Definition and basic properties

These results can be obtained by calculating the relevant cones.

For the problem (MPCC) it is known from the literature that a local minimizer is M-stationary if an MPCC-tailored constraint qualification holds. The next theorem is taken from [Flegel, Kanzow, 2006, Theorem 3.1]. For the definition of the MPCC-tailored Guignard constraint qualification, or MPCC-GCQ, we refer to [Flegel, Kanzow, 2006, Definition 2.1], where it is called MPEC-GCQ.

Theorem 2.5.3. Let x¯ ∈ R^d be a local minimizer of(MPCC) at which MPCC-GCQ holds. Then x¯ is M-stationary.

The proof of this theorem utilizes the limiting normal cone. We remark that MPCC-GCQ is a very weak constraint qualification and that MPCC-GCQ holds at every point if the functions g, h, G, H are affine functions.

The M-stationarity condition can also be used for numerical algorithms, see [Harder, Mehlitz, G. Wachsmuth, 2020].

Definition 2.6.1. For a setA⊂Ω we define its capacity via cap(A) := inf^{︁∥v∥²_H1

0(Ω)

⃓

⃓v∈H₀¹(Ω), v≥1 a.e. in an open neighborhood ofA^}︁. If there is no functionv ∈H₀¹(Ω) such that v ≥1 a.e. in an open neighborhood of A, then we set cap(A) =∞.

This definition can be found in [Bonnans, Shapiro, 2000, Definition 6.47], however, it is only used to define the capacity of Borel measurable sets. Our definition for general subsets of Ω can also be found in slightly different formulations in [Attouch, Buttazzo, Michaille, 2014, Definition 5.8.1] (although technically the case d = 1 is not covered there), or [Fukushima, Oshima, Takeda, 2010, p. 66]. In the literature, there is some variety of definitions for the capacity of sets. For example, in [Delfour, Zolésio, 2011, Definition 8.6.2] the capacity is first defined on compact subsets of Ω, then extended to open subsets via supremums over compact subsets and finally extended to arbitrary subsets of Ω via infimums over open sets. This approach is equivalent to our definition of capacity, seeLemma 2.6.4. Our version of the definition of the capacity is simpler than the mentioned definitions from the literature.

It is also possible to extend the definition of the capacity to the spaces W₀^1,p(Ω) for p∈[1,∞) by replacing∥v∥²

H₀¹(Ω) by ∥v∥^p

W₀^1,p(Ω) in the above definition. However, for our purposes it suffices to only consider the case ofH₀¹(Ω), i.e. p= 2.

There is an analogy that compares the capacity with the Lebesgue measure on Ω. For a Lebesgue measurable set A⊂Ω it is easy to see that

meas(A) := inf^{︁∥v∥²_L2(Ω)

⃓

⃓v∈L²(Ω), v≥1 a.e. in an open neighborhood ofA^}︁

holds. Thus, meas(·) relates to L²(Ω) similarly as cap(·) relates to H₀¹(Ω).

We give some examples to highlight the difference between measure and capacity. However, the calculations to prove these claims have been omitted.

Example 2.6.2. (a) Leta < ω₀ < bbe real numbers. Then for d:= 1 and Ω := (a, b) we have

cap({ω₀}) = 1

ω0−a+ 1 b−ω0

>0. (b) Ford≥2 and ω0 ∈Ω, the singleton{ω₀}has zero capacity.

(c) The domain Ω has infinite capacity.

As can be seen by Example 2.6.2 (a), there can be sets that have measure zero but a nonzero capacity. We also observe that inExample 2.6.2 (a) the capacity of a set can depend on the surrounding domain Ω.

In the next lemma, we will collect some basic properties of capacities, which will also

Lemma 2.6.3. (a) The empty set has zero capacity.

(b) For sets A, B with A ⊂ B ⊂ Ω we have cap(A) ≤ cap(B), i.e. the capacity is monotone.

(c) For sets A, B⊂Ω we have

cap(A∪B) + cap(A∩B)≤cap(A) + cap(B).

In particular, the inequality cap(A∪B)≤cap(A) + cap(B) holds for all subsets A, B⊂Ω, i.e. the capacity is subadditive.

