Week 7: Traces, duals and elliptic equations

We now similarly construct trace operators for bounded domains of class $C^{1}$.

Theorem 7.1. Suppose $\Omega$ is a bounded domain of class $C^{1}$. There is a unique bounded linear operator $T_{\partial\Omega}:H^{1,p}(\Omega)\rightarrow L^{p}(\Omega)$ such that for all $u\in C^{1}(\overline\Omega)$, $T_{\partial\Omega}u=\left.u\right|_{\partial\Omega}$. Moreover, if $\varphi\in C^{1}(\overline{\Omega})$, then for any $v\in H^{1,p}(\Omega)$, $T(v\cdot \varphi)=Tv\cdot T\varphi$.

Proof. Let $u\in C^{1}(\overline{\Omega})$. As before, we note that $\partial\Omega\subset \bigcup_{x\in \partial\Omega}\Phi_{x}^{-1}(B(0,\frac{1}{2}))$ for a suitable collection of $C^{1}$ diffeomorphisms $\{\Phi_{x}:N_{x}\rightarrow B(0,1)\}$ taking $x$ to $0$ and $\partial\Omega\cap N_{x}$ to $B(0,1)\cap \mathbb{R}^{n-1}$. By compactness, $\partial\Omega$ is contained in the finite union $\bigcup_{i=1}^{N}\Phi_{x_{i}}^{-1}(B(0,\frac{1}{2}))$ for some $\{x_{i}\}_{i=1}^{N}\subset\partial\Omega$.

Let $\{\psi_{i}\}_{i=1}^{N}$ be a partition of unity subordinate to this cover. Note that

$\int_{\partial\Omega}|u|^{p}=\int_{\partial\Omega}|\sum_{i=1}^{N}\psi_{i}\cdot u|^{p} \leq C(N,p)\cdot \sum_{i=1}^{N}\int_{\partial\Omega\cap N_{x_{i}}}|\psi_{i}u|^{p}.$

Now, setting $\varrho(x):=\sqrt{\det(\partial_{r}\Phi_{x_{i}}^{-1}\cdot \partial_{s}\Phi_{x_{i}}^{-1})_{r,s=1}^{n-1}}(x,0)$, which is strictly positive, we know that

$\begin{aligned}\int_{\partial\Omega\cap N_{x_{i}}}|\psi_{i}u|^{p} = \int_{B^{+}(0,1/2)}|\psi_{i}\cdot u|^{p}\circ \Phi_{x_{i}}^{-1}\cdot \varrho&\leq C(\Omega,n,p)\cdot \int_{B^{+}(0,1/2)}|\psi_{i}\cdot u|^{p}\circ \Phi_{x_{i}}^{-1} + |\D(\psi_{i}\cdot u\circ \Phi_{x_{i}}^{-1})|^{p}\\ &\leq C(\Omega,n,p)\cdot \int_{B^{+}(0,1/2)}\left(|u|^{p}+|\D u|^{p}\right)\circ \Phi_{x_{i}}^{-1}\cdot |\det \D\Phi_{x_{i}}^{-1}|\\ &=C(\Omega,n,p)\cdot \int_{\Phi_{x_{i}}^{-1}(B^{+}(0,1/2))}|u|^{p}+|\D u|^{p}\leq C\cdot ||u||_{1,p,\Omega}^{p}.\end{aligned}$

Altogether, we see that $||u||_{p,\partial\Omega}\leq C(\Omega,n,p)\cdot ||u||_{1,p,\Omega}$ so that the `restriction to the boundary mapping' $C^{1}(\overline{\Omega})\subset H^{1,p}(\Omega)\rightarrow L^{p}(\partial\Omega)$ is continuous. By Lemma 4.5, it extends uniquely to a continuous linear mapping $T:H^{1,p}(\Omega)\rightarrow L^{p}(\partial\Omega)$, which is what we sought to prove. The last statement is an immediate consequence of the definition of $T$.

The trace operator allows us to nicely characterise $H_{0}^{1,p}(\Omega)$.

Theorem 7.2. Suppose $\Omega$ is a bounded domain of class $C^{1}$. Then $u\in H_{0}^{1,p}(\Omega)$ iff $u\in H^{1,p}(\Omega)$ and $T_{\partial\Omega}u\equiv 0$.

Proof. ($\Rightarrow$) Clearly $u\in H^{1,p}(\Omega)$. Since $u=\lim_{m\rightarrow\infty}u_{m}$ in $W^{1,p}(\Omega)$ for $\{u_{m}\}\subset C_{0}^{1}(\Omega)$ so that $Tu_{m}\equiv 0$, we see that $T_{\partial\Omega}u=\lim_{m\rightarrow\infty}T_{\partial\Omega}u_{m}=0$, the limit being taken in $L^{p}(\partial\Omega)$.

