An optimal insulation problem

In this paper we consider a minimization problem which arises from thermal insulation. A compact connected set K, which represents a conductor of constant temperature, say 1, is thermally insulated by surrounding it with a layer of thermal insulator, the open set Ω \ K with K ⊂ Ω̄. The heat dispersion is then obtained as inf { ∫


Introduction
As energy saving and noise pollution are growing in importance year after year, insulation represents one of biggest challenge for environmental improvement. Even if insulation is one of the oldest and most studied problem in mathematical physics, the mathematics involved is still very tricky especially when one looks at shape optimization questions. In this paper we will consider a domain of given temperature, thermally insulated by surrounding it with a constant thickness of thermal insulator. As the geometry of the set varies (under prescribed geometrical constraints to be specified later) it is reasonable to ask for the best (or worst) choice (in terms of heat dispersion).
Let K be a compact set in R n and let Ω be a bounded connected open set of R n , with K ⊂Ω (here the set K represents a thermally conducting body, surrounded by an amount of insulating material Ω \ K). The heat dispersion is given by where β > 0 is a fixed parameter depending on the physical characteristics of the problem, and ∂ * Ω stands for the reduced boundary of Ω.
If Ω has Lipschitz boundary and K ⊂ Ω, then there exists a minimiser u of (1) which solves in the weak sense the Robin-Dirichlet problem in Ω \ K, and it holds that In what follows we will use D to denote a bounded open connected subset of Ω and in that case the notation I β (D, Ω) stands for I β (D, Ω). If D = Ω (no insulation), we set where P (D) stands for the classical perimeter in R n . One of the most intriguing questions in thermal insulation is to study the configurations of D and Ω for which I β (D, Ω) is minimal or maximal, under reasonable geometric constraints. Shape optimization problems of this or similar nature already appeared in several papers, as for instance in [1,2,3,4,5,7,10,12,13,14]. Related results are contained in [11].
In Section 2 we consider I β (D, Ω) when Ω is the Minkowski sum Ω = D + δB (see Section 2 for its definition), B is the unit ball centered at the origin, and δ is a positive number. This corresponds to accomplish the insulation of D by adding an insulation layer of constant thickness. For the sake of simplicity, we set In Section 3 we consider the planar case and prove that in the class of smooth domains D with given perimeter P , and fixed δ > 0, the disk maximizes I β,δ . Therefore, we deduce that the worst possible geometry (in terms of heat dispersion) is the symmetric one.
Thereafter, in Section 4, we prove that in higher dimension (n ≥ 3) balls still maximize I β,δ , but our result finds its natural generalization in the class of convex domains D with given (n − 1)−quermassintegral.
The study of the worst possible shape, the one with the highest heat dispersion, is justified by the idea that, in designing the optimal insulation, one has an a-priori bound of the heat dispersion in terms of geometric quantities alone.
However, it is of great interest as well, to study the existence of the best geometries: those which minimize heat dispersion. In the last section we make some remarks on that, and discuss few open problems. For instance we consider the minimization of I β,δ (D) in the class of sets of given perimeter or measure. Comparison with classical Capacitary problem (which corresponds to the case β → ∞) suggests that the minimization of I β,δ (D) when we vary D by preserving its measure, occurs when D is a ball. Even more complicated is the case where both Ω and D vary among sets of fixed volume. However such a case requires a careful formulation otherwise the problem can be ill-posed, in the sense that (against common sense) sometimes insulation might increase heat dispersion. In particular we show that, when D = B R (open ball of radius R), and β is small enough, there exists a positive constant δ 0 (which depends on β and R alone) such that for any bounded connected open set Ω, with Ω ⊃ B R and |Ω \ B R | < δ 0 , then

Insulating problem
Given an open, bounded, connected set D ⊂ R n , and Ω = D + δB, δ > 0, where B is the unit ball centered at the origin and D + δB stands for the Minkowski sum of the two sets, that is we are interested in the study of the properties of If Ω has Lipschitz boundary, then there exists a unique minimizer u of (3) such that in the sense that for all ϕ ∈ W 1,2 D (Ω). Then We will call I β,δ (D) the "heat dispersion" and we observe that if β = 0, the condition on ∂Ω in (4) becomes a Neumann condition: ∂u ∂ν = 0, while if β = +∞, the problem has to be meant with a Dirichlet condition on ∂Ω: u = 0.
Furthermore, by choosing ϕ = 1 in (3) as test function, we get where P (Ω) stands for the perimeter of Ω in R n .

Maximization of the heat dispersion: the planar case
In this section we consider the two dimensional case. For an open, bounded, connected set D we denote by D * the disk having the same perimeter of D. We consider Ω = D+δB and the disk Ω * = D * + δ * B, where B is the unit disk centered at the origin. If D is convex, the Steiner formulae state that |Ω| = |D| + P (D)δ + πδ 2 , P (Ω) = P (D) + 2πδ, If we ask that the area of the insulating material Ω \D remains constant, then Then, since P (D) = P (D * ) then δ = δ * and, as byproduct, P (Ω) = P (Ω * ). On the contrary, if δ = δ * , then |Ω| − |D| = |Ω * | − |D * |.
For a general bounded domain with piecewise C 1 boundary, it holds that which means that the area of the insulating material increases keeping fixed the perimeter P (D) and the thickness δ.
where D * is the disk having the same perimeter of D, that is P (D * ) = P (D).
Proof. Let v be the radial minimizer of I β,δ (D * ). We denote by R be the radius of D * , Being v radial, the modulus of the gradient of v is constant on the level lines of v. Let us consider the function Let d(x) be the distance of a point x from D and set ds.
By construction w ∈ H 1 (Ω) and, being G decreasing, it results: Hence w is a test function. Then and let By Steiner formula (7) we get for every w m < t ≤ 1. Hence, by (9), then, by co-area formula and (8), As regards the boundary terms, since by construction w = w m = v m on ∂Ω (see (8)), and P (Ω) = P (Ω * ), we have The n-dimensional case

