Given u∈Wk,p(Ω), one might naturally ask whether the extra 'regularity' in the form of weak differentiability implies that u∈Lq(Ω) for some q>p, or whether u is in fact actually differentiable in the classical sense. We now turn our attention to these questions, whose answers take the form of the so-called Sobolev embedding theorems.
Embeddings of Banach spaces
We recall some functional analytic notions relevant to our considerations.
Definition 4.1. Let (X,∣∣⋅∣∣X) and (Y,∣∣⋅∣∣Y) be normed spaces. A continuous embedding of X into Y is an injective, continuous linear mapping ι:X→Y, i.e. a one-to-one mapping such that there exists a constant C>0 with the following property: Whenever x∈X,
∣∣ι(x)∣∣Y≤C⋅∣∣x∣∣X.(*)
A continuous embedding ι:X→Y is said to be compact if whenever {xn}n=1∞⊂X is a bounded sequence, it admits a subsequence {xnk}k=1∞ such that {ι(xnk)}k=1∞ converges in Y, i.e. if ι maps bounded sets to relatively compact sets.
Example 4.2. Let 1≤p<q≤∞ and suppose Ω is bounded. The inclusion Lq(Ω)↪Lp(Ω) is a continuous embedding.
Example 4.3. If U⊂V⊂Rn and p∈[1,∞], the inclusion Lp(V)↪Lp(U), which is none other than the restriction mapping⋅∣U, is a continuous embedding. More generally, we have the continuous embedding Wk,p(V)↪Wk,p(U).
Example 4.4. The embedding Ck+1(Ω)↪Ck(U) is compact whenever U⋐Ω. If Ω is convex, we furthermore have the compactness of the embedding Ck+1(Ω)↪Ck(Ω).
It is often tedious to establish an inequality of the form (*) for all elements of X. Fortunately, as long as Y is complete, it suffices to consider a dense subspace of X. This is the content of the following lemma.
Lemma 4.5. Suppose that
Z⊂X is dense, i.e. Z=X;
Y is a Banach space; and
ι:Z→Y is a continuous linear mapping.
Under these conditions, ι extends uniquely to a continuous mapping ι:X→Y.
Proof. Since ι is a continuous embedding, we have that ∣∣ι(z)∣∣Y≤C⋅∣∣z∣∣X for all z∈Z, C>0 being a fixed constant. Now suppose x∈X and {zj}⊂Z is a sequence converging to x in X. This sequence is then Cauchy in X, and by linearity,
∣∣ι(zi)−ι(zj)∣∣Y≤C⋅∣∣zi−zj∣∣Xi,j→∞0.
Hence, {ι(zj)} is also Cauchy and therefore has a limit, which we call ι(x). It follows from a similar argument that this limit is independent of the chosen sequence.
Example 4.6. Let Ω be a bounded domain of class Ck, V⋑Ω and suppose E:Ck(Ω)→C0k(V)⊂H0k,p(V) is an extension operator as constructed in Lemmas 3.6 and 3.7 (cf. Remark 3.8). Since Ck(Ω) is dense in Hk,p(Ω) by definition and the inequality
∣∣Eu∣∣k,p,V≤C⋅∣∣u∣∣k,p,Ω
holds by construction, E is a continuous linear mapping. Therefore, it extends uniquely to a continuous mapping E:Hk,p(Ω)→H0k,p(V). Note that Eu∣Ω=u so that E is an embedding: Since Eum→Eu in Wk,p(V), it must also converge in Wk,p(Ω) by Example 4.2. However, Eum(x)=um(x) for all x∈Ω so that by uniqueness of limits, Eu∣Ω=u.
The Gagliardo-Nirenberg-Sobolev inequality
Notation. If u∈Wk,p(Ω) with k≥1, define the gradient of u as the vector Du:=(∂1u,…,∂nu)T. If k≥2, we have the HessianD2u:=(∂i∂ju)i,j=1n. More generally, for k>2, we define the kth-order gradient of u as the set
Dku:={Dαu:∣α∣=k}
and define its norm as ∣Dku∣:=(∑∣α∣=k∣Dαu∣2)1/2. We will also write ∣∣Dku∣∣p:=∣∣∣Dku∣∣∣p.
Exercise 4.7. There exists a constant C>0 depending only on n, k and p such that whenever u∈Wk,p(Ω),
C−1⋅∣α∣=k∑∫∣Dαu∣p≤∫∣Dku∣p≤C⋅∣α∣=k∑∫∣Dαu∣p.
We now focus on the case of W1,p(Ω) and ask whether an inequality of the form
∣∣u∣∣q≤C⋅∣∣Du∣∣p(**)
may be established for u∈C0∞(Rn), where C>0 is some constant independent of u. This inequality would then be useful in the case where Ω=Rn or Ω is a bounded domain of class C1, in which case C0∞(Rn) is dense in W1,p(Ω).
Exercise 4.8. Show that if (**) holds for p∈[1,n[ and n>1, then we must have that q=n−pnp.
Definition 4.9. Let n>1 and p∈[1,n[. The Sobolev conjugate of p is given by p∗:=n−pnp.
Note that p∗>p. We will now show that an inequality of the form (**) does indeed hold. We first state the following lemma.
Lemma 4.10 (generalised Hölder inequality). Suppose {pi}i=1m⊂[1,∞] are such that ∑i=1mpi1=1 and for each i∈{1,…,m}, fi∈Lpi(Ω). Then
∫U∣f1⋅⋯⋅fm∣≤i=1∏m∣∣fi∣∣pi.