(d) For a nondecreasing sequence of sets{A_i}_i∈NwithA_i ⊂Ω the capacity of the union of this sequence can be expressed via

cap^{(︂ ⋃︂}

i∈N

A_i^)︂= lim

i→∞cap(A_i), i.e. the capacity is continuous from below.

(e) For a sequence of sets {A_i}_i∈N withAi ⊂Ω the inequality cap^{(︂ ⋃︂}

i∈N

Ai

)︂≤^∑︂

i∈N

cap(Ai) holds, i.e. the capacity is countably subadditive.

(f) There exists a constant C > 0 (that depends only on Ω) such that for every Lebesgue measurable set A⊂Ω the inequality meas(A)≤Ccap(A) holds.

(g) If a set has zero capacity, then it is included in a Borel measurable set of measure zero.

Proof. Parts(a) and(b) follow directly from the definition.

For part(c), letvA, vB∈H₀¹(Ω) be such that vA≥1 a.e. on an open neighborhood of A and v_B≥1 a.e. on an open neighborhood of B. Then the function v∪ := max(v_A, v_B)∈ H₀¹(Ω) satisfiesv∪ ≥1 a.e. on an open neighborhood ofA∪B and the function v∩ :=

min(vA, vB) ∈H₀¹(Ω) satisfies v∩ ≥1 a.e. on an open neighborhood of A∩B. Due to Lemma 2.2.8 (d)we also have the equality∥v_∪∥²

H₀¹(Ω)+∥v_∩∥²

H₀¹(Ω) =∥v_A∥²

H₀¹(Ω)+∥v_B∥²

H₀¹(Ω). The claim follows by taking the infimum over the functions v_A, v_B that are admissible in the definition of the capacity of A and B.

We continue with part (d). From part(b)we obtain that the inequality cap^{(︂ ⋃︂}

i∈N

A_i^)︂≥ lim

i→∞cap(A_i)

holds. We also observe that supi∈Ncap(Ai) = limi→∞cap(Ai) holds. In order to show the remaining inequality, we can without loss of generality assume that supi∈Ncap(A_i)<∞ holds. Letε >0 be given. For each i∈N, let O˜_i⊂Ω be an open set such that A_i⊂O˜_i and cap(O˜_i)≤cap(Ai) + 2⁻ⁱεhold. It is possible to find such an open set O˜_i because the functionsv∈H₀¹(Ω) in the definition of the capacity for the setA_i have the property v≥1 a.e. on an open neighborhood ofA_i. Next, we recursively define a sequence{O_i}_i∈N of open subsets of Ω viaO1 :=O˜₁,Oi+1 :=Oi∪O˜_i+1 for all i∈N. Clearly,Ai⊂Oi is true for alli∈N. We claim that

cap(Oi)≤cap(Ai) + (1−2⁻ⁱ)ε (2.30) holds for alli∈N. Indeed, the claim is clearly true for i= 1. If the claim(2.30)is true fori∈N, then we can use part(c) to obtain

cap(Oi+1)≤cap(Oi) + cap(O˜_i+1)−cap(Oi∩O˜_i+1)

≤cap(A_i) + (1−2⁻ⁱ)ε+ cap(A_i+1) + 2⁻⁽ⁱ⁺¹⁾ε−cap(A_i)

= cap(A_i+1) + (1−2⁻⁽ⁱ⁺¹⁾)ε,

which shows that(2.30)also holds fori+ 1. According to the definition of cap(Oi), there exists a sequence{v_i}_i∈NinH₀¹(Ω) such thatv_i≥1 a.e. onO_iand∥v_i∥²

H¹₀(Ω)≤cap(A_i)+ε holds for alli∈N. Clearly,{v_i}_i∈N is a bounded sequence in H₀¹(Ω) and therefore has a subsequence that converges weakly inH₀¹(Ω) and strongly in L²(Ω) to some function v∈H₀¹(Ω). This subsequence has a further subsequence that converges pointwise almost everywhere tov. Without loss of generality we again denote this subsequence by {v_i}_i∈N. Because{O_i}_i∈N is a nondecreasing sequence of open sets, we obtain thatv≥1 holds a.e.

on the open setO:=^⋃︁i∈NO_i. Then the weak convergence in H₀¹(Ω) and ^⋃︁i∈NA_i⊂O imply

cap^{(︂ ⋃︂}

i∈N

Ai

)︂≤cap(O)≤ ∥v∥²_H1

0(Ω)≤lim inf_i→∞ ∥v_i∥²_H1

0(Ω)≤_i→∞lim cap(Ai) +ε.

