Asymptotic behavior of two semilinear elliptic free boundary problems

Thomas Vogel

Vol. 123 (1986), No. 2, 477–495

DOI: 10.2140/pjm.1986.123.477

Abstract

Given a bounded open set Ω ⊆ Rⁿ with C^2+α boundary and a monotone increasing function f(t) with f(0) = 0, this paper treats two related exterior free boundary problems:

Problem A: Given λ > 0, determine u ∈ C₀^2+α₁(Rⁿ − Ω) satisfying:

Δu	= λf(u) in Rn −Ω
u	= 1 on ∂Ω.	(1.1)

Problem B: Given c > 0, determine v ∈ C₀^2+α₁(Rⁿ − Ω) satisfying:

Δv	= f(v) in Rn −Ω
v	= c on ∂Ω.	(1.2)

In both problems, the free boundary is the boundary of the support of the sought function.

Mathematical Subject Classification 2000

Primary: 35R35

Secondary: 35B40, 35J65

Milestones

Received: 2 November 1984

Published: 1 June 1986