($\Leftarrow$) Assume the same setup as the proof of Theorem. The assertion that $T_{\partial\Omega}u\equiv 0$ is equivalent to the existence of $\{u_{m}\}_{m=1}^{\infty}\subset C^{1}(\overline{\Omega})$ such that $u_{m}\xrightarrow{m\rightarrow \infty}u$ in $W^{1,p}(\Omega)$ and $\left.u_{m}\right|_{\partial\Omega}\xrightarrow{m\rightarrow\infty} 0$ in $L^{p}(\partial\Omega)$, which implies in particular that $(\psi_{i}\cdot u_{m})\circ \Phi_{x_{i}}^{-1}\xrightarrow{m\rightarrow\infty}(\psi_{i}\cdot u)\circ \Phi_{x_{i}}^{-1}$ in $W^{1,p}(\mathbb{R}^{n}_{+})$ and $\left.(\psi_{i}\cdot u_{m})\circ \Phi_{x_{i}}^{-1}\right|_{\partial\mathbb{R}^{n}_{+}}\xrightarrow{m\rightarrow\infty}0$ in $L^{p}(\partial\mathbb{R}^{n}_{+})$. It therefore suffices to show that if $v\in H^{1,p}(\mathbb{R}^{n}_{+})$ is compactly supported in $\overline{\mathbb{R}^{n}_{+}}$ is such that $Tv\equiv 0$, then $v\in H_{0}^{1,p}(\mathbb{R}^{n}_{+})$.

Let $\{v_{m}\}\subset C_{0}^{1}(\overline{\mathbb{R}^{n}_{+}})$ be such that $v_{m}\rightarrow v$ in $W^{1,p}(\mathbb{R}^{n}_{+})$. We may assume that $\supp v_{m}\subset K\Subset\overline{\mathbb{R}^{n}_{+}}$ for all $m\in\mathbb{N}$ and a suitable compact set $K$. For $x\in\partial\mathbb{R}^{n}_{+}$ and $x^{n}\geq 0$ we compute that

$|v_{m}(x,x^{n})|\leq |v_{m}(x,0)| +\int_{0}^{x^{n}}\left|\frac{\partial v_{m}}{\partial x^{n}}(x,t)\right|dt=|Tv_{m}(x)| + \int_{0}^{x^{n}}\left|\frac{\partial v_{m}}{\partial x^{n}}(x,t)\right|dt$

so that by the elementary inequality $(a+b)^{p}\leq C(a^{p} + b^{p})$ and Hölder's inequality,

$\int_{\mathbb{R}^{n-1}}|v_{m}(x,x^{n})|^{p}dx \leq C(p)\left(\int_{\partial\mathbb{R}^{n}_{+}}|Tv_{m}|^{p} + (x^{n})^{p-1}\int_{\mathbb{R}^{n-1}\times [0,x^{n}]}|\partial_{n}v_{m}|^{p} \right).\tag{*}$

Now fix a function $\chi\in C_{0}^{\infty}(\mathbb{R})$ with $0\leq \chi\leq 1$, $\left.\chi\right|_{[0,1]}\equiv 1$ and $\supp\chi\subset [0,2]$. We claim that the function

$v_{m}'(x,x^{n}):=v(x,x^{n})\cdot (1-\chi(mx^{n})),$

whose support is contained in $\left(\mathbb{R}^{n} \times \left[\frac{2}{m},\infty\right[\right)\cap\supp v\Subset \mathbb{R}^{n}_{+}$ is an approximating sequence for $v$; mollifying this sequence would then establish the claim. Now, it is clear that

$\int_{\mathbb{R}^{n}_{+}}|v_{m}'-v|^{p}=\int_{K\cap\mathbb{R}^{n-1}\times[0,2/m]}|\chi(m^{1/p}x^{n})|^{p}\cdot |v(x)|^{p}dx\xrightarrow{m\rightarrow\infty}0,$

and similarly, $\int|\partial_{i}v_{m}'-\partial_{i}v|^{p}\xrightarrow{m\rightarrow\infty}0$ for $i<n$. For the $n$th derivative, note that

