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Gamma-Convergence for Beginners$

Andrea Braides

Print publication date: 2002

Print ISBN-13: 9780198507840

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198507840.001.0001

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(p.197) APPENDIX A SOME QUICK RECALLS

(p.197) APPENDIX A SOME QUICK RECALLS

Source:
Gamma-Convergence for Beginners
Publisher:
Oxford University Press

A.I Convexity

We recall that a function f : R N → (−∞, +∞] is convex if we have

(A.1)
f ( t z 1 + ( 1 t ) z 2 ) t f ( z 1 ) + ( 1 t ) f ( z 2 )
for all z 1,z 2R N and t ∈ (0,1).

  1. Remark A.1 (a) The convexity of f is equivalent to requiring that Jensen's inequality holds:

    (A.2)
    f ( X g d μ ) X f ( g ( x ) ) d μ
    for all probability spaces (X, μ) and measurable g : XR N.

  2. (b) If fC l(R N) then it is convex if and only if

    (A.3)
    f ( z ) f ( w ) + < f ( z ) , z w >
    for all z, wR N.

  3. (c) The supremum of a family of convex functions is convex.

  4. (d) If f is a convex function and f is finite at every point of an open set Ω then f is continuous on Ω and locally Lipschitz continuous on Ω.

  5. (e) If f is convex and there exist 1 ≤ p < ∞ and c > 0 such that 0 ≤ f(z) ≤ c(1 + |z|p) for all zR N, then f satisfies the local Lipschitz condition

    (A.4)
    | f ( z ) f ( w ) | c ( 1 + | z | p 1 + | w | p 1 ) | z w |
    for all z, wR N for some c′ depending only on c and p.

  6. (f) If fj : R NR is a sequence of locally equi-bounded convex functions then there exists a subsequence of (fj) converging uniformly on all compact subsets of R N.

The verification of statements (b) and (c) is immediate and is left as an exercise. Jensen's inequality is easily derived from the convexity of f when fC 1(R N). In that case by (b) we have

f ( X g d μ ) f ( g ( x ) ) + < f ( X g d μ ) , X g d μ g ( x ) >
and it is sufficient to integrate with respect to μ. If f is not C 1 we can proceed by approximation using (c) (the details of the proof, using, for example, Exercise (p.198) 1.10, are left to the reader). Conversely, convexity trivially follows from Jensen's inequality by choosing X = {0,1}, μ = tδ0 + (1 − t1, g(0) = z 1 and g(1) = z 2. To prove (d), if Ω = (a, b) is an interval of R then we may use the well-known monotonicity properties of the difference quotient of a convex function. Thus, if T > 0 and r,s ∈ [a+2T, b− 2T] with r < s. we have
f ( s ) f ( r ) s r f ( b T ) f ( s ) b T s 2 sup { f ( x ) : x [ a + T , b T ] } 1 T = : C T ,
so that f(s) − f(r) ≤ CT(sr). Symmetrically, we get f(s) − f(r) ≥ −CT(sr) and hence the local Lipschitz continuity of f. If N > 1 then the thesis is proven by arguing as above in each coordinate direction. The proof of (e) can be performed in the same way. Details are left as an exercise. Finally, (f) immediately follows from (d) and Ascoli Arzelà's Theorem.

A.2 Sobolev spaces

In all that follows (a, b) is a bounded open interval of R.

Definition A.2 (weak derivative). We say that uL 1(a, b) is weakly differentiable if a function gL 1 (a, b) exists such the following integration by parts formula holds:

(A.5)
a b u ϕ d t = a b g ϕ d t
for all ϕ C 0 1 ( a , b ) . If such g exists then it is called the weak derivative of u and is denoted by u′.

Remark A.3 The notion of weak derivative is an extension of the notion of classical derivative: if uC 1 (a, b) and its classical derivative belongs to L 1(a, b) then the classical derivative coincides with its weak derivative. The function x ↦ |x| is weakly differentiable in any (a, b) but uC 1(−1,1). and its weak derivative is the function xx/|x|, which in turn is not weakly differentiable in (−1, 1).

Definition A.4 (Sobolev spaces). Let p ∈ [1, ∞]; the Sobolev space W 1,P(a, b) is defined as the space of all weakly differentiable uLp(a, b) such that u′ ∈ Lp(a, b). The norm of u in W 1,p(a, b) is defined as

| | u | | W 1 , p ( a , b ) p = | | u | | L p ( a , b ) p + | | u | | L p ( a , b ) p .
The space W loc 1 , p ( R ) is defined as the space of uW 1,p(I) for all bounded open intervals IR.

Remark A.5 The Sobolev space W 1,p(a, b) equipped with its norm is a Banach space. This is easily checked upon identifying W 1,p(a, b) with the subspace of Lp(a, b) × Lp(a, b) of all pairs (u, u′) with uW 1,p(a, b). The same identification shows that W 1,p(a, b) is separable if 1 ≤ p < ∞.

