Week 4: Sobolev embedding theorems

Given uWk,p(Ω)u\in W^{k,p}(\Omega), one might naturally ask whether the extra 'regularity' in the form of weak differentiability implies that uLq(Ω)u\in L^{q}(\Omega) for some q>pq>p, or whether uu 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)(X,||\cdot||_{X}) and (Y,Y)(Y,||\cdot||_{Y}) be normed spaces. A continuous embedding of XX into YY is an injective, continuous linear mapping ι:XY\iota:X\rightarrow Y, i.e. a one-to-one mapping such that there exists a constant C>0C>0 with the following property: Whenever xXx\in X,

ι(x)YCxX.(*)||\iota(x)||_{Y}\leq C\cdot ||x||_{X}.\tag{*}

A continuous embedding ι:XY\iota:X\rightarrow Y is said to be compact if whenever {xn}n=1X\{x_{n}\}_{n=1}^{\infty}\subset X is a bounded sequence, it admits a subsequence {xnk}k=1\{x_{n_{k}}\}_{k=1}^{\infty} such that {ι(xnk)}k=1\{\iota(x_{n_{k}})\}_{k=1}^{\infty} converges in YY, i.e. if ι\iota maps bounded sets to relatively compact sets.

Example 4.2. Let 1p<q1\leq p<q\leq \infty and suppose Ω\Omega is bounded. The inclusion Lq(Ω)Lp(Ω)L^{q}(\Omega)\hookrightarrow L^{p}(\Omega) is a continuous embedding.

Example 4.3. If UVRnU\subset V\subset\mathbb{R}^{n} and p[1,]p\in[1,\infty], the inclusion Lp(V)Lp(U)L^{p}(V)\hookrightarrow L^{p}(U), which is none other than the restriction mapping U\left.\cdot \right|_{U}, is a continuous embedding. More generally, we have the continuous embedding Wk,p(V)Wk,p(U)W^{k,p}(V)\hookrightarrow W^{k,p}(U).

Example 4.4. The embedding Ck+1(Ω)Ck(U)C^{k+1}(\overline{\Omega})\hookrightarrow C^{k}(\overline{U}) is compact whenever UΩU\Subset\Omega. If Ω\Omega is convex, we furthermore have the compactness of the embedding Ck+1(Ω)Ck(Ω)C^{k+1}(\overline{\Omega})\hookrightarrow C^{k}(\overline{\Omega}).

It is often tedious to establish an inequality of the form (*) for all elements of XX. Fortunately, as long as YY is complete, it suffices to consider a dense subspace of XX. This is the content of the following lemma.

Lemma 4.5. Suppose that

Under these conditions, ι\iota extends uniquely to a continuous mapping ι:XY\iota:X\rightarrow Y.

Proof. Since ι\iota is a continuous embedding, we have that ι(z)YCzX||\iota(z)||_{Y}\leq C\cdot ||z||_{X} for all zZz\in Z, C>0C>0 being a fixed constant. Now suppose xXx\in X and {zj}Z\{z_{j}\}\subset Z is a sequence converging to xx in XX. This sequence is then Cauchy in XX, and by linearity,

ι(zi)ι(zj)YCzizjXi,j0.||\iota(z_{i})-\iota(z_{j})||_{Y}\leq C\cdot ||z_{i}-z_{j}||_{X}\xrightarrow{i,j\rightarrow\infty}0.

Hence, {ι(zj)}\{\iota(z_{j})\} is also Cauchy and therefore has a limit, which we call ι(x)\iota(x). It follows from a similar argument that this limit is independent of the chosen sequence.

Example 4.6. Let Ω\Omega be a bounded domain of class CkC^{k}, VΩV\Supset\Omega and suppose E:Ck(Ω)C0k(V)H0k,p(V)E:C^{k}(\overline\Omega)\rightarrow C_{0}^{k}(V)\subset H_{0}^{k,p}(V) is an extension operator as constructed in Lemmas 3.6 and 3.7 (cf. Remark 3.8). Since Ck(Ω)C^{k}(\overline\Omega) is dense in Hk,p(Ω)H^{k,p}(\Omega) by definition and the inequality

Euk,p,VCuk,p,Ω||Eu||_{k,p,V}\leq C\cdot ||u||_{k,p,\Omega}

holds by construction, EE is a continuous linear mapping. Therefore, it extends uniquely to a continuous mapping E:Hk,p(Ω)H0k,p(V)E:H^{k,p}(\Omega)\rightarrow H_{0}^{k,p}(V). Note that EuΩ=u\left.Eu\right|_{\Omega}=u so that EE is an embedding: Since EumEuEu_{m}\rightarrow Eu in Wk,p(V)W^{k,p}(V), it must also converge in Wk,p(Ω)W^{k,p}(\Omega) by Example 4.2. However, Eum(x)=um(x)Eu_{m}(x)=u_{m}(x) for all xΩx\in \Omega so that by uniqueness of limits, EuΩ=u\left.Eu\right|_{\Omega}=u.

