UNIVERSIT `
A DI PISA
Scuola di Dottorato in Scienze di Base ”Galileo Galilei” Dottorato in matematica
XXXI Ciclo
Shape Optimisation Problems for
Integral Functionals and
Regularity Properties of Optimal
Domains
Harish Shrivastava
Relatore:
Prof. Giuseppe Buttazzo
Co-ordinatore Del Dottorato:
Prof. Giovanni Alberti
Contents
Chapter 1. Introduction 3
1.1. Some Examples 5
1.2. Plan of the Thesis 8
Chapter 2. Preliminaries 13
2.1. Lower Semicontinuity Results and the Direct Method 14
2.2. p-Capacity 18
2.3. γ−convergence of Quasi open subsets of RN 20
2.4. Cases when the family of open subsets is γ-compact 28
2.5. Weak-γ Convergence for Quasi Open sets 31
2.6. Capacitary Measures 31
Chapter 3. Setting up the Problem and Existence Results 39
3.1. The Problem 39
3.2. Existence 40
Chapter 4. Regularity of Optimal Domain 45
4.1. Optimal Domain is Open 45
4.2. Optimal Domain has finite Perimeter 50
Chapter 5. Additional Regularity 55
5.1. The Free Boundary Problem 56
5.2. Distance like behaviour of the solution 57
5.3. Ahlfors Regularity of the Free Boundary 69
5.4. Penalized problem and Problem with measure constraint 71
Bibliography 75
CHAPTER 1
Introduction
The thesis is devoted to the study of shape optimisation problems. The main emphasis shall be on the problems involving a general integral function.
Shape optimisation provide a systematic framework for studying the optimal shapes described by various physical or mechanical models. The subject attends to shapes instead of functions. Ancient civilisations had been dealing with various questions on how to optimise shape of an object to make it most resistant, or most noiseless, or most streamlined, or lightest, or anything depending on the need. Our modern understanding of mathematics and development of computation technology has given the study of shape optimisation a whole new life.
The shape optimisation has become increasingly popular in academics and in-dustry, partly because of increase in the cost of raw materials, which has made optimisation of mechanical parts necessary from early stage of design.
A shape optimisation problem is easy to pose: we analyse a problem of the kind: find Ω∗ ⊂ D such that Ω∗ ∈ A and J(Ω∗) = min
Ω∈AJ (Ω)
for some set functional J (Ω) and A is a class of admissible subsets of D ⊂ RN. We
shall call D as the design set and J as the shape functional.
The shape functional J (Ω) is defined by the shape of the domain Ω. To obtain a function associated to the shape of the domain, we usually consider a PDE which involve a second order operator which acts on the domain Ω. Usually, we consider the following PDEs associated to the domain Ω:
−∆w = 1 or − ∆u = λu, with Dirichlet conditions on the boundary ∂Ω.
In the thesis, we shall be considering the p-Laplacian operator: ∆p. It is defined
as
∆pu = div(|∇u|p−2∇u),
and we shall be considering the cases for p ∈ (1, ∞). For a general case other than
p = 2, the absence of Hilbert space structure poses its own difficulties in proving
various results. We shall discuss the issue in greater details in coming chapters. The few possible ways in which a shape functional can be defined are the following:
• Shape functional of the form:
J (Ω) =
Z
Ω
j(x, u, ∇u) dx
where u solves some PDE associated to the domain Ω. • Shape functional defined by the energy of the form:
J (Ω) =
Z
Ω
j(x, uΩ, ∇uΩ) dx
where uΩ is one of the minimizers of the same functional
R
Ωj(x, v, ∇v) dx
among all v ∈ W01,p(Ω).
• Spectral functionals of the form J(Ω) = F (Λ(Ω)), where Λ(Ω) is collection of terms from the spectrum of Ω.
• There are other possible shape functionals like
J (Ω) =
Z
∂Ω
f (x, ν(x)) dHN −1.
This shape functional is studied to study the shape of the body which shall provide least resistance to a moving fluid.
The main focus of the thesis shall be on the functionals of the form:
J (Ω) =
Z
Ω
j(x, uΩ, ∇uΩ) dx,
with uΩ being the minimizer of
R
Ωj(x, v, ∇v) dx among all v ∈ W 1,p
0 (Ω). As we
shall be discussing in the Chapter 3, J (Ω) mentioned above can be re-written in the following form: J (Ω) = Z D h(x, uΩ, ∇uΩ) + a(x)χ{uΩ,0} dx, with h(x, 0, 0) = 0.
The domain Ω is Ω = {uΩ , 0} (we refer to Chapter 4). In this way we can
translate the problem of searching an optimal domain into problem of searching a function which minimizes an energy. The regularity properties of u nearby the set
∂{uΩ , 0} governs the regularity of the optimal shape Ω. We call the term containing
χ{uΩ,0} in the energy functional the ”Free Boundary term”. The set ∂{v , 0} the
”Free Boundary” of the function v. The above version of the problem leads us to apply the techniques by Caffarelli in the subject of free boundary problems to shape
1.1. SOME EXAMPLES 5
optimization. The free boundary aspect of shape optimization problems shall be discussed in the last chapter of the thesis.
We mostly deal with the following questions while studying the shape optimisa-tion.
Existence. To study the existence we make a deep study of the dependence of the shape
functional on the domain. We also study the compactness properties of the class of admissible shapes.
Regularity. We study the properties of optimal shape like connectedness, openness,
con-vexity, singularity etc.
1.1. Some Examples
1.1.1. Historical Examples. One classical example of shape optimisation
prob-lem is the following: Given a fence of fixed length, we wish to find the shape of the largest area which can be surrounded by that fence. Greeks knew that the answer to this problem is the circle.
In the language of modern mathematics, this translates as the isoperimetric prob-lem. We know the isoperimetric inequality, for a domain Ω with area |Ω| and perime-ter Per(Ω), we have
|Ω| ≤ 1
4π(Per(Ω))
2.
The equality holds in the case of a disc. Another equivalent to above problem is Queen Dido’s problem mentioned in Virgil’s Aeneid. The Queen was promised a piece of land which could be surrounded by a bullhide. Then the queen cut the hide into strips and surrounded the maximum possible land with the strips as its boundary.
The analogous to isoperimetric inequality for any dimension N is: |Ω|N −1 ≤ 1
NNω N
(Per(Ω))N,
ωN is the volume of the unit ball in RN. the equality holds when Ω is a sphere.
We had to wait until 19th century for a proof of the isoperimetric problem by Steiner, who assumed the existence of the optimal shape and proved that the optimal shape is a circle. The proof by Steiner was completed by Carath´eodory (we refer to [42]). H.A. Schwarz proved the isoperimetric inequality for the dimension 3, [41] and E. Schmidt proved it for a general case [40]. We also refer to [4], [5] for other proofs of the isoperimetric inequality.
We remark that the solution of the isoperimetric problem is not unique but unique up to displacement.
1.1.2. Soap Bubbles. The problem is to find the equilibrium surface taken by
a soap bubble or film. For example if one has a rigid frame and frame is dipped in soapy water, we are asked to determine the shape of the soap film fixed to the frame. Formalising the above problem in language of mathematics, we consider a curve γ in R3 and since we already know the surface energy of a film is directly proportional to its surface area, we study the problem:
find S∗, such that |S∗| = min{|S|, S is the surface surrounded by γ}.
|S| is the surface area of a surface S. S∗ is called the minimal surface associated to
the curve γ.
A generalisation to the soap film problem if the problem of capillary surfaces. Consider a tank containing some liquid. The liquid adheres to the wall by a capillary action. We need to find the shape of the liquid air interface inside the tank. Math-ematically, let D ⊂ R3 be a smooth and bounded set. Here, D corresponds to the
tank. Let V0 be the volume of the liquid. We denote by Ω, the domain occupied by
the liquid. Energy of the system consisting of the tank and the liquid is the sum of two energies: surface tension given by
E1(Ω) = Area(∂Ω ∩ D) + cos θArea(∂Ω ∩ ∂D)
and potential energy by gravity :
E2(Ω) = −
Z
Ω
k(x) dx
where θ is angle between the liquid and the tank. k(·) is a bounded function.