(p.199) Theorem A.6 (pointwise value of Sobolev functions). Let uW 1,p(a, b); then ũC([a, b]) exists such that ũ = u a.e. on (a, b) and

(A.6)
u ¯ ( y ) u ¯ ( x ) = x y u ( t ) d t
for all x, y ∈ [a, b]. We commonly identify u with its continuous representative ũ whenever pointwise values are taken into account.

Remark A.7 (boundary values). If uW 1,p(a, b) then the boundary values u(a) and u(b) are uniquely defined by the values ũ(a) and ũ(b), respectively. We may then extend a function uW 1,p(a, b) to a function u W loc 1 , p ( R ) by simply setting u(t) = u(a) for ta and u(t) = b for tb.

Theorem A.8 (equivalent definitions of Sobolev spaces). Let 1 < p ≤∞; then the following statements are equivalent:

  1. (i) uW 1,p(a, b);

  2. (ii) there exists C ≥ 0 such that | a b u ϕ d t | C ϕ L p ( a , b ) if ϕ C 0 1 ( a , b ) ;

  3. (iii) there exists C ≥ 0 such that for all I ⊂⊂ (a, b) and for all hR such that |h| ≤ dist (I, {a, b}) we have ‖τh uuLp(I)C|h|, where τh u(t) = u(th);

  4. (iv) there exsists a sequence (uj) in C ([a, b]) such that

    (A.7)
    lim j | | u j u | | W 1 , p ( a , b ) = 0

  5. (v) there exists a sequence (uj) in C 0 ( R ) such that (A.7) holds;

  6. (vi) there exists a sequence (uj) in C 0 ( R ) such that supju jW 1,p(a, b) < +∞ and limjujuLp(a, b)=0.

  1. Remark A.9 (a) The best constant C in (ii) and (iii) above is ‖u′‖Lp(a, b).

  2. (b) If p = 1 then (i) ⇒ (ii) ⇔ (iii). Note that the function xx/|x| satisfies (ii)−(vi) with p = 1 but does not belong to W 1,1(−1,1).

  3. (c) By (iii) we easily see that W 1,∞(a, b) coincides with the space Lip(a, b) of all Lipschitz functions on (a, b), and ‖u′‖L (a, b)) is the best Lipschitz constant for u.

Theorem A.10 (embedding results). There exists C = C(a, b) such that

(A.8)
| | u | | L ( a , b ) C | | u | | W 1 , p ( a , b ) .
Moreover we have the compact embeddings
(A.9)
W 1 , p ( a , b ) C 0 ( [ a , b ] )
for 1 < p ≤ ∞, and
(A.10)
W 1 , 1 ( a , b ) L q ( a , b )
for all q ≥ 1.

(p.200) Definition A.11 The space W 0 1 , p ( a , b ) is defined as the closure of C 0 ( a , b ) in the W 1,p-norm, or, equivalently, as the set of those uW 1,p(a, b) with boundary values u(a) = u(b) = 0.

Theorem A.12 (Poincaré's inequality). There exists a constant C = C(a, b) such that

(A.ll)
| | u | | W 1 , p ( a , b ) C | | u | | L p ( a , b )
for all uW 1,p(a, b) such that ũ(x) = 0 for some x ∈ [a, b]. In particular this holds for u W 0 1 , p ( a , b ) .

Note that if we apply a similitude of ratio p to the domain (a, b), the constant C is multiplied by p. This remark holds in any space dimension.

Definition A.13 Let u : (a, b)R be a measurable function. The total variation of u on (a, b) is defined as

(A.12)
Var ( u , ( a , b ) ) = inf { sup { i = 1 N | v ( t i + 1 ) v ( t i ) | : a < t 0 < < t N < b , N N } : v = u a .e o n ( a , b ) }
If Var(u, (a, b)) < +∞ then we say that u is a function of bounded variation. We simply write Var u if (a, b) is fixed.

Remark A.14 If uW 1,1(a, b) then Var ( u , ( a , b ) ) = a b | u | d t ; in particular, u is a function of bounded variation. Note that also v(x) = x/|x| is a function of bounded variation with Var(v,(−1,1)) = 2.

A.3 *Sets of finite perimeter

A classical problem in the Calculus of Variations is that of the computation of the set of least perimeter and given area. The attack of such a problem by the direct methods need a definition of surface area which is lower semicontinuous under a convergence of sets which ensures also a compactness property. It is clear that a bound on the area of a sequence of sets does not ensure any continuity, even though all sets are smooth. We give a quick introduction to this subject, sufficient to the exemplificatory use that we make in Chapter 15. We refer to the book of Ambrosio et al. (2000) for a complete treatment and to Morgan (1988) for a quick introduction.

