Embeddings of $W^{1,p}(E)$ into $L^q(E)$

Abstract

Let E be a domain in $\mathbb {R}^N$. An embedding from $W^{1,p}(E)$ into $L^q(E)$ is an estimate of the $L^q(E)$-norm of a function $u\in W^{1,p}(E)$, in terms of its $W^{1,p}(E)$-norm.

Appendices

Problems and Complements

1c Multiplicative Embeddings of $W^{1,p}_o(E)$

 

1.1.:

The following two Propositions are established by a minor variant of the arguments in § 2–4. Their significance is that u is not required to vanish, in some sense, on $\partial E$. Let $Q={\mathop {\textstyle {\prod }}\limits }_{j=1}^N(a_j,b_j)$ be a cube in $\mathbb {R}^N$.

 

Proposition 1.1c

Let $u\in W^{1,p}(Q)$ for some $p\in [1,N)$. Then $u\in L^{p^*}(Q)$, and

$$\begin{aligned} \Vert u\Vert _{p^*}\le \mathop {\textstyle {\sum }}\limits _{j=1}^N\Big (\frac{1}{N(b_j-a_j)} \Vert u\Vert _p+\frac{p(N-1)}{N(N-p)}\Vert u_{x_j}\Vert _p\Big ). \end{aligned}$$

The inequality continues to hold if $(b_j-a_j)=\infty$ for some j, provided we set $(b_j-a_j)^{-1}=0$.

Proposition 1.2c

Let $u\in W^{1,p}(\mathbb {R}^N)$ for some $p\in [1,N)$. Then $u\in L^{p^*}(\mathbb {R}^N)$ and

$$\begin{aligned} \Vert u\Vert _{p^*}\le \frac{p(N-1)}{N(N-p)} \mathop {\textstyle {\sum }}\limits _{j=1}^N\Vert u_{x_j}\Vert _p. \end{aligned}$$

 

1.2.:

There exists a function $u\in W_o^{1,N}(E)$, that is not essentially bounded.

1.3.:

The functional $u\rightarrow \Vert Du\Vert _p$ is a semi-norm in $W^{1,p}(E)$ and a norm in $W^{1,p}_o(E)$. Such a norm is equivalent to $\Vert u\Vert _{1,p}$, that is, there exists a constant $\gamma$ depending only upon N and p such that

$$\begin{aligned} \gamma ^{-1}\Vert Du\Vert _p\le \Vert u\Vert _{1,p}\le \gamma \Vert Du\Vert _p\quad \text { for all }\> u\in W^{1,p}_o(E). \end{aligned}$$

 

8c Embeddings of $W^{1,p}(E)$

 

8.1.:

Let $W^{1,p}(E)^*$ denote the dual of $W^{1,p}(E)$ for some $1\le p<\infty$ and let q be the Hölder conjugate of p. Prove that $L^q(E)^N\subset W^{1,p}(E)^*$. Give an example to show that inclusion is, in general strict.

8.2.:

Let E satisfy the cone condition, and let $W^{1,p}(E)^*$ denote the dual of $W^{1,p}(E)$ for some $1\le p<N$. Prove that $L^q(E)^N\subset W^{1,p}(E)^*$ where q is the Hölder conjugate of $p_*$. Give an example to show that inclusion is, in general strict.

8.3.:

Let E be the unit ball of $\mathbb {R}^N$ and consider formally the integral

$$\begin{aligned} W^{1,p}(E)\ni f\rightarrow T(f)=\int _E|x|^{-\alpha } f dx. \end{aligned}$$

Find the values of $\alpha$ for which this defines a bounded linear functional in $W^{1,p}(E)$. Note that the previous problem provides only a sufficient conditions. Hint: Compute $(x_i|x|^{-\alpha })_{x_i}$ weakly.

8.4.:

Let $E\subset$ be open set. Give an example of $f\in W^{1,p}(E)$ unbounded in every open subset of E. Hint: Properly modify the function in 17.9. of the Complements of Chap. 4.

 

1.1 8.1c Differentiability of Functions in $W^{1,p}(E)$ for $p>N$

A continuous function u defined in an open set $E\subset \mathbb {R}^N$, is differentiable at $x\in E$ if it admits a Taylor expansion of the form of (21.1) of Chap. 8 in the context of the Rademacher’s theorem.

Functions $u\in W^{1,p}(E)$ for $1<p<\infty$ are characterized as admitting a Taylor type expansion in the topology of $L^p(E)$ (Proposition 20.3 of Chap. 8). The embedding Theorem 8.1 implies that for $p>N$, such an expansion holds a.e. in E.