The Gagliardo-Nirenberg-Sobolev inequality

Notation. If uWk,p(Ω)u\in W^{k,p}(\Omega) with k1k\geq 1, define the gradient of uu as the vector Du:=(1u,,nu)T\D u:=(\partial_{1}u,\dots, \partial_{n}u)^{T}. If k2k\geq 2, we have the Hessian D2u:=(iju)i,j=1n\D^{2}u:=(\partial_{i}\partial_{j}u)_{i,j=1}^{n}. More generally, for k>2k>2, we define the kkth-order gradient of uu as the set

Dku:={Dαu:α=k}\D^{k}u:=\{\D^{\alpha}u:|\alpha|=k \}

and define its norm as Dku:=(α=kDαu2)1/2|\D^{k}u|:=\left( \sum_{|\alpha|=k}|\D^{\alpha}u|^{2} \right)^{1/2}. We will also write Dkup:= Dku p||\D^{k}u ||_{p}:=||\ |\D^{k}u|\ ||_{p}.

Exercise 4.7. There exists a constant C>0C>0 depending only on nn, kk and pp such that whenever uWk,p(Ω)u\in W^{k,p}(\Omega),

C1α=kDαupDkupCα=kDαup.C^{-1}\cdot \sum_{|\alpha|=k}\int |\D^{\alpha}u|^{p} \leq \int|\D^{k}u|^{p}\leq C\cdot \sum_{|\alpha|=k}\int|\D^{\alpha}u|^{p}.

We now focus on the case of W1,p(Ω)W^{1,p}(\Omega) and ask whether an inequality of the form

uqCDup(**)||u||_{q}\leq C\cdot ||\D u||_{p}\tag{**}

may be established for uC0(Rn)u\in C_{0}^{\infty}(\mathbb{R}^{n}), where C>0C>0 is some constant independent of uu. This inequality would then be useful in the case where Ω=Rn\Omega=\mathbb{R}^{n} or Ω\Omega is a bounded domain of class C1C^{1}, in which case C0(Rn)C_{0}^{\infty}(\mathbb{R}^{n}) is dense in W1,p(Ω)W^{1,p}(\Omega).

Exercise 4.8. Show that if (**) holds for p[1,n[p\in\left[1,n\right[ and n>1n>1, then we must have that q=npnpq=\frac{np}{n-p}.

Definition 4.9. Let n>1n>1 and p[1,n[p\in[1,n[. The Sobolev conjugate of pp is given by p:=npnpp^{\ast}:=\frac{np}{n-p}.

Note that p>pp^{\ast}>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,]\{p_{i}\}_{i=1}^{m}\subset[1,\infty] are such that i=1m1pi=1\sum_{i=1}^{m}\frac{1}{p_{i}}=1 and for each i{1,,m}i\in\{1,\dots,m\}, fiLpi(Ω)f_{i}\in L^{p_{i}}(\Omega). Then

Uf1fmi=1mfipi.\int_{U}|f_{1}\cdot\dots\cdot f_{m}|\leq \prod_{i=1}^{m}||f_{i}||_{p_{i}}.

Proof. Follows from Hölder's inequality by induction.

Lemma 4.11 (Gagliardo-Nirenberg-Sobolev). Suppose uC01(Rn)u\in C_{0}^{1}(\mathbb{R}^{n}). Then for any p[1,n[p\in\left[1,n\right[,

upp2n1npDup.||u||_{p^{\ast}}\leq \frac{p}{2}\cdot \frac{n-1}{n-p}\cdot ||\D u||_{p}.