The simplest way to have a definition of perimeter which is lower semicontinuous by the L 1-convergence of the sets is by relaxation: if ER N is of class C 1 define the perimeter 𝒫(E, Ω) of the set E inside the open set Ω in a classical way, and then for an arbitrary set, define

P ( E , Ω ) = inf { lim inf j P ( E j , Ω ) : χ E j χ E in L 1 ( Ω ) } .
(p.201) Another choice leading to the same definition is to start with Ej of polyhedral type, for example. This definition coincides with the distributional definition of perimeter
P ( E , Ω ) = sup { E div g d x : g ( C 0 1 ( Ω ) ) N , | g | 1 }
If 𝒫(E, Ω) < +∞ then we say that E is a set of finite perimeter in Ω. For such sets it is possible to define a notion of measure-theoretical boundary, where a normal is defined, so that we may heuristically picture those sets as having a smooth boundary. In order to make these concepts more precise we recall the definition of the k-dimensional Hausdorff measure (in this context we will limit ourselves to kN). If E is a Borel set in R N then we define
k ( E ) = sup δ > 0 ω k 2 k inf { i N ( diam E i ) k : diam E i δ , E i N E i } ,
where ωk is the Lebesgue measure of the unit ball in R k and diam B is the diameter of B. Note that the k-dimensional Hausdorff measure coincides with the elementarily-defined k;-dimensional surface measure on k-dimensional subspaces. In particular, the N-dimensional Hausdorff measure coincides with the Lebesgue on R N.

We say that xE is a point of density t ∈ [0,1] if there exists the limit

lim ρ 0 + | E B ρ ( x ) | ω n ρ n = t .
The set of all points of density t will be denoted by Et. If E is a set of finite perimeter in Ω then the De Giorgi's essential boundary of E, denoted by ∂*E, is defined as the set of points x ∈ Ω with density 1/2.

Theorem A.15 (De Giorgi's rectifiability theorem). Let ER N be a set of finite perimeter in Ω. Then ∂*E is rectifiable; i.e., there exists a countable familyi) of graphs of C 1 functions of (N−1) variables such that N 1 ( * E \ i = 1 Γ i ) = 0 . Moreover the perimeter of E in Ω´ ⊆ Ω is given by

P ( E , Ω ) = N 1 ( * E Ω ) .
By the previous theorem and the Implicit Function Theorem a internal normal νE(X) to E is defined at ℋN−1almost all points of ∂*E as the normal of the corresponding Γi. The following generalized Gauss-Green formula holds
(A.13)
E div g d x = * E < v , g > d N 1
holds for all g ( C 0 1 ( Ω ) ) N , which states that the distributional derivative of χE is a vector measure given by (p.202)
D χ E ( B ) = B v d N 1 ;
In particular, we have p(E, Ω) = |D χE|(Ω), the total variation of the measure D χE on Ω.

The following theorem essentially states that sets of finite perimeter are characterized as those sets (almost all) whose one-dimensional sections are finite unions of intervals (i.e. one-dimensional sets of finite perimeter). We use the notation for one-dimensional sections introduced in Chapter 15 and that for piecewise-constant functions introduced in Chapter 5.

Theorem A.16 (a) Let E be a set of finite perimeter in a smooth open set Ω ⊂ R N and let u = χE. Then for all ξ ∈ S n − 1 and for N − 1-a.a. y ∈ ∏ξ the function u ξ,y belongs to PCξ,y). Moreover, for such y we have

(A.14)
S ( u ξ , y ) = { t R : y + t ξ S ( u ) } ,
and for all Borel functions g
(A.15)
Π ξ t S ( u ξ , v ) g ( t ) d N 1 ( y ) = S ( u ) g ( x ) | < v E , ξ > | d N 1 ;
(b) Conversely, if E ⊂ Ω and for all ξ ∈ {e 1,…,e N} and for H N−1-a.a. y ∈ ∏ξ the function u ξ,y is piecewise constant in each interval of Ωξy and
(A.16)
Π ξ # ( S ( u ξ , y ) ) d n 1 ( y ) < + ,
then E is a set of finite perimeter in Ω.

The following theorem states that (if Ω is regular) sets of finite perimeter can be approximated by smooth sets in R N

Proposition A.17 Let Ω be a Lipschitz set. If E is a set of finite perimeter in Ω then there exists a sequence (Ej) of sets of finite perimeter in Ω, such that

(A.17)
lim j | E Δ E j | = 0 , lim j P ( E j , Ω ) = P ( E , Ω ) ,
and for every open set Ω´ with Ω ⊂⊂ Ω´ there exist sets Ej of class C in Ω´ and such that Ej ∩ Ω = Ej.

We finally recall the coarea formula on the open set AR N

(A.18)
A f ( x ) | D d | d x = + { d = t } f ( y ) d N 1 ( y ) d t
valid if d is a Lipschitz function and f a Borel function. Note that if d(x) = X j this is a particular case of Fubini's theorem.