Theorem 8.1c

Functions $u\in W^{1,p}_{{\text {loc }}}(E)$ for $N<p\le \infty$ are a.e. differentiable in E.

Proof

Assume first $N<p<\infty$. Since $Du\in L^p_{{\text {loc }}}(E)$,

For any such x fixed, set

$$\begin{aligned} v(y)=u(y)-u(x)-Du(x)\cdot (y-x)\qquad \text { for }\> y\in B_\rho (x). \end{aligned}$$

In particular $v(x)=0$. Apply (8.4) to the function $v(\cdot )$, with E being the ball $B_{|y-x|}(x)$. It gives

Thus for $|y-x|\ll 1$

$$\begin{aligned} \frac{\big |u(y)-u(x)-Du(x)\cdot (y-x)\big |}{|y-x|} =O(|y-x|). \end{aligned}$$

Let now $p=\infty$. The previous arguments are local and $L^\infty _{{\text {loc }}}(E)\subset L^p_{{\text {loc }}}(E)$. Hence one can always assume $N<p<\infty$. $\blacksquare$

Remark 8.1c

The theorem is an extension of the Rademaker’s theorem to functions in $W^{1,p}(E)$ for $N<p\le \infty$. In particular for $p=\infty$ it provides an alternative proof of the Rademaker’s theorem.

14c Compact Embeddings

 

14.1.:

Theorem 14.1 is false if E is unbounded. To construct a counterexample, consider a sequence of balls $\{B_{\rho _j}(x_j)\}$ all contained in E such that $|x_j|\rightarrow \infty$ as $j\rightarrow \infty$. Then construct a function $\varphi \in W_o^{1,p}\big (B_{\rho _o}(x_o)\big )$ such that its translated and rescaled copies $\varphi _j$, all satisfy $\Vert \varphi _j\Vert _{1,p; B_{\rho _j}(x_j)}=1$ for all $j\in \mathbb {N}$. The sequence $\{\varphi _j\}$ does not have a subsequence strongly convergent in any $L^q(E)$ for all $q\ge 1$.

14.2.:

Theorem 14.1 is false for $q=p^*$ for all $p\in [1,N)$. Construct an example.

 

17c Traces and Fractional Sobolev Spaces

1.1 17.1c Characterizing Functions in $W^{1-\frac{1}{p},p}(\mathbb {R}^N)$ as Traces

Proposition 17.1c

Every function $\varphi \in W^{1-\frac{1}{p},p}(\mathbb {R}^N)$ has an extension $u\in W^{1,p}(\mathbb {R}^{N+1}_+)$, such that the trace of u on the hyperplane $t=0$, coincides with $\varphi$ and

$$\begin{aligned} \Vert u(\cdot ,t)\Vert _{p,\mathbb {R}^N}&\le \Vert \varphi \Vert _{p,\mathbb {R}^N}\qquad \text {for all }\> t>0\end{aligned}$$

(17.1c)

$$\begin{aligned} \Vert Du\Vert _{p,\mathbb {R}^{N+1}_+}&\le \gamma \Vert \!|\varphi |\!\Vert _{1-\frac{1}{p},p;\mathbb {R}^N} \end{aligned}$$

(17.2c)

where $\gamma$ depends only on N and p.

Proof

Assume $N\ge 2$ and let

$$\begin{aligned} F(x-y;t)=\frac{1}{(N-1)\omega _{N+1}} \frac{1}{\big (|x-y|^2+t^2\big )^{\frac{N-1}{2}}} \end{aligned}$$

be the fundamental solution of the Laplace equation in $\mathbb {R}^{N+1}$ with pole at (x, 0), introduced in § 14 of Chap. 8. Let also

be the Poisson integral of $\varphi$ in $\mathbb {R}^{N+1}_+$ introduced in § 18.2c of the Complements of Chap. 6. Since the kernel $H=2F_t$ has mass one for all $t>0$ and is harmonic in $\mathbb {R}^N\times \mathbb {R}^+$ we may regard $2F_t$ as a mollifying kernel following the parameter t. Therefore (17.1c) follows from the properties of the mollifiers and in particular Proposition 18.2c of the Complements of Chap. 6. Again, since $H=2 F_t$ has mass one

