(p.197) APPENDIX A SOME QUICK RECALLS
(p.197) APPENDIX A SOME QUICK RECALLS
A.I Convexity
We recall that a function f : R ^{N} → (−∞, +∞] is convex if we have

Remark A.1 (a) The convexity of f is equivalent to requiring that Jensen's inequality holds:
(A.2)for all probability spaces (X, μ) and measurable g : X → R ^{N}.$$f\left({\int}_{X}g\text{}d\mu \right)\le {\int}_{X}f\left(g\left(x\right)\right)d\mu $$ 
(b) If f ∈ C ^{l}(R ^{N}) then it is convex if and only if
(A.3)for all z, w ∈ R ^{N}.$$f\left(z\right)\le f\left(w\right)+<f\prime \left(z\right),\text{}zw>$$ 
(c) The supremum of a family of convex functions is convex.

(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.

(e) If f is convex and there exist 1 ≤ p < ∞ and c > 0 such that 0 ≤ f(z) ≤ c(1 + z^{p}) for all z ∈ R ^{N}, then f satisfies the local Lipschitz condition
(A.4)for all z, w ∈ R ^{N} for some c′ depending only on c and p.$$\leftf\left(z\right)f\left(w\right)\right\le c\prime \left(1+{\leftz\right}^{p1}+{\leftw\right}^{p1}\right)\leftzw\right$$ 
(f) If f_{j} : R ^{N} → R is a sequence of locally equibounded convex functions then there exists a subsequence of (f_{j}) 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 f ∈ C ^{1}(R ^{N}). In that case by (b) we have
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 u ∈ L ^{1}(a, b) is weakly differentiable if a function g ∈ L ^{1} (a, b) exists such the following integration by parts formula holds:
Remark A.3 The notion of weak derivative is an extension of the notion of classical derivative: if u ∈ C ^{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 u ∉ C ^{1}(−1,1). and its weak derivative is the function x ↦ x/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 u ∈ L^{p}(a, b) such that u′ ∈ L^{p}(a, b). The norm of u in W ^{1,p}(a, b) is defined as
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 L^{p}(a, b) × L^{p}(a, b) of all pairs (u, u′) with u ∈ W ^{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 u ∈ W ^{1,p}(a, b); then ũ ∈ C([a, b]) exists such that ũ = u a.e. on (a, b) and
Remark A.7 (boundary values). If u ∈ W ^{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 u ∈ W ^{1,p}(a, b) to a function $u\in {W}_{\text{loc}}^{1,p}\left(R\right)$ by simply setting u(t) = u(a) for t ≤ a and u(t) = b for t ≥ b.
Theorem A.8 (equivalent definitions of Sobolev spaces). Let 1 < p ≤∞; then the following statements are equivalent:

(i) u ∈ W ^{1,p}(a, b);

(ii) there exists C ≥ 0 such that $\left{\int}_{a}^{b}u\varphi \prime dt\right\le C{\Vert \varphi \Vert}_{{L}^{p\prime}\left(a,b\right)}$ if $\varphi \in {C}_{0}^{1}\left(a,b\right)$;

(iii) there exists C ≥ 0 such that for all I ⊂⊂ (a, b) and for all h ∈ R such that h ≤ dist (I, {a, b}) we have ‖τ_{h} u − u‖_{Lp(I)} ≤ Ch, where τ_{h} u(t) = u(t−h);

(iv) there exsists a sequence (u_{j}) in C ^{∞} ([a, b]) such that
(A.7)$${\mathrm{lim}}_{j}{\left\left{u}_{j}u\right\right}_{{W}^{1,p}\left(a,b\right)}=0$$ 
(v) there exists a sequence (u_{j}) in ${C}_{0}^{\infty}\left(R\right)$ such that (A.7) holds;

(vi) there exists a sequence (u_{j}) in ${C}_{0}^{\infty}\left(R\right)$ such that sup_{j} ‖u _{j}‖ W ^{1,p}(a, b) < +∞ and lim_{j} ‖u_{j} − u‖_{Lp(a, b)}=0.

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

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

(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
(p.200) Definition A.11 The space ${W}_{0}^{1,p}\left(a,b\right)$ is defined as the closure of ${C}_{0}^{\infty}\left(a,b\right)$ in the W ^{1,p}norm, or, equivalently, as the set of those u ∈ W ^{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
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
Remark A.14 If u ∈ W ^{1,1}(a, b) then $\text{Var}\left(u,\left(a,b\right)\right)={\int}_{a}^{b}\leftu\prime \rightdt$; 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 E ⊂ R ^{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
We say that x ∈ E is a point of density t ∈ [0,1] if there exists the limit
Theorem A.15 (De Giorgi's rectifiability theorem). Let E ⊂ R ^{N} be a set of finite perimeter in Ω. Then ∂*E is rectifiable; i.e., there exists a countable family (Γ_{i}) of graphs of C ^{1} functions of (N−1) variables such that ${\mathscr{H}}^{N1}\left(\partial *E\backslash {\cup}_{i=1}^{\infty}{\text{\Gamma}}_{i}\right)=0$. Moreover the perimeter of E in Ω´ ⊆ Ω is given by
The following theorem essentially states that sets of finite perimeter are characterized as those sets (almost all) whose onedimensional sections are finite unions of intervals (i.e. onedimensional sets of finite perimeter). We use the notation for onedimensional sections introduced in Chapter 15 and that for piecewiseconstant 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 ℋ ^{N − 1}a.a. y ∈ ∏_{ξ} the function u _{ξ,y} belongs to PC(Ω_{ξ,y}). Moreover, for such y we have
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 (E_{j}) of sets of finite perimeter in Ω, such that
We finally recall the coarea formula on the open set A ⊂ R ^{N}