Proof. We will first prove this for the case p=1p=1. By the fundamental theorem of calculus, it is clear that for each xRnx\in\mathbb{R}^{n} and i{1,,n}i\in\{1,\dots,n\},

u(x)=xiiu dzixiiudzi,|u(x)|=\left|\int_{-\infty}^{x^{i}}\partial_{i}u\ dz^{i}\right|\leq \int_{-\infty}^{x^{i}}|\partial_{i}u| dz^{i},

where the argument of iu\partial_{i}u in both integrands is (x1,,xi1,zi,xi+1,,xn)(x^{1},\dots, x^{i-1},z^{i},x^{i+1},\dots,x^{n}). Similarly, u(x)xiiudzi|u(x)|\leq \int_{x^{i}}^{\infty}|\partial_{i}u|dz^{i} so that

u(x)12Riu(x)dxi.|u(x)|\leq \frac{1}{2}\int_{\mathbb{R}}|\partial_{i}u(x)|dx^{i}.

Multiplying these inequalities together for all i{1,,n}i\in\{1,\dots, n\} and taking the (n1)(n-1)th root of both sides,

2n/(n1)u(x)n/(n1)i=1n(Riu(x)dxi)1/(n1).\rightsquigarrow 2^{n/(n-1)}|u(x)|^{n/(n-1)} \leq \cdot \prod_{i=1}^{n}\left(\int_{\mathbb{R}}|\partial_{i}u(x)|d x^{i} \right)^{1/(n-1)}.

We now integrate both sides with respect to x1x^{1}:

2n/(n1)Ru(x)n/(n1)dx1(R1u(x)dx1)1/(n1)Ri=2n(Riu(x)dxi)1/(n1)dx1.\rightsquigarrow 2^{n/(n-1)}\int_{\mathbb{R}}|u(x)|^{n/(n-1)}dx^{1}\leq \left(\int_{\mathbb{R}}|\partial_{1}u(x)|d x^{1} \right)^{1/(n-1)}\cdot \int_{\mathbb{R}}\prod_{i=2}^{n}\left(\int_{\mathbb{R}}|\partial_{i}u(x)|d x^{i} \right)^{1/(n-1)}dx^{1}.

We now apply the generalised Hölder inequality to the integral of the product with p1,,pn1=1n1p_{1},\dots,p_{n-1}=\frac{1}{n-1} to obtain

2n/(n1)Ru(x)n/(n1)dx1(R1u(x)dx1)1/(n1)i=2n(RRiu(x)dxidx1)1/(n1).2^{n/(n-1)}\int_{\mathbb{R}}|u(x)|^{n/(n-1)}dx^{1}\leq \left(\int_{\mathbb{R}}|\partial_{1}u(x)|d x^{1} \right)^{1/(n-1)}\cdot \prod_{i=2}^{n}\left(\int_{\mathbb{R}}\int_{\mathbb{R}}|\partial_{i}u(x)|d x^{i} dx^{1}\right)^{1/(n-1)}.

Integrating now with respect to x2x^{2}, we obtain

2n/(n1)R2u(x)n/(n1)dx1dx2(R22u(x)dx1dx2)1/(n1)R[(R1udx1)1/(n1)i=3n(R2iu(x)dxidx1)1/(n1)]dx2. \begin{gathered} 2^{n/(n-1)}\iint_{\mathbb{R}^{2}}|u(x)|^{n/(n-1)}dx^{1}dx^{2}\\ \leq \left(\iint_{\mathbb{R}^{2}}|\partial_{2}u(x)|d x^{1}dx^{2} \right)^{1/(n-1)}\cdot\int_{\mathbb{R}}\left[\left(\int_{\mathbb{R}}|\partial_{1}u|dx^{1} \right)^{1/(n-1)}\cdot \prod_{i=3}^{n}\left(\iint_{\mathbb{R^{2}}}|\partial_{i}u(x)|d x^{i} dx^{1}\right)^{1/(n-1)}\right]dx^{2}.\end{gathered}

Applying the generalised Hölder inequality to the product of n1n-1 functions in the latter integral, we arrive at the inequality

2n/(n1)R2u(x)n/(n1)dx1dx2i=12(R2iu(x)dx1dx2)1/(n1)i=3n(R3iu(x)dxidx1dx2)1/(n1).2^{n/(n-1)}\iint_{\mathbb{R}^{2}}|u(x)|^{n/(n-1)}dx^{1}dx^{2}\leq \prod_{i=1}^{2} \left(\iint_{\mathbb{R}^{2}}|\partial_{i}u(x)|d x^{1}dx^{2} \right)^{1/(n-1)}\cdot \prod_{i=3}^{n}\left(\iiint_{\mathbb{R}^{3}}|\partial_{i}u(x)|dx^{i}dx^{1}dx^{2}\right)^{1/(n-1)}.