$$\begin{aligned} \frac{\partial }{\partial t}\int _{\mathbb {R}^N} F_t(x-y;t)dy=0\quad \text { and }\quad \frac{\partial }{\partial x_i}\int _{\mathbb {R}^N}F_t(x-y;t)dy=0 \end{aligned}$$

for $i=1,\dots ,N$. Therefore denoting by $\eta$ any one of the components of (x, t)

$$\begin{aligned} \frac{\partial }{\partial \eta }\varPhi (x,t)=\int _{\mathbb {R}^N} \frac{\partial ^2}{\partial \eta \partial t}F(x-y;t)[\varphi (y)-\varphi (x)]dy \end{aligned}$$

and by direct calculation

$$\begin{aligned} |D\varPhi (x,t)|\le \gamma \int _{\mathbb {R}^N} \frac{|\varphi (x)-\varphi (y)|}{(|x-y|+t)^{N+1}}dy \end{aligned}$$

for a constant $\gamma$ depending only upon N. Integrate in dy by introducing polar coordinates with pole at x and radial variable $\rho t$. If $\mathbf {n}$ denotes the unit vector spanning the unit sphere in $\mathbb {R}^N$

$$\begin{aligned} |D\varPhi (x,t)|\le \gamma \int _{|\mathbf {n}|=1}d\mathbf {n} \int _0^\infty \frac{\rho ^{N-1}}{(1+\rho )^{N+1}} \frac{|\varphi (x+\rho t\mathbf {n})-\varphi (x)|}{t}d\rho . \end{aligned}$$

By the continuous version of Minkowski’s inequality

$$\begin{aligned} \Vert D\varPhi \Vert _{p,\mathbb {R}^{N+1}_+}\le \gamma&\int _{|\mathbf {n}|=1} d\mathbf {n}\int _0^\infty \frac{\rho ^{N-1}}{(1+\rho )^{N+1}}d\rho \\&\quad \times \Big (\int _0^\infty \frac{\Vert \varphi (\cdot +\rho t\mathbf {n}) -\varphi (\cdot )\Vert ^p_{p,\mathbb {R}^N}}{t^p}dt\Big )^{\frac{1}{p}}. \end{aligned}$$

The last integral is computed by the change of variable $r=\rho t$. This gives

$$\begin{aligned} \Vert D \varPhi \Vert _{p,\mathbb {R}^{N+1}_+}&\le \gamma \Big (\int _0^\infty \frac{\rho ^{N-\frac{1}{p}}}{(1+\rho )^{N+1}}d\rho \Big )\\&\quad \times \int _{|\mathbf {n}|=1}\Big (\int _0^\infty r^{N-1}\frac{\Vert \varphi (\cdot +r\mathbf {n}) -\varphi (\cdot )\Vert ^p_{p,\mathbb {R}^N}}{r^{N+p-1}}\Big )^{\frac{1}{p}}d\mathbf {n}\\&\le \gamma (N,p)\Big (\int _{\mathbb {R}^N}\int _{\mathbb {R}^N} \frac{|\varphi (x)-\varphi (y)|^p}{|x-y|^{N+p-1}}dxdy\Big )^{\frac{1}{p}}\\&=\gamma (N,p)\Vert \!|\varphi |\!\Vert _{1-\frac{1}{p},p;\mathbb {R}^N}. \end{aligned}$$

The extension claimed by the proposition can be taken to be

$$\begin{aligned} u(x,t)=e^{-t/p}\varPhi (x,t). \end{aligned}$$

To prove that $\varphi$ is the trace of u, consider a sequence $\{\varphi _n\}\subset C_o^\infty \!\left( {\mathbb {R}^N}\right)$ that approximate $\varphi$ in the norm of $W^{s,p}(\mathbb {R}^N)$ for $s=1-1/p$. Such a sequence exists by virtue of Proposition 15.2. Then construct the corresponding Poisson integrals $\varPhi _n$ and let $u_n=e^{-t/p}\varPhi _n$. Writing

$$\begin{aligned} u_n-u= 2e^{-t/p}\int _{\mathbb {R}^N}F_t(x-y;t)[\varphi _n(y)-\varphi (y)]dy \end{aligned}$$

and applying (17.1c)–(17.2c) proves that $u_n\rightarrow u$ in $W^{1,p}(\mathbb {R}^{N+1}_+)$. By the definition of trace this implies that $u(\cdot ,0)=\varphi$.$\blacksquare$

Combining Proposition 17.1 and this extension procedure, gives the following characterization of the traces.