Sinceε >0 was arbitrary, the claim follows.

For part(e) we note that a repeated application of part(c) yields cap^(︂

i

⋃︂

j=1

Aj

)︂≤

i

∑︂

j=1

cap(Aj)

for alli∈N. Then the claim follows by applying part (d) to the nondecreasing sequence {︁⋃︁_i

j=1A_j^}︁_i∈

N.

For part(f), letA⊂Ω be Lebesgue measurable. We only need to consider the case where cap(A)<∞. Let v∈H₀¹(Ω) be a function such thatv≥1 a.e. in an open neighborhood

of A. Then, by the Poincaré inequality, we have

meas(A)≤ ∥v∥²_L2(Ω) ≤C∥v∥²_H1 0(Ω),

where the constant only depends on Ω. Taking the infimum over allv∈H₀¹(Ω) such that v≥1 a.e. in an open neighborhood ofA yields the desired inequality.

Finally, for a setA⊂Ω with zero capacity it can be concluded from the definition, that there is a sequence {O_i}_i∈N of open supersets of Awith arbitrarily small capacity. By part (f), those supersets must have arbitrarily small Lebesgue measure. Hence, A is included in the Borel measurable set ^⋂︁i∈NO_i which has measure zero. This concludes the proof of part(g).

Note that Lemma 2.6.3 (g)is not just a special case of Lemma 2.6.3 (f), because the measurability of the set is not an assumption inLemma 2.6.3 (g).

We further note that the capacity satisfies the definition of an outer measure (which is defined precisely via the properties shown in parts (a),(b), and (e) of Lemma 2.6.3).

Thus, the capacity can be considered as a generalization of a measure.

In Definition 2.2.15 we introduced regularity properties for measures. We can show outer regularity for arbitrary sets and inner regularity for open sets in a straightforward way.

Lemma 2.6.4. (a) The capacity is outer regular in the sense that cap(A) = inf{cap(O)|A⊂O⊂Ω, O open} holds for all A⊂Ω.

(b) The capacity is inner regular for open sets in the sense that cap(O) = sup{cap(K)|K ⊂O⊂Ω, K compact} holds for all open setsO ⊂Ω.

Proof. Part (a)follows directly from the definition of the capacity because the functions v∈H₀¹(Ω) in the definition of the capacity for the setA have the property v≥1 a.e. on an open set O ⊃A.

We recall that every open setO⊂Ω has a compact exhaustion, i.e. there exists a sequence {K_i}_i∈N of compact subsets of O such thatO = ^⋃︁i∈NKi. Then part (b) follows from Lemma 2.6.3 (d).

In the next definition, we introduce some notions that are similar to the expressions

“almost everywhere” and “for almost all”, but are based on capacities instead of mea-sures.

Definition 2.6.5. (a) We say that a property P that depends onω∈Ω holds quasi-everywhere (q.e.) on a set A⊂Ω if the set{ω ∈A|P(ω) does not hold}has zero capacity. We will also say thatP(ω) holds forquasi-all (q.a.) ω∈A, which has the same meaning. If no such setA is specified in this context, we mean the setA= Ω.

(b) For setsA, B⊂Ω we writeA⊂_q B if cap(A\B) = 0.

(c) For sets A, B⊂Ω we writeA=q B if cap^(︁(B\A)∪(A\B)^)︁= 0.

(d) We say that a setA⊂Ω isunique up to a set of zero capacity with respect to a property P if A satisfies P and we have A =q B for every set B ⊂ Ω that also satisfiesP.

We give some simple results to show that the relations “=q” and “⊂_q” behave as one might expect.

Corollary 2.6.6. (a) For A, B ⊂ Ω we have A =q B if and only if A ⊂_q B and B⊂_qA.

(b) ForA, B⊂Ω we haveA⊂_q B ifA⊂B and A=q B ifA=B.

(c) The relation “=q” is an equivalence relation.

(d) The relation “⊂_q” is reflexive and transitive.

These statements follow directly from a combination ofLemma 2.6.3 (a),Lemma 2.6.3 (c), and the definitions of “=q” and “⊂_q”.

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