Integrating with respect to x3,,xnx^{3},\dots, x^{n} and applying the same argument each time, we arrive at the inequality

2n/(n1)Rnun/(n1)i=1n(Rniu)1/(n1)(RnDu)n/(n1),2^{n/(n-1)}\int_{\mathbb{R}^{n}}|u|^{n/(n-1)}\leq \prod_{i=1}^{n}\left(\int_{\mathbb{R}^{n}}|\partial_{i}u|\right)^{1/(n-1)}\leq \left(\int_{\mathbb{R}^{n}}|\D u|\right)^{n/(n-1)},

i.e. un/(n1)12Du1||u||_{n/(n-1)}\leq \frac{1}{2} ||\D u||_{1}. For the case p>1p>1, we consider uγ|u|^{\gamma} in place of uu for γ>1\gamma>1, which is again an element of C01(Rn)C_{0}^{1}(\mathbb{R}^{n}). The case p=1p=1 together with the fact that Duγ=γuγ1DuD|u|^{\gamma}=\gamma |u|^{\gamma-1}|\D u| implies that

(Rnuγnn1)(n1)/n12γRnuγ1Du.\left(\int_{\mathbb{R}^{n}}|u|^{\frac{\gamma\cdot n}{n-1}}\right)^{(n-1)/n}\leq \frac{1}{2}\gamma\cdot \int_{\mathbb{R}^{n}}|u|^{\gamma-1}\cdot |\D u|.

Applying Hölder's inequality to the right-hand integral, we obtain

(Rnuγnn1)(n1)/n12γ(Rnuq(γ1))1/qDup,\left(\int_{\mathbb{R}^{n}}|u|^{\frac{\gamma\cdot n}{n-1}}\right)^{(n-1)/n}\leq \frac{1}{2}\gamma\cdot \left(\int_{\mathbb{R}^{n}}|u|^{q(\gamma-1)} \right)^{1/q}\cdot ||\D u||_{p},

where q=pp1q=\frac{p}{p-1}. Choosing γ\gamma such that nn1γ=q(γ1)γ=p(n1)np\frac{n}{n-1}\gamma=q\cdot(\gamma-1)\Leftrightarrow \gamma =\frac{p\cdot (n-1)}{n-p} and dividing both sides by the right-hand uu integral yields the inequality

unp/(np)12p(n1)npDup.||u||_{np/(n-p)}\leq \frac{1}{2}\cdot \frac{p(n-1)}{n-p}\cdot ||\D u||_{p}.

We now establish an embedding theorem using this inequality.

Corollary 4.12 (Sobolev embedding theorem). If p[1,n[p\in\left[1,n\right[ and VRnV\subset\mathbb{R}^{n} is open, then the inclusion

H01,p(V)Lp(V)H_{0}^{1,p}(V)\subset L^{p^{\ast}}(V)

holds and the corresponding inclusion mapping is continuous. In particular, if Ω\Omega is a bounded domain of class C1C^{1}, then W1,p(Ω)Lp(Ω)W^{1,p}(\Omega)\subset L^{p^{\ast}}(\Omega) and this embedding is continuous.

Proof. First note that Gagliardo-Nirenberg-Sobolev immediately implies that whenever uC01(Rn)u\in C_{0}^{1}(\mathbb{R}^{n}), the inequality

upC(n,p)DupC(n,p)u1,p(***)||u||_{p^{\ast}}\leq C(n,p)\cdot ||\D u||_{p}\leq C(n,p) ||u||_{1,p}\tag{***}

holds. Now, consider the inclusion ι:C01(V)H01,p(V)Lp(Ω)\iota:C_{0}^{1}(V)\subset H_{0}^{1,p}(V)\rightarrow L^{p^{\ast}}(\Omega) defined by uuu\mapsto u, which by (***) is continuous. By Lemma 4.5, it extends uniquely to a continuous linear mapping ι:H01,p(V)Lp(Ω)\iota:H_{0}^{1,p}(V)\rightarrow L^{p^{\ast}}(\Omega). Since both {un}\{u_{n}\} and {ι(un)=un}\{\iota(u_{n})=u_{n}\} converge in Lloc1(V)\lloc^{1}(V) by Hölder's inequality, they must converge to the same limit so that ι(u)=u\iota(u)=u for all uH01,p(V)u\in H_{0}^{1,p}(V), i.e. ι\iota is a continuous embedding. The latter claim follows from choosing VV such that ΩV\Omega\Subset V and considering the following chain of continuous embeddings:

W1,p(Ω)H0k,p(V)Lp(V)Lp(Ω)W^{1,p}(\Omega)\hookrightarrow H_{0}^{k,p}(V)\hookrightarrow L^{p^{\ast}}(V)\hookrightarrow L^{p^{\ast}}(\Omega)

Remark 4.13 (Poincaré inequality). Suppose Ω\Omega is bounded and p[1,n[p\in[1,n[. The Sobolev embedding theorem together with Example 4.2 immediately implies that for q<pq<p^{\ast}, the embedding H01,p(Ω)Lq(Ω)H_{0}^{1,p}(\Omega)\subset L^{q}(\Omega) is continuous. However, we can extract more information out of the Gagliardo-Nirenberg-Sobolev inequality: If uH01,p(Ω)u\in H_{0}^{1,p}(\Omega), then combining the inequality

upC(n,p)Dup||u||_{p^{\ast}}\leq C(n,p)||\D u||_{p}

with Hölder's inequality uqμ(Ω)1/q1/pup||u||_{q}\leq \mu(\Omega)^{1/q - 1/p^{\ast}}\cdot ||u||_{p^{\ast}}, we obtain that

uqC(n,p,Ω)Dup.||u||_{q}\leq C(n,p,\Omega)\cdot ||\D u||_{p}.

Applying this in the case q=pq=p and appealing to Exercise 4.7, we deduce that there is a constant C>0C>0 depending only on nn, pp and Ω\Omega such that

(i=1nΩiup)1/pu1,pC(i=1nΩiup)1/p;\left( \sum_{i=1}^{n}\int_{\Omega}|\partial_{i}u|^{p} \right)^{1/p}\leq ||u||_{1,p}\leq C\cdot \left( \sum_{i=1}^{n}\int_{\Omega}|\partial_{i}u|^{p} \right)^{1/p};

this implies that the quantity

u0,1,p:=(i=1nΩiup)1/p||u||_{0,1,p}:=\left( \sum_{i=1}^{n}\int_{\Omega}|\partial_{i}u|^{p} \right)^{1/p}

defines a norm on H01,p(Ω)H_{0}^{1,p}(\Omega) and it is an equivalent norm to 1,p||\cdot||_{1,p}. In the case p=2p=2, this norm arises from the inner product

<u,v>0,1:=i=1nΩiuiv.\left<u,v \right>_{0,1}:=\sum_{i=1}^{n}\int_{\Omega}\overline{\partial_{i}u}\cdot \partial_{i}v.

Arguing by induction, we deduce more generally that H0k,p(Ω)H_{0}^{k,p}(\Omega) may be equipped with the equivalent norm

u0,k,p:=(α=kΩDαup)1/p||u||_{0,k,p}:=\left(\sum_{|\alpha|=k}\int_{\Omega}|\D^{\alpha}u|^{p} \right)^{1/p}

which, in the case p=2p=2, arises from the inner product

<u,v>0,k:=α=kΩDαuDαv.\left<u,v\right>_{0,k}:=\sum_{|\alpha|=k}\int_{\Omega}\overline{\D^{\alpha}u}\cdot \D^{\alpha}v.

Remark 4.14 (the case p=np=n). Suppose Ω\Omega is bounded and uW1,n(Ω)u\in W^{1,n}(\Omega). By Hölder's inequality, uW1,p(Ω)u\in W^{1,p}(\Omega) for any p[1,n[p\in[1,n[. Thus, if Ω\Omega is a domain of class C1C^{1} or uW01,p(Ω)u\in W_{0}^{1,p}(\Omega), we deduce that uLp(Ω)u\in L^{p^{\ast}}(\Omega) for any p[1,n[p\in [1,n[, i.e. uLq(Ω)u\in L^{q}(\Omega) for any q[1,[q\in[1,\infty[. Unfortunately, this is the best we can do: On the one hand, the constant CC appearing in the Gagliardo-Nirenberg-Sobolev inequality may be shown to tend to \infty as pnp\nearrow n. On the other hand, the function

u:B(0,1)Rxloglog(1+1x)\begin{aligned}u:B(0,1)&\rightarrow\mathbb{R}\\ x&\mapsto \log\log(1+\frac{1}{|x|})\end{aligned}

is in W1,n(B(0,1))W^{1,n}(B(0,1)) for all n>1n>1, but it is unbounded.