The principle of least energy states that the shape of optimal shape Ω is the one which minimises the total energy. i.e. we solve the shape optimisation problem:
find Ω∗ ⊂ D such that
E1(Ω∗) + E2(Ω∗) = min
n
E1(Ω) + E2(Ω) : Ω ⊂ D, |Ω| = V0
o
.
Remark 1.1. One can prove that the optimal shape Ω has a constant mean curvature everywhere over the free boundary ∂Ω ∩ D and ∂Ω makes a constant angle with the wall of the tank.
1.1. SOME EXAMPLES 7
1.1.3. Eigenvalue Problem. Eigenvalues are useful to define the functional
depending on the shape of a domain. The problem of minimizing an eigenvalue (e.g. eigenvalue of Laplace operator with some boundary conditions) has brought many serious works in early 20th century.
The question of minimising an eigenvalue was raised for the first time in Lord Rayleigh’s book ”The theory of sound” which posed a problem :
minnλ1(Ω) : |Ω| = A
o
,
where λ1 is the first eigenvalue of Dirichlet Laplacian. A is a fixed number A ∈ (0, ∞).
Some explicit computations and physical evidences in the book led to the conjucture that the minimiser of first eigenvalue of Dirichlet Laplacian among all domains of fixed measure should be a ball of measure A.
Faber ( [27]) and Krahn ( [36]) solved the Rayleigh’s conjecture in 1920s using a symmetrisation technique. The minimisation problem for second Dirichlet eigenvalue
λ2 followed as an easy corollary shown by Krahn ( [37]). The minimiser for λ2 is
union of two disjoint balls.
For shape optimisation problem for minimisation of λk, k ≥ 3 is more difficult.
In 1993, Buttazzo and Dal Maso ( [15]) obtained a proof of existence of the optimal domain for the third Dirichlet eigenvalue. They obtained the existence result as a particular case of a more general problem :
min{F (Ω) | |Ω| = A},
here F is a non-increasing set function, non-increasing with respect inclusions. We shall also be using same arguments as Buttazzo and Dal Maso for the proof of exis-tence in the thesis. We refer the reader to Chapter 3.
Other widely studied eigenvalue problems which involve operators other than the Laplacian are:
• Find Ω such that
−∆2u = Γ(Ω)u in Ω, u = 0 on ∂Ω, ∂u ∂n = 0 on ∂Ω.
This is called the clamped plate problem. Rayleigh conjectured that the ball should be the solution of the clapmed plate problem. N. Nadirashvili proved the conjecture for dimension 2 ( [39]). R. Benguria and M. Ashbaugh proved it for dimension 3 ( [2]). For higher dimensions, the problem is still open.
• The buckling plate problem, which we state as :
Find Ω such that
−∆2u = Γ(Ω)∆u in Ω, u = 0 on ∂Ω, ∂u ∂n = 0 on ∂Ω.
P´olya and Szeg¨o conjectured that the ball is a solution when the candidate domains have a fixed volume. They could also prove the conjecture while assuming that the first eigenvalue is non-negative. M. Ashbaugh and D. Bucur ( [3])proved the existence of solution in class of simply connected open sets. Regularity is still an open problem.
1.2. Plan of the Thesis
Chapter 2 covers the preliminary results and the definitions which shall be re-quired to have a better understanding of forthcoming chapters. We shall start with the definition and various characterisations of the lower semicontinuity (see Defini-tion 2.2) of a funcDefini-tional defined on a Sobolev space. Given an open set Ω ⊂ D, we define the Sobolev space W1,p(Ω) as :
W1,p(Ω) = nf ∈ Lp(Ω) : ∇f ∈ Lp(Ω)o,
here Lp(Ω) is the set of p-integrable functions on Ω, i.e. :
Lp(Ω) =nf : Ω → R : Z Ω |f |pdx < +∞o Lp(Ω) is a normed space: ||f ||Lp(Ω)= Z Ω |f |pdx1/p
∇f corresponds to the weak derivatives of the function f . Weak derivative is defined as follows:
Definition 1.2. The function g : Ω → R is a weak derivative of f in direction ν
(represented as ∂νf ) if the following holds:
Z Ω f ∂νφ dx = − Z Ω gφ dx ∀φ ∈ Cc∞(Ω).
In this case, we say ∂νf = g in weak sense.
The Sobolev space W01,p(Ω) is defined as closure of Cc∞(Ω) in the W1,p(Ω) norm
||u||W1,p(Ω)= ||u||Lp(Ω)+ ||∇u||Lp(Ω)
1.2. PLAN OF THE THESIS 9
Remark 1.3. In the above definitions we have assumed that the domain Ω is an open set. But we shall see in Section 2.2 that we can use notion of capacity to define Sobolev space W01,p(Ω) on the sets which are not open (but quasi open, see Definition 2.22). We refer to Chapter 2 for more details.
We remark that proving the lower semicontinuity is one of the two major steps for proving the existence result in calculus of variations. Other important step is proving the compactness of the topology involved.
In the Section 2.2 we have introduced the notion of capacity of a set. Capacity of a set is in a sense analogue of measure, but unlike measure, capacity of a set keeps track of the first derivative of the functions concentrated around the given set. The statement which holds true on the set apart from the sets of capacity zero is said to hold true ”quasi everywhere”.
The notion of capacity is an important tool to characterise the γ-topology of quasi open sets (see Definition 2.47), which analyse the mapping Ω → uΩ ∈ W01,p(Ω). Here,
uΩ solves a PDE (2.3.1) associated to the shape of the set Ω. For our case, we have
adopted the definition of γ-topology as equivalent to the Γ-topology of associated
p-torsion functionals (see Definition 2.47).
We conclude the preliminary chapter by introducing the notion of p-capacitary measures. In the famous example by Cioranescu and Murat (see [18]), we have a given sequence of open subsets {Ωn}, constructed by deleting the holes of appropriate size,
periodically from a ball (we refer to Example 2.70 for details). The corresponding sequence {un} of solutions of the PDE
−∆u = f in Ωn,
un∈ H01(Ωn)
converges weakly to a function solving the following PDE:
−∆u + cu = f in D,
un ∈ H01(D)
for some appropriate constant c. The completion of γ topology of the family of quasi open sets is studied in [21] for p = 2 and in [22] for a general value of p. It states that the solutions of the PDE:
−∆pu = f in Ωn,
converge weakly to u such that: −∆pu + µu = f in Ωn, un ∈ W01,p(Ωn)
where µ is a capacitary measure. The set of p-capacitary measures is the completion of quasi open sets in the γ-topology. Hence, the shape optimisation problems which do not attain a quasi open set as a minimiser, attains a minimiser as a capacitary measure.
In the Chapter 4 we have presented the shape optimisation problem which we shall be dealing in the most of the part of the thesis. The problem is minimisation of an integral functional, which satisfy some assumptions (f 1), (f 2), and (f 3) to ensure that that the functional attains a minimiser in the family of admissible subsets of the design set D. (we refer to Section 3.2).
We have presented two proofs of the existence of the minimiser of the problem (3.1.2). The first proof works for rather general class of shape optimisation prob-lems, which include problems involving integral functionals considered in the thesis, but also the problems involving spectral functinals or any other set functional lower semicontinuous with respect to γ topology of quasi open sets (see defintion 2.47)
The second proof, which shall also appear in [16] works only for a restricted class of integral functionals. Although, the proof involves simpler machinery. We discuss a counter example of non-existence of an optimal domain for a shape optimisation problem in the form of a control problem. Even a very shape optimisation simple problem in the form stated in Example 3.2 can fail to have an optimal domain.