Theorem 17.1c

A function $\varphi$ defined and measurable in $\mathbb {R}^N$ belongs to $W^{s,p}(\mathbb {R}^N)$ for some $s\in (0,1)$ if and only if it is the trace on the hyperplane $t=0$ of a function $u\in W^{1,p}(\mathbb {R}^{N+1}_+)$ where $p=1/(1-s)$.

18c Traces on $\partial E$ of Functions in $W^{1,p}(E)$

Theorem 18.2c

Let E be a bounded domain in $\mathbb {R}^N$ with boundary $\partial E$ of class $C^1$ and with the segment property. A function $\varphi \in W^{1-\frac{1}{p},p}(\partial E)$ for some $p>1$ admits an extension $u\in W^{1,p}(E)$ such that the trace of u on $\partial E$ is $\varphi$.

Theorem 18.3c

Let E be a bounded domain in $\mathbb {R}^N$ with boundary $\partial E$ of class $C^1$ and with the segment property. A function $\varphi$ defined and measurable on $\partial E$ belongs to $W^{s,p}(\partial E)$ for some $s\in (0,1)$ if and only if it is the trace on $\partial E$ of a function $u\in W^{1,p}(E)$ where $p=1/(1-s)$.

1.1 18.1c Traces on a Sphere

The embedding inequalities of Proposition 18.1 might be simplified if E is of relatively simple geometry, such as a ball or a cube in $\mathbb {R}^N$. Let $B_R$ the ball of radius R about the origin of $\mathbb {R}^N$ and let $S_R=\partial B_R$.

Proposition 18.1c

Let $u\in W^{1,p}(B_R)$ for some $p\in [1,\infty )$. Then

$$\begin{aligned} \Vert u\Vert ^p_{p,S_R}\le \frac{N}{R}\Vert u\Vert _p^p+p\Vert u\Vert _p^{p-1} \Big \Vert \frac{\partial u}{\partial |x|}\Big \Vert _p. \end{aligned}$$

(18.3c)

If $p\in [1,N)$ then

$$\begin{aligned} \Vert u\Vert _{m,S_R}^m\le N \kappa _N^{\frac{1}{N}}\Vert u\Vert _{p^*}^m +m\Vert u\Vert _{p^*}^{m-1}\Big \Vert \frac{\partial u}{\partial |x|} \Big \Vert _p \end{aligned}$$

(18.4c)

where

$$\begin{aligned} p^*=\frac{Np}{N-p}\qquad \text { and }\qquad m=\frac{N-1}{N}p^*. \end{aligned}$$

Proof

We may assume that $u\in C^1(\bar{B}_R)$. Having fixed $q\ge p\ge 1$, set

$$\begin{aligned} \theta =\max \Big \{p;1+q\Big (1-\frac{1}{p}\Big )\Big \} \quad \text { so that }\quad p\le \theta \le q. \end{aligned}$$

Then for any unit vector $\mathbf {n}$

$$\begin{aligned} R^N&|u(R\mathbf {n})|^\theta =\int _0^R \frac{\partial }{\partial \rho }(\rho ^N|u(\rho \mathbf {n})|^\theta )d\rho \\&=N\int _0^R\rho ^{N-1}|u(\rho \mathbf {n})|^\theta d\rho +\theta \int _0^R\rho ^N|u(\rho \mathbf {n})|^{\theta -1} \frac{\partial u(\rho \mathbf {n})}{\partial \rho }{\text {sign }}u(\rho \mathbf {n})d\rho . \end{aligned}$$

Integrating in $d\mathbf {n}$ over the unit sphere $|\mathbf {n}|=1$ gives

$$\begin{aligned} R\int _{\mathbf {n}=1}&|u(R\mathbf {n})|^\theta R^{N-1}d\mathbf {n}= N\int _{B_R}|u|^\theta dx+\theta \int _{B_R} |x||u|^{\theta -1}\frac{\partial u}{\partial |x|}{\text {sign }}u dx\\&\le N\mu (B_R)^{1-\frac{\theta }{q}}\Big (\int _{B_R}|u|^q dx\Big )^{\frac{\theta }{q}}+\theta R\Big ( \int _{B_R}|u|^{(\theta -1)\frac{p}{p-1}}dx \Big )^{\frac{p-1}{p}}\Big \Vert \frac{\partial u}{\partial |x|}\Big \Vert _p. \end{aligned}$$

Inequalities (18.3c) and (18.4c) follow form this for $q=p$ and $q=p^*$.$\blacksquare$