Valores y concepciones nativas de recursos naturales

The maximal function space considered is the space of distributions on D, i.e. the (linear) dual space of the space C∞

c (D) of complex-valued smooth functions compactly supported

in D. We define weak or distributional (partial) derivatives ∂j on the space of distributions

by ‘duality’, i.e. ∂ju is the (unique) distribution v defined by

Z D vφ= − Z D u∂φ ∂xj ∀φ ∈ C_c∞(D), (1.19)

where the classical or strong derivative appears on the right-hand side (∂φ

∂xj(x) =

limh→0

φ(x+hej)−φ(x)

h for ej a unit vector in the xj direction). Integrals initially being

defined for functions only, for a distribution v we interpret R

Dvφ as notation for the

duality pairing, i.e. R

vφ gives result of applying the linear map v : C_c∞(D) → C to φ.

We naturally associate to a measurable function f the distribution ˜f : C_c∞(D) → C,

˜

f(φ) = R

Df φ, consistently with the above notation. Any distribution v can be asso-

ciated with at most one function f up to a Lebesgue null set (i.e. if ˜f = v = ˜g then f = g Lebesgue almost everywhere7_{), and whenever such a function associated with} the distribution ∂j˜u exists, it will also be called the weak derivative of the function u.

Integration by parts shows that if the classical derivative ∂u

∂xj exists then it is also the

weak derivative. Henceforth all derivatives, whether denoted ∂j or _∂x∂_j, will be defined in

a weak sense unless otherwise specified, and we will not distinguish between a function and its associated distribution.

Integration by parts remains true for weak derivatives via the divergence theorem (see e.g. [17] eq. (38)).

Theorem 1.6 (The divergence theorem). For a domain D with outward unit normal ν

on the boundary ∂D, Z D ∇ · F = Z ∂D F · ν,

where ∇ · denotes the divergence operator on vector fields F , ∇ · F = Pd

j=1∂jFj.

7_{Proved as follows: let φ ∈ C}∞

c (Rd) be a ‘bump function’, with φ = 0 if |x| > 1, φ(x) ≥ 0 for all x, and

R

Rdφ = 1. Then φε= ε

−d_φ(ε−1_{(·)) is called a mollifier. It can be shown that}R

Df (x)φε(a − x) dx → f (a)

as ε → 0 for Lebesgue-almost all a ∈ D, provided that f is locally integrable (which means that the integral is well-defined for ε sufficiently small) – see [27, Appendix C.4]. But R

Df (x)φε(a − x) dx =

R

Dg(x)φε(a − x) dx for all a and all ε sufficiently small by assumption, hence taking limits we see

1.6 Elliptic PDEs: a brief introduction 27

Remark. We can for example apply to F = u ∇ v − v ∇ u to see that

Z D u∆ v = Z D v∆ u + Z ∂D u∂v ∂ν − v ∂u ∂ν,

hence the description as ‘integration by parts’ (∇ denotes the usual gradient operator and ∆ the Laplacian). More generally, writing

∂α = ∂ |α| ∂xα1 1 . . . ∂x αd d ≡ ∂α1 1 · · · ∂ αd d , (1.20)

for a multi-index α = (α1, . . . , αd), αj ∈ N ∪ {0} for j ≤ d, of order |α| =Pj≤dαj, and

assuming u, v are compactly supported in D to avoid having to define the appropriate boundary operators (cf. [53, Chapter II, Theorem 2.1, p114]), we have

Z D u∂αv = (−1)|α| Z D v∂αu.

The Sobolev spaces Hr_{(D), r ∈ R, are constructed to capture the notion of the}

number of weak derivatives a function has: the derivative ∂j maps Hr+1(D) → Hr(D)

continuously for L2_{–Sobolev spaces H}r_{(D), r ≥ 0 (indeed, it is immediate from the}

definition below of Hr_{(D) that ∥∂}

ju∥_Hr_{(U )} ≤ ∥u∥_Hr+1_{(U )} for r ≥ 0; for r < 0 see Theorem

12.1 in [53] Chapter 1, p71). The space H0_{(D) = L}2_{(D) is the Lebesgue space of functions}

L2(D) = {f : D → C measurable s.t. ∥f∥2_L2_(D) :=

Z

D

|f(x)|2 < ∞},

where functions which are equal almost everywhere are understood to be identified, and for r ∈ R we define as follows.

Definitions (Hr_{(D), H}r

0(D), Hlocr (D)). Definitions are drawn from Lions & Magenes

[53]. For r ∈ N ∪ {0} we define

Hr(D) = {f ∈ L2(D) s.t. ∂αf ∈ L2(D) for all multi-indices α satisfying |α| ≤ r}. Hr_{(D) is a Hilbert space, equipped with the inner product}

⟨f, g⟩_Hr_(D)= X |α|≤r Z D ∂αf · ∂αg∗,

and associated norm ∥·∥Hr_(D) (see [53, Chapter I, §1.1, p1]). We use the analogous

For r ∈ R, r ≥ 0 we define Hr(D) via interpolation (see [53, Chapter I, §9.1, p40] for

details).

Hr

0(D) is defined as the ∥·∥Hr_(D)–closure of C_c∞(D) ⊂ Hr(D) (see [53, Chapter I,

§11.1, p55]), and the spaces Hr(D), r < 0 are defined as the (topological) dual spaces,

equipped with the dual norms,

Hr(D) = (H₀r(D))∗ = {distributions f s.t. ∥f∥_Hr_(D) = sup g∈H₀|r|(D), ∥g∥ H|r|(D)=1 Z f g∗ < ∞}, r <0. (See [53, Chapter I, §12.1, p70].) Hr

loc(D) is defined as the set of distributions f such that fφ ∈ Hr(D) for all φ ∈ C ∞

c (D)

(see [53, Chapter II, §3.2, p125]) or, equivalently, as the set of distributions f such that

f |U ∈ Hr(U) for all domains U b D, where the symbol b is read ‘compactly contained’

and U b D means the closure ¯U is a subset of the interior int D = D.

A fundamental result of PDE theory is Poincaré’s inequality, which says that the H1

norm is equivalent to the H1 seminorm on the subset H1

0(D) ⊂ H1(D). There is also a

version for functions whose average on D is zero, but only the given version will be used in this thesis.

Theorem 1.7 (Poincaré’s inequality). There exists a constant C = C(D) such that for

all u ∈ H1 0(D),

∥u∥_H1_(D) ≤ C∥∇ u∥_L2_(D).

Proof. See Corollary 6.31 in [3]. Roughly, the idea of the proof is that the fundamental theorem of calculus shows that a classically differentiable function cannot take values much larger than those of its derivative if the boundary values are zero, and this extends to H1

0(D) functions by density of Cc∞(D) (density can be proved using convolution with

mollifiers, similarly to the footnote in Section1.6.1).