In the Chapter 4 we have discussed the conditions under which the optimal shape is open and when the optimal shape has a finite perimeter. We have used the regu-larity theory of quasi minimisers by Giaquinta and Giusti dveloped in [30] and [31]. In [30], Giaquinta and Giusti deals with the regularity theory of quasi minimiser of an integral functional, whose integrand satisfy a condition of the type (5.2.2). They study the regularity of quasi minimiser defined in the Section 4.1.1, instead of actual minimiser of the functional. And this was done without passing through the Euler equation. The techniques used by Giaquinta and Giusti in [30] and [31] permits us to give regularity results when the integrand is less regular. For example, without assuming the growth conditions on the derivatives of the integrand, i.e. when the integrand is not Gateaux differentiable. We also discuss a counter example where the optimal domain of an integral problem is not an open set, but just a quasi open set.
1.2. PLAN OF THE THESIS 11
We stress on the remark that it is easier to construct counter examples for the shape optimisation problems with a measure constraint on the candidate domains, i.e. the problems of the form (4.1.3). We also did the same thing in Example 3.2 to prove that no optimal domain exists. But, on the other hand, we have proved various theo-rems proving existence and regularity for the shape optimisation problems involving integral functionals with penalisation on the measure of the candidate domain (the problems of the form (3.2.4)). We refer to the Section 5.4 for the discussion on how the problems of form (3.2.4) and (4.1.3) are related.
In Section 4.1 we have used the technique used by Dorin Bucur in [10] in which he proved the minimiser if the kth eignvalue of the Dirichlet laplacian has a finite perimeter. He proved that a local shape subsolution has a finite perimeter. We adapted the proof to our case of a more general intergral functional. We proved that the optimal shape of the problem (3.1.2) has finite perimeter.
The main idea in the proof is that the link between the integration of gradient of a function and the perimeter of its free boundary comes from the co-area formulae.
In the final chapter, we make an attempt to study the smoothness of the optimal shape by adapting the proofs by Alt-Caffarelli in [1] and [17]. We again study the penalised form of shape optimisation problem for the regularity. As mentioned in the chapter, for a shape optimisation problem in the form of constrained problem, we can very easily construct counter examples of the problems which have an optimal domain as irregular as we please. On the other hand, we have proved that the penalising term in the shape optimisation problem of the form (3.2.4) forces some regularity on the solution. That is why, the two forms of shape optimisation problems: penalised one (3.2.4) and the one with constraint (4.1.3) are generally not equivalent.
One way of forcing regularity on the solution in a constrained shape optimisation problem is to ensure the saturation condition mentioned in [8], [32], [9] which states that:
Given D ⊂ RN, open, smooth and bounded. For the minimisation problem minn Z Ω (1 2|∇u| 2 − f u) dx : |{u , 0}| ≤ mo,
there exists no function u∗ ∈ H1
0(D) such that −∆u∗ = f in D, u∗ ∈ H1 0(D) and |{u ∗ , 0}| ≤ m.
The counter examples which we have constructed in the thesis do not satisfy the saturation condition mentioned above.
CHAPTER 2
Preliminaries
In this chapter, we shall be discussing adequate topology of subsets in order to find an optimal domain of shape functional. In most of the cases, the class of open sets is not big enough to find an optimal set of the functional. For this reason we introduce a broader family of sets, which also include the open sets and also we define a metrisable topology on that class of sets.
Moreover, we shall see in Section 2.6, that in many cases we cannot expect the existence of an optimal domain. We define the family of capacitary measures, which is even broader class and includes sets characterised as measures (we refer to (2.6.2)), the family of capacitary measures on a set is a complete set.
However, the sequence of quasi open subsets subsets defined in the Definition 2.22 may converge to a set in the same family provided they satisfy some stability conditions which we shall discuss in section
But, before defining the topology of subsets/measures we look at the preliminary definitions and other tools. Reader is referred to [11], [33], [14] for detailed exposure to upcoming notions.
Remark 2.1. We shall always be considering the subset D ⊂ RN as an open, smooth and bounded. And any set or family of sets addressed in the thesis shall be by default, considered as subset of D, we shall address them otherwise only if there is a scope of an ambiguity.
2.1. Lower Semicontinuity Results and the Direct Method
The most classical way to obtain the existence of a minimizer is ”the direct method”, which consists of proving that the functoinal (say F ) is coercive and lower semicontinuous with respect to some suitable topology. Coerciveness is an easy prop-erty to prove in most of the situations, we may say that the coerciveness is a quan-titative property.
On the other hand, lower semicontinuity depends on the qualitative properties of the integrand f (x, s, z) if the functional F is defined on a suitable Sobolev space as:
F (u) =
Z
D
f (x, u, ∇u) dx.
First we define the lower semicontinuous function and a few equivalent notions: Definition 2.2. Let X be a topological space, and x ∈ X.
We denote N (x) as set of all open neighbourhoods of x. We say that a function f : X → ¯R is lower semi continuous at a point x ∈ X if for every t ∈ R, with t < F (x), there exists U ∈ N (x) such that t < F (y) for every y ∈ U . We say F is lower semicontinuous on X if F is lower semicontinuous at each point x ∈ X.
Remark 2.3. We define an upper semicontinuous function in a similar way by replacing the sign < with > in previous definition.
Remark 2.4. A function which is both upper continuous and lower semi-continuous, is in fact continuous.
Remark 2.5. From the above definition we can easily derive another equivalent definition lower semicontinuous functions: F : X → ¯R is lower semicontinuous at x if and only if
F (x) ≤ sup
U ∈N (x)
inf
y∈UF (y),
which implies, F is lower semicontinuous if for every sequence {xn} converging to x
in X
F (x) ≤ lim inf
n→∞ F (xn). (2.1.1)
The converse is not true in general, but it holds true is X satisfy first axiom of countability, i.e. the neighbourhood system of every point has a countable base. (we refer to [19]).
2.1. LOWER SEMICONTINUITY RESULTS AND THE DIRECT METHOD 15
Definition 2.6. If F satisfy (2.1.1), the functional F is called sequentially lower
semi-continuous. Sequentially upper semi-continuous function is defined in similar way.
Proposition 2.7. If X satisfy the first axiom of countability. Let F : X → ¯R be
a function and let x ∈ X. The following facts are equivalent: (1) F is lower semicontinuous at x.
(2) F (x) ≤ lim infn→∞F (xn) for every sequence {xn} converging to x in X.
(3) F (x) ≤ limn→∞F (xn) for every sequence {xn} converging to x in X and
such that limn→∞F (xn) exists and is less than ∞.
Other useful characterisation of lower semicontinuity comes from the property of epigraph and sub/super level sets of the function,
Definition 2.8 (Sub leve sets, Epigraphs). We define:
(1) For every F : X → ¯R and for every t ∈ R we define
{F ≥ t} = {x ∈ X : F (x) ≥ t}.
The sets {F > t}, {F ≤ t},{F < t} are defined in similar way by putting the corresponding inequalities on right hand side.
(2) The epigraph of F is defined as:
epi(F ) = {(x, t) ∈ X × R : F (x) ≤ t}.
Definition 2.9. A set A ⊂ X is sequentially closed if it contains all the limits
of the converging sequences. That is,
∀ {xn} ⊂ A such that xn → x ⇒ x ∈ A.
A sequentially open set is a set whose compliment is sequentially closed.
Definition 2.10. A set A is sequentially compact if every sequence in A has a
converging subsequence with its limit in A. i.e.
∀ {xn} ⊂ A, ∃ {xnk}k∈N such that xnk → x and x ∈ A.
Proposition 2.11. Let F : X → ¯R be a functional. The following properties are
equivalent:
(1) F is lower semicontinuous (resp. sequentially lower semicontinuous) on X; (2) for every t ∈ R the set {F > t} is open (resp. sequentially open) in X;
(3) for every t ∈ R the set {F ≤ t} is closed (resp. sequentially closed) in X; (4) epi(F ) is closed (resp. sequentially closed) in X × R.
