Sobolev spaces on metric measure spaces 

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Shape optimization problems

A shape optimization problem is a variational problem, in which the family of competitors consists of shapes, i.e. geometric objects that can be chosen to be metric spaces, manifolds or just domains in the Euclidean space. The shape optimization problems are usually written in the form n o min F(Ω) : Ω∈A , (1.1.1)
where
F is a cost functional,
A is an admissible family (set, class) of shapes.
If there is a set Ω ∈ A which realizes the minimum in (1.1.1), we call it an optimal shape, optimal set or simply a solution of (1.1.1). The theory of shape optimization concerns, in par-ticular, the existence of optimal domains and their properties. These questions are of particular interest in the physics and engineering, where the cost functional F represents some energy we would like to minimize and the admissible class is the variety of shapes we are able to produce. We refer to the books [21], [71] and [72] for an extensive introduction to the shape optimization problems and their applications.
We are mainly interested in the class of shape optimization problems, where the admissible family of shapes consists of subsets of a given ambient space D. In this case we will sometimes call the variables Ω ∈ A domains instead of shapes. The set D is called design region and can be chosen to be a subset of Rd, a differentiable manifold or a metric space. A typical example of an admissible class is the following:n o A = Ω : Ω ⊂ D, Ω open, |Ω| ≤ c , where D is a bounded open set in Rd, |•| is the Lebesgue measure and c is a positive real number. The cost functionals F we consider are defined on the admissible class of domains A through the solutions of some partial differential equation on each Ω ∈ A. The typical examples of cost functionals are:
• energy functionals �
F(Ω) = Ω g x, u(x), ∇u(x) dx,
where g is a given function and u ∈ H01(Ω) is the weak solution of the equation −Δu = f in Ω , u ∈ H01(Ω),
where f is a fixed function in L2(D) and H01(Ω) is the Sobolev space of square integrable functions with square integrable distributional gradient on Ω. spectral functionals
F(Ω) = F λ1(Ω), . . . , λk(Ω) ,
where F : Rk → R is a given function and λk(Ω) is the kth eigenvalue of the Dirichlet Laplacian on Ω, i.e. the kth smallest number such that the equation −Δuk = λk(Ω)uk in Ω , uk ∈ H01(Ω), has a non-trivial solution. o

Why quasi-open sets?

