## 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.

##### Department

##### Description

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