The family of lower semicontinuous functions also satisfy stability property: Proposition 2.12. We have:
• Let {Fi}i∈I be a family of lower semicontinuous (resp. sequentially lower
semicontinuous) functions on X. Then the function F : X → ¯R defined as F (x) = supi∈IFi(x) is lower semi continuous (resp. sequentially lower
semicontinuous) on X. If I is finite, then the function G : X → ¯R defined by G(x) = infi∈IFi(x) is lower semi continuous (resp. sequentially lower
semi continous) on X.
• If F and G are lower semi continuous (resp. sequentially lower semi
contin-uous) on X and is F + G is well defined (i.e. (−∞, −∞) , (F (x), G(x)) ,
(∞, ∞) for every x ∈ X), then F + G is lower semi continuous (resp.
se-quentially lower semi continuous) on X.
2.1.1. Conditions for Lower Semicontinuity. The following theorem (see
[13], Theorem 4.1.1) states that given of a functinal F (u), defined as integral func-tional, the optimal condition for the lower semicontinuity in weak W1,p(D) topology
is that the integrand f (x, s, z) is convex with respect to z.
Theorem 2.13. Given that D has a Lipschitz boundary, and f : D × R × RN → [0, ∞] be such that f (x, ·, ·) is lower semicontinuous for almost every x ∈ D and
f (x, s, ·) is convex for almost every x ∈ D and ∀s ∈ R. Then F (u) =
Z
D
f (x, u, ∇u) dx, (2.1.2)
is sequentially lower semicontinuous in the weak topology of W1,1(D).
Remark 2.14. The above theorem is an immediate consequence of theorem 2.3.1 in [13] stating that the above result is true for f : D × R × RN → (−∞, ∞] satisfying
inf-compactness property:
for every sequence {un} converging in Lp(D) and every sequence
{vn} weakly converging in Lq(D) (weak∗ if q = ∞) such that for a
suitable constant c > 0 Z D f+(x, un, vn) dx ≤ c + Z D f−(x, un, vn) dx ∀n ∈ N,
2.1. LOWER SEMICONTINUITY RESULTS AND THE DIRECT METHOD 17
we can show that in this case F (u) defined in (2.1.2) is sequentially lower semicontin-uous in Lp(D) × w − Lq(D). (Strong topology of Lp(D) and weak topology of Lq(D),
(w∗ if q = ∞)).
The first proof of the result was given by De Giorgi in [24] for Carath´eodory integrands f ≥ 0. The version of the theorem in [13] is due to Ioffe [34]. Ioffe’s proof uses simpler ideas like Dunford-Pettis characterisation of weakly compact subsets of
L1(D) (see [13], Theorem 2.2.4) and Mazur’s theorem for weakly compact sequences.
The De Giorgi approach was, by a suitable use of measurable selection lemmas, to find appropriate approximations of the form
f (x, z, s) = sup{an(x, s) + hbn(x, s), zi : n ∈ N},
where an and bn are Caratheodory integrands.
We conclude this section with few commentaries on the direct method on calculus of variations.
Definition 2.15. We say that a function F : X → ¯R is coercive (resp.
sequen-tially coercive) on X, if the closure of the set {F ≤ t} is countably compact (resp. sequentially compact) in X for every t ∈ R.
Remark 2.16. We have:
• If F is coercive ( resp sequentially coercive) and G ≥ F on X then G is also coercive (resp. sequentially coercive) on X.
• If F is coercive (resp. sequentially coercive) on X, then every sequence in X with lim supn→∞F (xn) < ∞ has a cluster point (resp. convergent
subsequence) in X. The converse holds true in case when F is lower semi-continuous or if X is metrizable.
Definition 2.17. A minimum point (or minimizer) for F in X is a point x ∈ X
such that F (x) ≤ F (y) for every y ∈ X, i.e. F (x) = inf
y∈XF (y).
A minimizing sequence {xn} for F in X is a sequence such that
inf
y∈XF (y) = limn∈∞F (xn).
Note that every function F has a minimizing sequence.
Theorem 2.18. Assume that the function F : X → ¯R is coercive and lower
semi continuous (resp. sequentially coercive and sequentially lower semicontinuous). Then,
(1) F has a minimum point in X.
(2) If {xn} is a minimizing sequence of F in X and x is a cluster point of {xn}
(resp. x is the limit of subsequence of xn) then x is a minimum point of F
in X.
(3) If F is not identically +∞, then every minimizing sequence for F has a cluster point (resp. a convergent subsequence).
Remark 2.19. We can ensure that the minimizer of functional F is unique, under the assumption that F : X → ¯R is strictly convex, i.e. F . ∞ in X and
F (tx + (1 − t)y) < tF (x) + (1 − t)F (y)
for every t ∈ (0, 1) and for every x, y ∈ X such that x , y, F (x) < ∞, and F (y) < ∞
2.2. p-Capacity
Definition 2.20 (p-Capacity). Given a set E ⊂ RN, the p-Capacity of E is
defined as follows:
capp(E) = inf
Z
RN
(|∇u|p+ |u|p) dx : u ∈ UE
where UE is the class of functions
UE =
n
u ∈ W1,p(RN) : u ≥ 1 in a neighbourhood of Eo.
Let D ⊂ RN and E ⊂ D; the relative p-capacity of a set E with respect to D is defined as
capp(E, D) = inf
Z
D
(|∇u|p+ |u|p) dx : u ∈ UE
.
Remark 2.21. Notice that if E b D and cap(E) = 0, then capp(E; D) = capp(E).
We shall mostly be dealing with set of zero capacity or with sequence of subsets with capacity tending to zero.
We say that a property P (x) is true ”p-quasi everywhere” or ”p-q.e.” if it is true for all x except for a set of p-capacity zero.
Definition 2.22. (p-quasi open and p-quasi closed sets) We say a subset A ⊂ RN
is quasi open if we can approximate it by family open sets such that p-capacity of their symmetric difference with A vanishes to zero. Precisely, A ⊂ RN is quasi open
2.2. p-CAPACITY 19
if ∃ {A} ⊂ RN, open such that capp(A∆A) → 0 as → 0. We define quasi closed
sets in same sense.
Definition 2.23. We represent A(D) with set of all quasi open subsets of D. A(D) = {Ω ⊂ D : Ω is quasi open} .
Remark 2.24. In absence of ambiguity, we shall use the terms capacity (cap(A)), quasi open/closed, quasi everywhere, in place of separately mentioning capacity, p-quasi open/closed, etc.
Definition 2.25. (Quasi Continuous Functions) A function f : D → R is quasi
continuous if there exists a family of continuous functions {f} such that cap({f ,
f }) < and → 0.
Remark 2.26. It is well known that Sobolev functions u ∈ W1,p(D) have a quasi continuous representative, uniquely defined up to a set of capacity Zero. (see [25], Theorem 11.1.3) It is given by,
˜ u(x) = lim →0 Z B(x,) u(y) dy.
We shall identify any sobolev function u ∈ W1,p(D) with its quasi continuous
representative. This aids in imposing the point-wise conditions on u q.e. x ∈ D. For example, we can define capacity as:
cap(E, D) = inf
Z
D
|∇u|pdx : u ∈ W1,p(D) : u ≥ 1 q.e.in E.
Also, for defining the Sobolev spaces, we use the point-wise property of the quasi-continuous representative,
W01,p(E) = u ∈ W1,p(RN) : u = 0 q.e. in RN \ E.
We can also define a quasi open set as: A ⊂ RN is quasi open if ∃u ∈ W1,p(RN) (a
quasi continuous function) such that A = {u > 0}.