Proof. Follows from Hölder's inequality by induction.
Lemma 4.11 (Gagliardo-Nirenberg-Sobolev). Suppose u∈C01(Rn). Then for any p∈[1,n[,
∣∣u∣∣p∗≤2p⋅n−pn−1⋅∣∣Du∣∣p.
Proof. We will first prove this for the case p=1. By the fundamental theorem of calculus, it is clear that for each x∈Rn and i∈{1,…,n},
∣u(x)∣=∣∣∣∣∣∫−∞xi∂iudzi∣∣∣∣∣≤∫−∞xi∣∂iu∣dzi,
where the argument of ∂iu in both integrands is (x1,…,xi−1,zi,xi+1,…,xn).
Similarly, ∣u(x)∣≤∫xi∞∣∂iu∣dzi so that
∣u(x)∣≤21∫R∣∂iu(x)∣dxi.
Multiplying these inequalities together for all i∈{1,…,n} and taking the (n−1)th root of both sides,
i.e. ∣∣u∣∣n/(n−1)≤21∣∣Du∣∣1. For the case p>1, we consider ∣u∣γ in place of u for γ>1, which is again an element of C01(Rn). The case p=1 together with the fact that D∣u∣γ=γ∣u∣γ−1∣Du∣ implies that
(∫Rn∣u∣n−1γ⋅n)(n−1)/n≤21γ⋅∫Rn∣u∣γ−1⋅∣Du∣.
Applying Hölder's inequality to the right-hand integral, we obtain
where q=p−1p. Choosing γ such that n−1nγ=q⋅(γ−1)⇔γ=n−pp⋅(n−1) and dividing both sides by the right-hand u integral yields the inequality
∣∣u∣∣np/(n−p)≤21⋅n−pp(n−1)⋅∣∣Du∣∣p.
We now establish an embedding theorem using this inequality.
Corollary 4.12 (Sobolev embedding theorem). If p∈[1,n[ and V⊂Rn is open, then the inclusion
H01,p(V)⊂Lp∗(V)
holds and the corresponding inclusion mapping is continuous. In particular, if Ω is a bounded domain of class C1, then W1,p(Ω)⊂Lp∗(Ω) and this embedding is continuous.
Proof. First note that Gagliardo-Nirenberg-Sobolev immediately implies that whenever u∈C01(Rn), the inequality
∣∣u∣∣p∗≤C(n,p)⋅∣∣Du∣∣p≤C(n,p)∣∣u∣∣1,p(***)
holds. Now, consider the inclusion ι:C01(V)⊂H01,p(V)→Lp∗(Ω) defined by u↦u, which by (***) is continuous. By Lemma 4.5, it extends uniquely to a continuous linear mapping ι:H01,p(V)→Lp∗(Ω). Since both {un} and {ι(un)=un} converge in Lloc1(V) by Hölder's inequality, they must converge to the same limit so that ι(u)=u for all u∈H01,p(V), i.e. ι is a continuous embedding. The latter claim follows from choosing V such that Ω⋐V and considering the following chain of continuous embeddings:
W1,p(Ω)↪H0k,p(V)↪Lp∗(V)↪Lp∗(Ω)
Remark 4.13 (Poincaré inequality). Suppose Ω is bounded and p∈[1,n[. The Sobolev embedding theorem together with Example 4.2 immediately implies that for q<p∗, the embedding H01,p(Ω)⊂Lq(Ω) is continuous. However, we can extract more information out of the Gagliardo-Nirenberg-Sobolev inequality: If u∈H01,p(Ω), then combining the inequality
∣∣u∣∣p∗≤C(n,p)∣∣Du∣∣p
with Hölder's inequality ∣∣u∣∣q≤μ(Ω)1/q−1/p∗⋅∣∣u∣∣p∗, we obtain that
∣∣u∣∣q≤C(n,p,Ω)⋅∣∣Du∣∣p.
Applying this in the case q=p and appealing to Exercise 4.7, we deduce that there is a constant C>0 depending only on n, p and Ω such that
defines a norm on H01,p(Ω)and it is an equivalent norm to ∣∣⋅∣∣1,p. In the case p=2, this norm arises from the inner product
⟨u,v⟩0,1:=i=1∑n∫Ω∂iu⋅∂iv.
Arguing by induction, we deduce more generally that H0k,p(Ω) may be equipped with the equivalent norm
∣∣u∣∣0,k,p:=⎝⎛∣α∣=k∑∫Ω∣Dαu∣p⎠⎞1/p
which, in the case p=2, arises from the inner product
⟨u,v⟩0,k:=∣α∣=k∑∫ΩDαu⋅Dαv.
Remark 4.14 (the case p=n). Suppose Ω is bounded and u∈W1,n(Ω). By Hölder's inequality, u∈W1,p(Ω) for any p∈[1,n[. Thus, if Ω is a domain of class C1oru∈W01,p(Ω), we deduce that u∈Lp∗(Ω) for any p∈[1,n[, i.e. u∈Lq(Ω) for anyq∈[1,∞[. Unfortunately, this is the best we can do: On the one hand, the constant C appearing in the Gagliardo-Nirenberg-Sobolev inequality may be shown to tend to ∞ as p↗n. On the other hand, the function
u:B(0,1)x→R↦loglog(1+∣x∣1)
is in W1,n(B(0,1)) for all n>1, but it is unbounded.