In this section, we consider nthe shape optimization problem
min E(Ω) : Ω ⊂ D, Ω open, |Ω| = 1 , (1.2.1)
where D ⊂ Rd is a bounded open set (a box) of Lebesgue measure |D| ≥ 1 and E(Ω) is the Dirichlet Energy of Ω, i.e.
E(Ω) = min Ω u dx : u ∈ H01(Ω) . (1.2.2)
In the terms of the previous section, we consider the shape optimization problem (1.1.1) with admissible set n o
A= Ω: Ω⊂D, Ω open, |Ω|=1 , and cost functional E(Ω) = − 1 Z
Ω wΩ dx, (1.2.3)
where wΩ is the weak solution of the equation
− ΔwΩ = 1 in Ω , wΩ ∈ H01(Ω). (1.2.4)
Indeed, wΩ is the unique minimizer in H 1
Z(Ω) of the functionalZ E(Ω) = Ω |∇wΩ|2 dx − Ω wΩ dx. (1.2.5)
On the other hand, using wΩ as a testZ function in (1.2.4),Z we have that Ω |∇wΩ|2 dx = Ω wΩ dx, (1.2.6) which, together with (1.2.5), gives (1.2.3).
Remark 1.2.1. The functional T (Ω) = −E(Ω) is called torsion energy or just torsion. We will call the function wΩ energy function or sometimes torsion function.
Before we proceed, we recall some well-known properties of the energy functions.
(Weak maximum principle) If U ⊂ Ω are open sets, then 0 ≤ wU ≤ wΩ. In particular, the Dirichlet Energy is decreasing with respect to inclusion
E(Ω) ≤ E(U) ≤ 0.
(Strong maximum principle) wΩ > 0 on Ω. Indeed, for any ball B = Br(x0) ⊂ Ω, by the weak maximum principle, we have wΩ ≥ wB. On the other hand, wB can be written explicitly as wB(x) = r2 − |x − x0|2 , 2d which is strictly positive on Br(x0).
(A priori estimate) The energy function wΩ is bounded in H01(Ω) by the constant depending only on the Lebesgue measure of Ω. Indeed, by (1.2.6) and the H¨older inequality, we have 2 ≤ kwΩkL ≤ |Ω| d+2 kwΩk L d−2 ≤ Cd|Ω| d+2 k∇wΩkL , (1.2.7) k∇wΩkL2 2d 2d 1 2d 2 where Cd is the constant in the Gagliardo-Nirenberg-Sobolev inequality in Rd.
We now try to solve the shape optimization problem (1.2.11) by a direct method. Indeed, let Ωn be a minimizing sequence for (1.2.11) and let, for simplicity, wn := wΩn . By the estimate (1.2.7), we have k∇wnk ≤ Cd, ∀n ∈ N.
By the boundedness of D, the inclusion H01(D) ⊂ L2(D) is compact and so, up to a subsequence, we may suppose that wn converges to w ∈ H01(D) strongly in L2(D). Suppose that Ω = {w > 0} is an open set. Then, we have semicontinuity of the Dirichlet Energy E(Ω) ≤ lim inf E(Ω ). (1.2.8) n →∞ n
Indeed, since w H1 (Ω), we have that Z ∈ 0 1 Z Z E(Ω) ≤ Ω |∇w|2 dx − Ω w dx ≤ lim inf Ω |∇ w n| 2 dx − w dx
2 n
n→∞ Ω
= lim inf E(Ωn).
n→∞
semicontinuity of the Lebesgue measure
|Ω| ≤ lim inf |Ωn|. (1.2.9)
n →∞
This follows by the Fatou Lemma and the fact that Ω ≤ lim inf Ωn , (1.2.10) n →∞ where Ω is the characteristics function of Ω. Indeed, by the strong maximum principle, we have that Ωn = {wn > 0}.
On the other hand, we may suppose, again up to extracting a subsequence, that wn converges to w almost everywhere. Thus, if x ∈ Ω, then w(x) > 0 and so wn(x) > 0 definitively, i.e. x ∈ Ωn definitively, which proves (1.2.10).
Let Ω ⊂ D be an open set of unit measure, containing Ω. Then, we have that Ω ∈ A and, by the monotonicity of E and (1.2.8), e e lim inf E(Ω ), E(Ω) ≤ E(Ω) ≤ n→∞ n i.e. Ω is an optimal domain for (1.2.11). In conclusion, we obtained that, under the assumption that {w > 0} is an open set, the shape optimization problem (1.2.11) has a solution. Unfor-tunately, at the moment, since w is just a Sobolev function, there is no reason to believe that {w > 0} is open. In fact the proof of this fact would require some regularity arguments which can be quite involved even in the simple case when the cost functional is the Dirichlet Energy E. Similar arguments applied to more general energy and spectral functionals can be compli-cated enough (if even possible) to discourage any attempt of providing a general theory of shape optimization.
An alternative approach is relaxing the problem to a wider class of admissible sets. The above considerations suggest that the class of quasi-open sets, i.e. the level sets of Sobolev functions, is a good candidate for a family, where optimal domains may exist. Indeed, it was first proved in [33] that the shapen optimization problem o min E(Ω) : Ω ⊂ D, Ω quasi-open, |Ω| = 1 , (1.2.11) has a solution. After defining appropriately the Sobolev spaces and the PDEs on domains which are not open sets, we will see that the same proof works even in the general framework of a metric measure spaces and for a large class of cost functionals decreasing with respect to the set inclusion. For example, one mayn prove that there is a solution of the oproblem min λk(Ω) : Ω ⊂ D, Ω quasi-open, |Ω| = 1 , (1.2.12) where λk(Ω) is variationally characterized as R R|∇u|2 dx (Ω). Indeed, if Ωn is a λk(Ω) = min max , K⊂H01(Ω) u∈K,u6=0 u2 dx where the minimum is taken over all k-dimensional subspaces K �of H1 minimizing sequence, then we consider the vectors (un1, . . . , unk) ∈ H01(Ωn) k of eigenfunctions, orthonormal in L2. We may suppose that for each j = 1, . . . , k there is a function uj ∈ H01(D) such that unj → uj in L2. Arguing as in the case of the Dirichlet Energy, it is not hard to prove that the (quasi-open) set = {uj 6= 0}, j=1 is a solution of (1.2.12).