Remark 2.27. An important point to be noted is, we always consider p ≤ N because for p > N , even singletons have positive capacity. Hence anything true
2.3. γ−convergence of Quasi open subsets of RN
Before looking at the topology of the quasi open sets, we recall definition and some basic properties of Γ-convergence of functionals, which has turned out to be an important tool in studying the variational problems. For the detailed exposure to the subject, reader may refer to [6] and [19] We only mention the details useful for the characterisation of relaxed Dirichlet problem. The notion of γ-convergence of subsets and Γ-convergence of functionals are connected.
2.3.1. Γ-convergence and K-convergence. Just like in Section 1.2, X is a
topological space, the class of all open neighbourhoods of x ∈ X shall be denoted by N (x). Let {Fn} be a sequence of functions from X into ¯R.
Definition 2.28 (Γ−convergence). The Γ-lower limit and Γ-upper limit of the
sequence (Fn) are the functions from X into ¯R defined by: (Γ − lim inf
n→∞ Fn)(x) = supU ∈N (x)lim infn→∞ y∈Uinf Fn(y),
(Γ − lim sup
n→∞ Fn)(x) = supU ∈N (x)lim supn→∞ y∈Uinf Fn(y).
If there exists F : X → ¯R such that Γ − lim infn→∞Fn = Γ − lim supn→∞Fn = F ,
then we write F = Γ − limn→∞Fn and we say that the sequence {Fn} Γ-converges to
F (in X) or F is the Γ-limit of {Fn} in X.
Remark 2.29. Clearly we have, Γ − lim inf
n→∞ Fn≤ Γ − lim supn→∞ Fn.
Hence {Fn} Γ-converges to F if and only if
Γ − lim inf
n→∞ Fn≥ F ≥ Γ − lim supn→∞ Fn.
Remark 2.30. We can also define upper Γ-limit and lower Γ-limit of a sequence {Fn} as
(Γ − lim inf
n→∞ Fn)(x) = supU ∈B(x)lim infn→∞ y∈Uinf Fn(y),
(Γ − lim sup n→∞ Fn)(x) = sup U ∈B(x) lim sup n→∞ inf y∈UFn(y),
where B(x) is the base of neighbourhood system of x in X.
Remark 2.31. If the two sequence {Fn} and {Gn} coincide in an open set U of
2.3. γ−CONVERGENCE OF QUASI OPEN SUBSETS OF RN 21
Remark 2.32. When the sequence {Fn} is constant, either with respect to x or
with respect to n, we have:
• If the functions {Fn(x)} are independent of x, i.e., for every n ∈ N there
exists a constant an∈ ¯R such that Fn(x) = an for every x ∈ X, then:
(Γ − lim inf
n→∞ Fn)(x) = lim infn→∞ an; (Γ − lim supn→∞ Fn)(x) = lim supn→∞ an.
• If {Fn(x)} are independent of n, i.e. there exists F : X → ¯R such that
Fn(x) = F (x) for every x ∈ X and for every n ∈ N, then:
Γ − lim inf
n→∞ Fn= Γ − lim supn→∞ Fn= sc −
F,
where sc−F is the lower semicontinuous envelop of {Fn} defined below.
Definition 2.33. Let F : X → ¯R be a function. Then the lower semicontinuous
envelop of F (sc−F ) is defined for every x ∈ X by
(sc−F )(x) = sup
G∈G(F )
G(x), where,
G(F ) =nall lower semi-continuous functions G on X such that G(y) ≤ F (y) for every y ∈ Xo. We can equivalently define sc−F as:
(sc−F )(x) = sup
U ∈N (x)
inf
y∈UF (y).
Remark 2.34. Lower semicontinuous envelop of a function F is the greatest lower semicontinuous function majored by F . If F and G are two functions coinciding in an open set U then sc−F = sc−G in U .
We shall list the connection of Γ-convergence with the Kuratowski convergence (in short K-convergence) of the epigraphs. For this reason, Γ-convergence is also called epi-convergence.
Definition 2.35 (K-convergence of sequence of subsets of X). Let {En} be a
sequence of subsets of a topological space X.
The K-lower limit of denoted by K − lim infn→∞En, is the set of all points x ∈ X
with the following property:
for every U ∈ N (x) there exists k ∈ N such that U ∩ En , {φ} for
The K-upper limit, denoted by K − lim supn→∞En is the set of all points x ∈ X
with the following property:
for every U ∈ N (x) and for every k ∈ N there exists n ≥ k such that U ∩ En , {φ}.
That is if there exists E ⊂ X such that E = K − lim sup
n→∞ En= K − lim infn→∞ En,
then we write E = K − limn→∞En and we say that the sequence {En} converges to
E in the sense of Kuratowski or K-converges to E.
Remark 2.36. Other (more easily understood) way of saying that a sequence of subsets {En} K-converges to E if following situation is satisfied:
(a) For every x ∈ E and for every U ∈ N (x) there exists k ∈ N such that
U ∩ En , {φ} for every n ≥ k.
(b) For every x ∈ X \E there exists U ∈ N (x) and k ∈ N such that U ∩En= {φ}
for every n ≥ k.
Theorem 2.37. Let {Fn} be a sequence of functions from X into ¯R, and let
F0 = Γ − lim inf
n→∞ Fn; F
00 = Γ − lim sup n→∞ Fn.
Then,
epi(F0) = K − lim sup
n→∞
epi(Fn); epi(F00) = K − lim infn→∞ epi(Fn),
where K-limits are taken in the product topology of X × R. In particular {Fn}
Γ-converges to F in X if and only if {epi(Fn)} K-converges to epi(F ) in X × R.
Remark 2.38. Another characterization of Γ-convergence in terms of K-conver-gence is : Fn Γ − → F ⇐⇒ {F ≤ s} = \ t>s K − lim sup n→∞ {Fn ≤ t} = \ t>s K − lim inf n→∞ {Fn ≤ t} for every s ∈ R.
Next, we list some properties of Γ-convergence in comparison to pointwise conver-gence of the sequence functions. Note that, in gerneral, pointwise converconver-gence and Γ-convergence are independent.
Theorem 2.39. Given a sequence {Fn} of functions from X to ¯R, the following
2.3. γ−CONVERGENCE OF QUASI OPEN SUBSETS OF RN 23
(1) Γ − lim infn→∞Fn≤ lim infn→∞Fn and Γ − lim supn→∞Fn≤ lim supn→∞Fn.
That is, if Fn Γ
−
→ F and Fn→ G pointwise, then F ≤ G.
(2) If {Fn} converges to F uniformly, then Fn Γ − → sc−F . If F is lower semi continuous, then Fn Γ − → F .
(3) If {Fn} is an increasing sequence, then Γ − limn→∞Fn = limn→∞sc−Fn =
supn∈Nsc−Fn.
(4) If {Fn} is decreasing and Fn→ F pointwise, then Fn Γ
−
→ sc−F .
The pointwise convergence of a sequence of functions and Γ-convergence are equiv-alent provided the sequence is equi-lower semicontinuous as defined below:
Definition 2.40. We say that the sequence {Fn} is equi-lower semicontinuous at
a point x ∈ X if for every > 0 ther exists U ∈ N (x) such that Fn(y) ≥ Fn(x) −
for every y ∈ U and for every n ∈ N. We say {Fn} is equi-lower semicontinuous on
X if {Fn} is equi-lower semicontinuous at each x ∈ X. Equicontinuity is defined in
similar way.
Theorem 2.41. Assume {Fn} is equi-lower semicontinuous on X, then {Fn}
Γ-converges to F in X if and only if {Fn} converges to F pointwise in X. If X is
a normed vector space, and {Fn} is a sequence of convex functions on X. Suppose
{Fn} is equi bounded around x ∈ X then,
(Γ − lim inf
n→∞ Fn)(x) = lim infn→∞ Fn(x), (Γ − lim sup]n→∞ Fn)(x) = lim supn→∞ Fn(x).
