution. References T. Horiuchi: Existence of bounded solutions for semilinear degenerate elliptic equations with absorption term. Proc. Japan Acad., 74, Ser. A (1998).