Calculus of Variations

In this paper we study a classical optimization problem in heat conduction, which may briefly be described as follows: given a surface ∂D in R, and a positive function φ defined on it (the temperature distribution), we want to surround ∂D with a prescribed volume of insulating material so as to minimize the loss of heat in a stationary situation. Mathematically speaking, we want to find a function u, which corresponds to the temperature inD . The function u is harmonic whenever it is positive and the volume of the support of u is equal to 1. The quantity to be minimized, the flow of heat, is a continuous family of convex function of uμ along ∂D. Our paper was motivated by a series of remarkable papers [1], [2] and [3]. The first two articles study the constant temperature distribution, i.e., φ ≡ C on ∂D. All of them treated the linear case, i.e, Γ (x, t) = t. The linear setting allows, in [1] and [2], to reduce the quantity to be minimized to the Dirichlet integral. Even in the linear case, the nonconstant temperature distribution, problem studied in [3], presents several new difficulties. The ultimate goal of this article is to study the nonlinear case with nonconstant temperature distribution. The nonlinearity treated in this article has physical importance: problems with a monotone operator like the typewe study in this paper arise in questions of domain optimization for electrostatic configurations. The nonlinearity over uμ presents several new difficulties as well. For instance, even to provide a reasonable mathematical model, one faces the problem that it does not make sense to compute normal derivatives of H1-functions. In [3], this