That is, if {Fn} is equibounded sequence of convex functions in normed vector space
X then,
Fn→ F p.w. ⇐⇒ Fn Γ
− → F. We list some more properties of Γ-convergence
Theorem 2.42 (Properties of Γ− convergence). The following properties holds
for Γ convergence
(1) Every Γ limit is a lower semi-continuous functional.
(2) If {Fn} is equi-coercive on X, that is for every t ∈ R the set ∪n{Fn ≤ t} is
relatively compact in X, and {Fn} Γ converges to F , then F is also coercive
and admits a minimum on X. (3) Fn
Γ
−
Moreover, under the additional hypothesis that Fn and F have the unique
minimiser, (say xn and x, respectively)
Fn Γ
−
→ F ⇒ xn → x.
(4) (Compactness) From every sequence {Fn} of the functionals on X it is
pos-sible to extract a subsequence which Γ converges to a functional on X. (5) Let Φ : ¯R → ¯R be a continuous increasing function. Then
Γ − lim inf n→∞ (Φ ◦ Fn) = Φ ◦ (Γ − lim infn→∞ Fn), Γ − lim sup n→∞ (Φ ◦ Fn) = Φ ◦ (Γ − lim sup n→∞ Fn). This means, Fn Γ − → F ⇒ Φ ◦ Fn Γ − → Φ ◦ F .
(6) Given {Fn} and {Gn} two sequences of functions on X. The following
in-equations hold true provided the sums occurring in them are well defined:
Γ − lim inf
n→∞ (Fn+ Gn) ≥ Γ − lim infn→∞ Fn+ Γ − lim infn→∞ Gn,
Γ − lim sup n→∞ (Fn+ Gn) ≥ Γ − lim sup n→∞ Fn + Γ − lim inf n→∞ Gn. In particular, Fn Γ − → F , Gn Γ − → G and (Fn+ Gn) Γ − → H then, F + G ≤ H,
provided the sums Fn+ Gn and F + G are well defined on X.
A Γ-converging sequence is stable under addition to another sequence of functions provided the sequence added to original is continuously convergent in X which we are defining below:
Definition 2.43. As sequence of functionals, {Gn} is continuously convergent
to a functional G if for every x ∈ X and for every neighbourhood V of G(x) in ¯R
there exists k ∈ N and U ∈ N (x) such that Gn(y) ∈ V for every h ≥ k and for every
y ∈ U .
Theorem 2.44. Suppose that {Gn} is continuously convergent to a function G.
Also, Gn and G are everywhere finite in X. Then,
Γ − lim inf n→∞ (Fn+ Gn) = Γ − lim infn→∞ Fn+ G, Γ − lim sup n→∞ (Fn+ Gn) = Γ − lim sup n→∞ Fn + G.
2.3. γ−CONVERGENCE OF QUASI OPEN SUBSETS OF RN 25 In particular, Fn Γ − → F ⇒ Fn+ Gn Γ − → F + G.
Note that if there exists a fuctional G such that, Gn = G for all n ∈ N on X.
Then {Gn} satisfy hypothesis of the theorem.
Γ-limits and K-limts can be expressed in terms of convergent sequences, we shall mention the case when X satisfy the first axiom of countability
Definition 2.45 (Sequential Characterization of Γ-limits). Assume X satisfy the
first axiom of countability. Then the functional F0 = Γ − lim infn→∞Fn is
charac-terised by the following properties:
(a) For every x ∈ X and for every sequence {xn} converging to x in X it is
F0(x) ≤ lim inf
n→∞ Fn(xn).
(b) For every x ∈ X there exists a sequence {xn} converging to x in X such that
F0(x) = lim inf
n→∞ Fn(xn).
The function F00= Γ − lim supn→∞Fnis characterized by the following properties:
(c) For every x ∈ X and for every sequence {xn} converging to x in X it is
F00(x) ≤ lim sup
n→∞ Fn(xn).
(d) For every x ∈ X there exists a sequence {xn} converging to x in X such that
F00(x) = lim sup
n→∞ Fn(xn).
Therefore, Fn Γ
−
→ F if and only of following conditions are satisfied:
(e) For every x ∈ X and for every sequence {xn} converging to x in X it is
F (x) ≤ lim inf
n→∞ Fn(xn).
(d) For every x ∈ X there exists a sequence {xn} converging to x in X such that
F (x) = lim sup
n→∞
Fn(xn).
2.3.2. The γ-topology of quasi-open subsets. Given D ⊂ RN open, smooth
and bounded. We are interested in the study of dependence of the solution uA to the
differential equation:
and A ∈ A(D) where we denote A(D) as set of quasi open subsets of D. We rewrite the above PDE as problem of minimisation of an energy functional,
minn1 p Z D |∇u|pdx −Z D f u dx : u ∈ W01,p(A)o.
The above minimum problem can be re-written in the following from: minn1 p Z D |∇u|pdx −Z D f u dx + IA(u) : u ∈ W01,p(D) o where IA(u) = 0 if u ∈ W01,p(A), +∞ Otherwise. Remark 2.46. We denote : JA(u) = 1 p Z D |∇u|pdx −Z D f u dx + IA(u), (2.3.2)
Definition 2.47 (γ−convergence of quasi open sets). A sequence Ωn of quasi
open subsets of RN is said to γ-converge to Ω if the functionals
JΩn(u) = 1 p Z D |∇u|pdx −Z D f u dx + IΩn(u),
Γ−converge to the functional
JΩ(u) = 1 p Z D |∇u|pdx −Z D f u dx + IΩ(u).
in the metric space W01,p(Ω) endowed with Lp(D) topology.
Since, from Theorem 2.42, Γ− convergence of functionals is stable under pertur-bations by continuous functionals and the map u →R
Df u is continuous in W1,p(D),
hence we can say Ωn γ−converge to Ω if the sequence of functionals
1 2 Z D |∇u|pdx + I(Ω n),
Γ-converge to the functional 1 p Z D |∇u|pdx + I(Ω n).
Moreover, we know that by properties of Γ−convergence we can conclude Ωn, γ−
converge to Ω then for every f ∈ Lp(D), u
Ωn,f solving
−∆pu = f in Ωn, u ∈ W01,p(Ωn)
(uΩn,f is identified as element of L
p(D) by extending it by zero outside Ω
n) converge
in Lp(D) to u
Ω,f solving
2.3. γ−CONVERGENCE OF QUASI OPEN SUBSETS OF RN 27
Remark 2.48. We shall be denoting by uA,f as the minimiser of the functional
JA(u) mentioned in (2.3.2). The function uA,f also solve the PDE (2.3.3) for Ω = A.
We also remark that the notion of γ-convergence is independent of the choice of f (we refer to [22]). That is why, we can use the term uAunless the function f involved
do not cause ambiguity.
Definition 2.49 (Mosco Convergence of spaces). If we define weak upper and
strong lower limits of the sequence of subspaces W01,p(Ωn) (subspaces of W01,p(D)) as
w − lim sup n→∞ (W01,p(Ωn)) = u ∈ W01,p(D) : ∃unk ∈ W 1,p 0 (Ωnk), unk * u and s − lim inf n→∞ (W 1,p 0 (Ωn)) = u ∈ W01,p(D) : ∃un∈ W01,p(Ωn), un→ u
where the convergences * and → are respectively the weak and strong ones in W01,p(D).
We call the sequence of subspaces W01,p(Ωn) of W01,p(D) converge to W 1,p 0 (Ω) in Mosco sense if • (M1) W01,p(Ω), ⊂ s − lim infn→∞(W01,p(Ωn)), • (M2)w − lim supn→∞(W01,p(Ωn)) ⊂ W 1,p 0 (Ω).
