Regularity of free boundary in variational problems

Abstract

We study the existence and geometric properties of an optimal configurations to a variational problem with free boundary. More specifically, we analyze the nonlinear optimization problem in heat conduction which can be described as follows: given a surface ∂D ⊂ R n and a positive function ϕ defined on it (temperature distribution of the body D), we want to find an optimal configuration Ω ⊃ ∂D (insulation), that minimizes the loss of heat in a stationary situation, where the amount of insulating material is prescribed. This situation also models problems in electrostatic, potential flow in fluid mechanics among others. The quantity to be minimized, the flow of heat, is given by a monotone operator on the flux uµ. Mathematically speaking, let D ⊂ R n be a given smooth bounded domain and ϕ: ∂D → R+ a positive continuous function. For each domain Ω surrounding D such that Vol.(Ω \ D) = 1, we consider the potential associated to the configuration Ω, i.e., the harmonic function on Ω\D taking boundary data u

∂D ≡ ϕ and u

∂Ω ≡ 0, and compute J(Ω) := Z ∂D Γ(x,uµ(x))dσ, vii where µ is the inward normal vector defined on ∂D and Γ is a continuous family of convex functions. Our goal is to study the existence and geometric properties of an optimal configuration related to the functional J. In other words, our purpose is to study the problem: minimize { J(u) := Z ∂D Γ(x,uµ(x))dσ : u: DC → R, u = ϕ on ∂D, ∆u = 0 in {u > 0} and Vol.(supp u) = 1 } Among other regularity properties of an optimal configuration, we prove analyticity of the free boundary up to a small singular set. We also establish uniqueness and symmetry results when ∂D has a given symmetry. Full regularity of the free boundary is obtained under these symmetry conditions imposed on the fixed boundary.