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Compactness and monotonicity assumptions in the shape optimization

In the previous section we sketched the proofs of the existence of an optimal domain for the problems (1.2.11) and (1.2.12). The essential ingredients for these existence results were the following assumptions:
The compactness of the inclusion H01(D) ⊂ L2(D) in the design region D;
The monotonicity of the cost functional F.
In Chapter 2 we prove a general existence result under the above assumptions, even in the case when D is just a metric space endowed with a finite measure. Nevertheless, non-trivial shape optimization problems can be stated without imposing these conditions. For example, by a standard symmetrization argument

Table of contents :

Chapter 1. Introduction and examples 
1.1. Shape optimization problems
1.2. Why quasi-open sets?
1.3. Compactness and monotonicity assumptions in the shape optimization
1.4. Lipschitz regularity of the state functions
Chapter 2. Shape optimization problems in a box 
2.1. Sobolev spaces on metric measure spaces
2.2. The strong-γ and weak-γ convergence of energy domains
2.3. Capacity, quasi-open sets and quasi-continuous functions
2.4. Existence of optimal sets in a box
Chapter 3. Capacitary measures 
3.1. Sobolev spaces in Rd
3.2. Capacitary measures and the spaces H1 μ
3.3. Torsional rigidity and torsion function
3.4. PDEs involving capacitary measures
3.5. The γ-convergence of capacitary measures
3.6. The γ-convergence in a box of finite measure
3.7. Concentration-compactness principle for capacitary measures
Chapter 4. Subsolutions of shape functionals 
4.1. Introduction
4.2. Shape subsolutions for the Dirichlet Energy
4.3. Interaction between energy subsolutions
4.4. Subsolutions for spectral functionals with measure penalization
4.5. Subsolutions for functionals depending on potentials and weights
4.6. Subsolutions for spectral functionals with perimeter penalization
4.7. Subsolutions for spectral-energy functionals
Chapter 5. Shape supersolutions and quasi-minimizers 
5.1. Introduction and motivation
5.2. Preliminary results
5.3. Lipschitz continuity of energy quasi-minimizers
5.4. Shape quasi-minimizers for Dirichlet eigenvalues
5.5. Shape supersolutions of spectral functionals
5.6. Measurable sets of positive curvature
5.7. Subsolutions and supersolutions
Chapter 6. Spectral optimization problems in Rd 
6.1. Optimal sets for the kth eigenvalue of the Dirichlet Laplacian
6.2. Spectral optimization problems in a box revisited
6.3. Spectral optimization problems with internal constraint
6.4. Optimal sets for spectral functionals with perimeter constraint
6.5. Optimal potentials for Schr¨odinger operators
6.6. Optimal measures for spectral-torsion functionals
6.7. Multiphase spectral optimization problems
Chapter 7. Appendix: Shape optimization problems for graphs 
7.1. Sobolev space and Dirichlet Energy of a rectifiable set
7.2. Sobolev space and Dirichlet Energy of a metric graph
7.3. Some examples of optimal metric graphs
Bibliography

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