We can also write more simplified definition of Mosco convergence:
Definition 2.50. The sequence of subspaces {W01,p(Ωn)}, subspaces of W01,p(D)
converge in sense of Mosco to W01,p(Ω) if the following two condition holds:
(M1) ∀ u ∈ W01,p(Ω) there exists un∈ W01,p(Ωn) such that un → u in W01,p(D),
(M2) ∀ u ∈ W01,p(D) such that if there exists a subsequence {unk}, such that unk ∈
W01,p(Ωnk) and unk * u weakly in W
1,p
0 (D), then u ∈ W 1,p 0 (Ω).
The following are important properties of γ−convergence which we use in many situations.
Theorem 2.51. We have:
(1) Topology induced by γ convergence of quasi open sets is metrizable. (2) It is known that if uΩn,f → uΩ,f in L
p(D) if and only if u
Ωn,1 → uΩ,1 in
Lp(D).
(3) Hence we can define γ−distance between the sets Ω1 and Ω2 as
(4) Let RΩ be the operator mapping the data f ∈ Lp(D) to uΩ,f ∈ W01,p(Ω). If
sequence Ωnis such that, Ωn γ
−
→ Ω, then the corresponding resolvent operators
RΩn converge to RΩ point-wise. in the topology L(L
p(D)).
(5) In fact, for p = 2, the convergence of {uΩn} to uΩ is strong in H
1
0(D) topology
and the resolvent RΩn converge to RΩ in L(L
2(D)) topology. Hence the
corresponding spectrum of RΩn converge to spectrum of RΩ.
Unfortunately, γ−convergence of quasi open sets is not compact, we have another notion of weak-γ convergence which leads to a compact topology on the family of quasi open subsets.
2.4. Cases when the family of open subsets is γ-compact
Generally, a sequence {Ωn} of quasi open sets need not have a γ limit as quasi
open set. The limit of the functionals in Definition 2.47 may contain an additional term which is a capacitary measure (see Section 2.6). We also refer to the counter example by Cioranescu and Murat in [18] also discussed in Example 2.70. However, we can ensure the γ-compactness by introducing some constraints in the admissible class of open sets, expressed in terms of Sobolev capacity.
In this section, we shall list the main results for compactness of open subsets of
D. There are two kind of approaches: first one consider the ”compact convergence”
of the sequence in which a ”stability condition” is imposed on the limit domian. The second one considers the Hausdorff complementary topology in the class of open sets, and moving domain is constrained in terms of capacity. The geometric cases when compactness occurs are given in terms of uniform cone conditions.
Definition 2.52 (Hausdorff complemetary topology). Let D be a bounded subset
of RN and we set
O = {Ω : Ω ⊂ D, Ω open},
denoted by τ the Hausdorff complementary topology on O is given by metric: dHc(Ω1, Ω2) = d(Ωc1, Ωc2),
here, d(A, B) is the Hausdorff distance defined as: d(A, B) = sup
x∈A
d(x, B) ∨ sup
x∈B
d(x, A), where d(x, E) = inf{|x − y| : y ∈ E}.
2.4. CASES WHEN THE FAMILY OF OPEN SUBSETS IS γ-COMPACT 29
We mention that the topology τ , (O, dHc) is compact topology. Also the first
Mosco condition (M1) is fulfilled for every sequence Ωn Hc
−→ Ω. In general the second mosco condition (M2) doesnt hold true when Ωn
Hc
−→ Ω. Geometrical constraints play a crucial rule for this case. Then from thesis (4) in Theorem 2.42, we can ensure Ωn
γ
− → Ω.
Let us look at a counter example where Hc convergence sequence does not imply
γ-convergence.
Example 2.53. Let p = 2. If {x1, x2, ...} is enumeration of points with rational
co-ordinates in D = (0, 1) × (0, 1). Define Ωn= D \ {x1, x2....} we get
Ωn Hc −→ {∅} and Ωn γ − → D since cap(D \ Ωn) = 0.
We list below the class of domains in which the γ-convergence is equivalent to the
Hc-convergence (from the strongest constraint to the weakest one):
• The class Oconvex ⊂ O of convex sets.
• The class Ounif cone ⊂ O of domains satisfying the uniform exterior cone
property. (i.e. Ω ∈ Ounif cone if ∀ x0 ∈ ∂Ω; there exists a closed cone, with
uniform height and opening, with vertex in x0, lying in compliment of Ω).
• The class Ounif f lat cone of domains satisfying uniform flat cone conditions,
i.e. the condition same as above but weaker in the sense that the cone can be of dimension N − 1.
• The class Ocap density ⊂ O of domains satisfying a uniform density condition.
i.e. Ω ∈ Ocap density if ∃ c, r > 0 such that for every x ∈ ∂Ω, we have
∀t ∈ (0, r) cap(Ω
c∩ B
t(x), B2t(x))
cap(Bt(x), B2t(x))
≥ c where Bs(x) denote ball of radius s and centre x.
• The class Ounif wiener ⊂ O of domains satisfying: Ω ∈ Ounif wiener if for every
x ∈ ∂Ω. Z R r cap(Ωc∩ Bt(x), B2t(x)) cap(Bt(x), B2t(x)) !p−11 dt t ≥ w(r, R, x) for every 0 < r < R < 1
where w : (0, 1) × (0, 1) × B → R+ is such that :
(1) limn→∞w(r, R, x) = ∞, locally uniformly on x.
(2) w is lower semi continuous in the third variable.
Other important constraint in terms of topology was studied by ˇSve´rak for N = 2 and consists of the following:
For N = 2, the class of all open subsets of D for which the number of connected components of ¯D \ Ω is uniformly bounded.
The following inclusion can be established:
Oconvex ⊂ Ounif cone ⊂ Ounif f lat cone ⊂ Ocap density ⊂ Ounif wiener.
A result analogous to that of ˇSve´rak in [43] was proved by Bucur and Trebeschi in [12] :
Theorem 2.54. Let N ≥ p > N − 1 and set
Ol(D) = {Ω ⊂ D | #Ωc ≤ l},
and for A ⊂ D, #A is number of connected components of Ac. Consider a sequence
Ωn∈ Ol(D), and Ωn Hc −→ Ω then Ω ∈ Ol(D) and Ωn γ − → Ω.
Keldysh [35] studied a stability condition for the limit domain provided the con-vergence is compact (defined below).
Definition 2.55. The sequence {Ωn} is said to compactly convergent to Ω if for
every compact set K ⊂ Ωc∪ Ω, there exists n
K ∈ N, such that for all n ≥ nK,
K ⊂ Ωn∪ Ωcn.
Definition 2.56. Set Ω is p-stable, if for every u ∈ W1,p(RN) vanishing a.e. in Ωc belongs to W1,p
0 (Ω). Using Hedberg result in [35], this is same as saying
∀u ∈ W1,p
(RN), u = 0 a.e. in Ωc⇒ u = 0 q.e. in Ωc,
Keldysh proves that Ωn γ
−
→ Ω if {Ωn} compactly converges to Ω and Ω is p-stable.
Another simple characterization of p-stable domain is :
Theorem 2.57. A bounded open set Ω is p-stable if and only if or every x ∈ R
and r > 0, we have
cap(Br(x) \ Ω, B2r(x)) = cap(Br(x) \ ¯Ω, B2r(x)).
Remark 2.58. Open sets with cracks are not p-stable. Hence the above result is not very useful in shape optimization. We generally cannot exclude the appearance of singularity as cracks in γ-limits of the minimising sequence.
Apart from the γ topology, we oftenly use the weak-γ topology which is a complete topology in family of quasi open subsets.
2.6. CAPACITARY MEASURES 31
2.5. Weak-γ Convergence for Quasi Open sets
Definition 2.59. We say a sequence of quasi open subsets {Ωn} converge in
weak-γ topology to a quasi open set Ω if there exists u ∈ W1,p(D) such that u Ωn
converge weakly to u in W1,p(D) and Ω = {u , 0}.
Theorem 2.60 (sequential compactness of Weak-γ convergence). Weak-γ
con-vergence is sequentially compact.
