Qualitative behavior of solutions of elliptic free boundary problems

Kirk Lancaster

Vol. 154 (1992), No. 2, 297–316

DOI: 10.2140/pjm.1992.154.297

Abstract

A general free boundary problem is investigated and the qualitative behavior of the fixed boundary is compared with that of the fixed boundary. As an illustration, consider the following situation. Let Γ^∗ be a given Jordan curve in R². For each Jordan curve Γ in R² which surrounds Γ^∗, we let Ω = Ω(Γ^∗,Γ) be the region between Γ^∗ and Γ. Let Q be the second-order elliptic operator given by

where a, b, c depend on x, y, u_x, and u_y and ac−b² > 0. Consider the free boundary problem of finding a curve Γ and a function u ∈ C²(Ω) ∩ C¹(Ω ∪ Γ) ∩ C^∘(Ω) such that

and, for a fixed λ > 0,

where Ω = Ω(Γ^∗,Γ). Suppose Γ and u constitute a solution of this free boundary problem. Using curves of constant gradient direction, the geometry of the free boundary Γ is compared to the geometry of the fixed boundary Γ^∗. In particular, Γ is shown to have a “simpler” geometry than does Γ^∗.

Mathematical Subject Classification 2000

Primary: 35R35

Secondary: 35J25

Milestones

Received: 13 March 1991

Published: 1 June 1992