Preliminaries on convex sets
Here we list some basic facts on convex sets. All the notions and results can be found, for example, in [6,16]. Let K be a nonempty, bounded, convex set in R n and let δ > 0.
where B is the unit ball in R n centered at the origin, whose measure is denoted by ω n . The coefficients W j (K) are the so-called quermassintegrals of K. It immediately follows that If K as C 2 boundary, with nonzero Gaussian curvature, the quermassintegrals are related to the principal curvatures of ∂K. Indeed, in such a case Here H j denotes the j−th normalized elementary symmetric function of the principal curvatures of ∂K, that is H 0 = 1 and where κ 1 (x), ..., κ n−1 (x) are the principal curvatures at a point x ∈ ∂K. In particular, by (12) and (13) we get also that where H 1 (x) is the mean curvature of ∂K at a point x.
The Aleksandrov-Fenchel inequalities state that where the inequality is replaced by an equality if and only if K is a ball.
In what follows, we use the Aleksandrov-Fenchel inequalities for particular values of i and j. When i = 0 and j = 1, we have the classical isoperimetric inequality: Moreover, if i = k − 1, and j = k, we have Let us denote by K * a ball such that W n−1 (K) = W n−1 (K * ). Then by Aleksandrov-Fenchel inequalities (14), for 0 ≤ i < n − 1

Maximization of the heat dispersion in convex domains
where D * is the ball having the same W n−1 quermassintegral of D, that is W n−1 (D) = W n−1 (D * ).
We denote by v m = v(R + δ) = min Ω * v and by max Ω * v = v(R) = 1. Being v radial, the modulus of the gradient of v is constant on the level lines of v.
Let us consider the function Let d(x) be the distance of a point x ∈ Ω from D and set ds.
By construction w ∈ H 1 (Ω) and being G decreasing it results: Then Being W n−1 (D) = W n−1 (D * ), using the Steiner formula and (15), we get for w m < t ≤ 1 and ρ = G −1 (t) then, by co-area formula and (16), As regards the boundary behavior, since by construction w = w m = v m on ∂Ω (see (16)), and P (Ω) ≤ P (Ω * ), we have

Remarks and Open Problems
As regards the infimum of I β,δ (D) among sets with fixed perimeter, it is easy to show that, without any geometrical restriction, such an infimum is zero. For example, for n = 2, even limiting the analysis to open connected sets is not enough to bound the infimum away from zero. To this aim we consider the sequence of sets Q k = 0, 1 k 2 .
Then we construct the set D k by removing from Q k k 2 disjoint closed squares of side 1 4k 2 − 1 k 3 each (k ≥ 5). Then for k sufficiently large, it holds that I δ,β (D k ) = I δ,β (Q k ). Hence I β,δ (D k ) → 0 as k → +∞, while for any k P (D k ) = 1.
The class of simply connected sets suffers the same problem. However, among planar convex sets of given perimeter, any minimising sequence for I β,δ (D) admits, by Blaschke-Santaló Theorem, a subsequence which converges (possibly degenerating into a segment) to some convex body of same perimeter (to be understood as twice its length, or Minkowski content, in case of a segment). Since the functional I β,δ (·) is continuous under converging sequence of convex bodies, a minimum is achieved. This leads to the following Open Problem 1. Find the minimum of I β,δ (·) in the class of convex planar sets of given perimeter.

Another closely related question is
Open Problem 2. Prove or disprove in any dimension that the minimum of I β,δ (·) among sets of given volume exists.
For β → ∞ the functional I β,δ (D) converges to the capacity of the setD with respect to D + δB. For the capacity one can employ standard techniques, like the Schwarz symmetrization, to deduce that a minimum is achieved on balls. This suggests that, at least for large β, balls might provide a positive answer to Open Problem 2.
Yet another important question is Again, comparison with Capacitary problem and common sense suggest that the minimum is achieved when Ω and D are concentric balls. The fact that one can not prescribe the exact volume of Ω but only an upper bound is due to the some counterintuitive behaviour of the functional I β (D; Ω) which is very peculiar and that can be summarised in next two Propositions.
In particular, if n = 2, computing the heat dispersion I β,δ (B R ), we have We recall that ν is the outer normal to B R+δ \B R . Let us observe that We have Therefore in the regime β "small", insulation increases the heat dispersion if the insulator thickness is below a certain threshold value. A more careful analysis brings to a sharper result.
To prove this result, we first need the subsequent Lemma 5.1. We introduce first the following notation.
We first need the following Lemma.
Proof. By the isoperimetric inequality it holds that Hence ∆P ≥ nω 1 n n (ω n R n + ∆V ) 1− 1 n − nω n R n−1 and then, if ∆V ≤ δ 0 , we have that where C is the constant in (17), and this complete the proof.
On the other hand, Choosing ε = 1 β it holds that Therefore, being R < 1 β , for δ 0 sufficiently small the constant C is larger than β. Hence the inequality (20) implies (18).