Proof. Let Ωn be a sequence of quasi open subsets of D. the corresponding
solutions, uΩn ∈ W
1,p
0 (D) has a weak limit in W1,p(D) (Since D is bounded). Let
uΩn * u. We can assign Ω = {u , 0} hence Ω is the weak-γ limit.
Theorem 2.61. Given a(x) ∈ L1(D), a(x) ≥ 0 a.e. x ∈ D. The set functional
A →R
A a(x) dx is lower semi-continuous in weak-γ convergence.
Proof. If Ωn → Ω in weak-γ sense. Then uΩn
W1,p
−−−* u where u is such that Ω = {u , 0}. and we can find a subsequence of uΩn such that uΩn → u point-wise.
If x ∈ Ω ⇒ u(x) > 0. Hence, for n sufficiently large, uΩn(x) > 0 ⇒ x ∈
Ωn for n large. Use Fotou’s Lemma to have:
Z
Ω
a(x)dx ≤ lim inf
n
Z
Ωn
a(x) dx,
which concludes the proof.
Optimal domain of a given energy need not exist even in the class of quasi open sets. As mentioned above, γ−convergence of quasi open sets of D is not compact. It is useful to note that, for a set functional which is monotonically decreasing and γ−lower semicontinuous, it is also weak-γ lower semicontinuous. Owing to compactness of weak-γ, we can ensure the existence of an optimal domain in class of admissible quasi open sets.
For a given sequence Ωn, the weak limit of uΩn,1 may not be of form uΩ,1. But
it is proved that the weak limit satisfies a different differential equation involving ”capacitary measures”. We shall define the notion in following section.
2.6. Capacitary Measures
Given a sequence of quasi open sets {Ωn}, it need not γ-converge of a quasi open
sequence in γ topology. i.e. we look for a completion of family of Quasi Open sets endowed with γ topology.
The completion of the family of quasi open sets A(D) endowed by γ topology was studied with Dal Maso and Mosco in [21] for the case p = 2, for general p we refer to [22].
Definition 2.62. A capacitary measure µ on D is a non negative Borel measure
which can possibly be ∞ valued, for which the following hold true: (1) µ(B) = 0 for every Borel set B ⊂ D with cap(B) = 0,
(2) µ(B) = infnµ(U ) : U quasi open, B ⊂ U for every set B ⊂ Do.
We shall denote M0(D) as set of capacitary measures on D.
Remark 2.63. We stress on the fact that a capacitary measure µ need not be a finite measure, it may take values ∞ on some subsets.
Example 2.64. µ = a(x)HN with a ∈ Lp(D) is an example of capacitary measure. We can identify a quasi open set also as a capacitary measure, to see this let us try to understand the space M0(D) of capacitary measures, which also include the
identification of quasi open subsets.
For µ ∈ M0(D) we define the space Xµ(D) = W01,p(D)∩Lp(D, µ). Xµis a Hilbert
space with the norm,
||u||pXµ = Z D |∇u|pdx + Z D updµ.
The dual of W01,p(D), i.e. the space W−1,p0(D) can be considered as subspace of the dual of Xµ(D) represented as Xµ0(D). We define the action for f ∈ W
−1,p0
(D) as:
< f, v >X0
µ(D)=< f, v >W−1,p0(D) ∀v ∈ Xµ(D).
When f ∈ Lp0(D), we simply define
< f, v >X0
µ(D)=
Z
D
f v dx ∀v ∈ Xµ(D).
And let uµ∈ Xµ(D) be the solution of the PDE
− ∆pu + µu|u|p−2 = f in D, u ∈ Xµ(D), (2.6.1)
with the weak formulation
u ∈ Xµ(D); Z D |∇u|p−2∇u.∇v dx + Z D u|u|p−2v dµ = Z D f v dx ∀ v ∈ Xµ(D).
2.6. CAPACITARY MEASURES 33
See that if Ω is a quasi open subset, we can identify the set Ω with the measure
µ = ∞D\Ω ∈ M10(D). ∞D\Ω is a capacitary measure such that :
∞D\Ω(E) = ∞ if cap(E ∩ Ω) = 0, ∞D\Ω(E) = 0 if cap(E ∩ Ω) > 0. (2.6.2) In the case when µ = ∞D\Ω it is easy to verify that the PDE (2.6.1) reduces to the
PDE coresponding to the set Ω:
−∆pu = f in Ω,
u ∈ W01,p(Ω).
This is how, we say that the capacitary measure µ = ∞D\Ω represents the quasi open
set Ω.
We shall be briefly discussing how the γ-limit of quasi open sets is equal to a capacitary measure, but only for simple case of p = 2. The general case for 1 < p ≤ N can be found in [22].
Example 2.65. Let f ∈ L2(D) and let {Ωn} be a sequence of quasi open subsets
of D. We denote by un the solution of the following PDE:
−∆un= f in Ωn, un ∈ H01(Ωn). (2.6.3) Suppose that the function wn is the solution on Ωn for the PDE (2.6.3), with f ≡ 1.
We know that, there exists u ∈ H1
0(D) and w ∈ H01(D) such that, up to a subsequence
(still denoted by {un} and {wn}) we have :
un* u and wn* w weakly in H01(D).
Let ϕ ∈ Cc∞(D), we shall be taking wnϕ as a test function in the weak formulation
for the PDE (2.6.3):
Z D f wnϕdx = Z D ∇un∇(wnϕ)dx = Z D ∇un∇ϕwndx + Z D ∇un∇wnϕdx = Z D ∇un∇ϕwndx − Z D un∇wn∇ϕdx − < ∆wn, ϕun >H−1(D)×H1 0(D) = Z D ∇un∇ϕwndx − Z D un∇wn∇ϕdx + Z D unϕdx,
in the limit n → ∞, we get:
Z D f ϕw dx = Z D ∇u∇ϕw dx − Z D u∇w∇ϕ dx + Z D uϕ dx.
We have, − Z D u∇w∇ϕdx = Z D ∇u∇wϕdx + < ∆w, uϕ >H−1(D)×H1 0(D).
We calculate the second term on the right hand side: − Z D u∇w∇ϕdx = Z D ∇u∇wϕdx + < ∆w, uϕ >H−1(D)×H1 0(D).
Note that 1+∆wnare positive Radon measure on D, hence in the limit, 1+∆w ≥ 0 in
D0(D). Let us call 1 + ∆w = ν, ν ∈ H−1(D). Define a Borel measure µ as following:
µ(B) = +∞ if cap(B ∩ {w = 0}) > 0, R B 1 wdν if cap(B ∩ {w = 0}) = 0. We can write Z D ∇u∇(ϕw)dx + Z D uϕwdµ = Z D f ϕwdx.
In the above sketch, the following facts need to be verified rigorously. We refer the reader to [22] for the details:
(1) u = 0 where w = 0. (2) u ∈ Xµ(D).
(3) The set {ϕw : ϕ ∈ Cc∞(D)} is dense in Xµ(D).
This means the weak limit of the sequence {un} from the equation (2.6.3) solves the
PDE:
−∆u + µu = f in D,
un∈ Xµ(D).
We can define the γ topology in the family of capacitary measures M0(D) as:
Definition 2.66. Given a sequence {µn} in M0(D), we say that µn γ
−
→ µ when
the functions uµn ∈ Xµn which are solutions to the PDE:
−∆pu + µnu|u|p−2= 1 in D,
u ∈ Xµn(D),
converges weakly in W01,p(D) to solution uµ ∈ Xµ(D) of the PDE:
−∆pu + µu|u|p−2= 1 in D,
u ∈ Xµ(D).
Remark 2.67. Given a sequence of p−quasi open sets {Ωn}, the capacitary
mea-sure µ as constructed in Example 2.65 shall be
µ(B) = +∞ if cap(B ∩ {w = 0}) > 0, R B 1 wp−1dν if cap(B ∩ {w = 0}) = 0,
