Capillary surfaces in wedge domains
Time and place
1 PM on Thursday, October 14th, 2010; NAC 6-113
Kirk Lancaster (Wichita State University)
Abstract
Let Ω be a Lipschitz domain in IRn and O ∈ ∂Ω.
If ∂Ω is not smooth at O, how does a nonparametric capillary (or prescribed mean curvature) hypersurface S0 = {(x, f (x)) : x ∈ Ω} satisfying contact angle boundary conditions behave near O?
In this case, f ∈ C 2 (Ω) satisfies the boundary value problem
Nf= H(·, f (·)) in Ω T f · ν = cos γ (a.e.) on ∂Ω,
where T f = √∇f/1+|∇f |2, N f = ∇ · T f, ν is the exterior unit normal C^0(Ω × IR) and γ = γ(x) ∈ [0, π]. (For simplicity, we assume H(x, t) is a weakly increasing function of t for each x ∈ Ω and therefore exclude the case of “negative gravity” (e.g. pendant drops); however this assumption is not critical for us.)
After a brief “historical” discussion, we will largely focus on the case n = 2 and discuss the existence of radial limits, the recent proof of the Concus-Finn conjecture and the “central fan question.” We will also discuss some examples when n > 2.
One starting point in examining S_0 is to think of the closure S in IRn+1 of S0 as a (generalized) manifold with boundary and parametrize it in appropriate coordinates. When n = 2, these coordinates are isother- mal coordinates. When n > 2, we suspect that harmonic coordinates will prove to be useful.