$\begin{aligned}\int_{\mathbb{R}^{n}_{+}}|\partial_{n}v_{m}'-\partial_{n}v|^{p}&=\int_{K\cap \mathbb{R}^{n-1}\times[0,2/m]}|\chi(mx^{n})\cdot \partial_{n}v(x) + mv(x)\cdot \chi'(mx^{n})|^{p}\\ &\leq C(p,\chi)\left(\int_{K\cap \mathbb{R}^{n-1}\times[0,2/m]}|\chi(mx^{n})|^{p}\cdot |\partial_{n}v|^{p} + m^{p}\int_{K\cap \mathbb{R}^{n-1}\times[0,2/m]}|v(x)|^{p} \right). \end{aligned}$

Now, the first parenthetical term tends to $0$ as $m\rightarrow\infty$ as before. To handle the latter term, we use (*):

$\begin{aligned} m^{p}\int_{K\cap \mathbb{R}^{n-1}\times[0,2/m]}|v(x)|^{p}&=\lim_{l\rightarrow\infty} m^{p}\int_{K\cap \mathbb{R}^{n-1}\times[0,2/m]}|v_{l}(x)|^{p}\\ &\leq C(p)\lim_{l\rightarrow\infty}\left( 2m^{p-1}||Tv_{l}||_{p} + m^{p}\int_{0}^{2/m}(x^{n})^{p-1} \int_{\mathbb{R}^{n-1}\times[0,x^{n}]}|\D v_{l}|^{p}\right). \end{aligned}$

Since $Tv=\lim_{l\rightarrow\infty}Tv_{l}$ in $L^{2}(\partial\mathbb{R}^{n}_{+})$, the former parenthetical term tends to $0$ as $l\rightarrow\infty$. On the other hand, we may estimate the latter term from above by $2^{p}/p \cdot \int_{\mathbb{R}^{n-1}\times [0,2/m]}|\D v_{l}|^{p}$, which tends to $\frac{2^{p}}{p}\cdot \int_{K\cap \mathbb{R}^{n-1}\times [0,2/m]}|\D v|^{p}$ as $l\rightarrow\infty$. Altogether,

$m^{p}\int_{K\cap \mathbb{R}^{n-1}\times[0,2/m]}|v(x)|^{p}\leq C(p)\cdot\frac{2^{p}}{p}\cdot \int_{K\cap\mathbb{R^{n-1}\times [0,2/m]}}|\D v|^{p}\xrightarrow{m\rightarrow\infty}0.$

The spaces $H^{-k,p}(\Omega)$

In this section, we more explicitly describe the dual spaces to Sobolev spaces, focussing on the case of $H^{k,p}_{0}(\Omega)$.

Throughout this section, we assume that $p\in\left]1,\infty\right[$, $q:=\frac{p}{p-1}$ and write $N:=N(n,k)$ for the number of multiïndices $\alpha\in\mathbb{Z}^{n}$ with $|\alpha|\leq k$. Clearly $N(n,k)\leq (k+1)n$.

Before describing the dual to $H_{0}^{k,p}(\Omega)$, let $\{f_{\alpha}\}_{|\alpha|\leq k}\subset L^{q}(\Omega)$ and consider the linear mapping $F:H_{0}^{k,p}(\Omega)\rightarrow\mathbb{K}$ defined by

This expression is guaranteed to be finite by Hölder's inequality; in fact,

so that $F\in\left(H_{0}^{k,p}(\Omega) \right)^{\ast}$. This motivates the following definition.

Definition 7.3. The space $H^{-k,q}(\Omega)\subset(H^{k,p}_{0}(\Omega))^{\ast}$ is given by

It turns out that linear functionals on $H_{0}^{k,p}(\Omega)$ of the form (#) are actually the most general ones.

Theorem 7.4. $(H_{0}^{k,p}(\Omega))^{\ast}=H^{-k,p}(\Omega)$, i.e. for any $F\in (H_{0}^{k,p}(\Omega))^{\ast}$, there exist $\{f_{\alpha}\}_{|\alpha|\leq k}\subset L^{q}(\Omega)$ such that for all $u$,

and $||F||=\left(\sum_{|\alpha|\leq k}\int_{\Omega}|f_{\alpha}|^{q} \right)^{1/q}$.

Before proving this theorem, we recall the following fundamental result from functional analysis, which says that we can always extend continuous linear functionals in such a way as to preserve their norm.

Theorem 7.5 (Hahn-Banach). If $X$ is a Banach space, $U\subset X$ a linear subspace and $\varphi\in U^{\ast}$, then there exists a $\varphi'\in X^{\ast}$ such that $\left.\varphi'\right|_{U}\equiv \varphi$ and $||\varphi'||_{X^{\ast}}=||\varphi||_{U^{\ast}}$.

We shall also make use of the normed space $((L^{p}(\Omega))^{N},||\cdot||_{p})$ with norm given by

It is straightforward to check that $(L^{p}(\Omega))^{N}$ is a Banach space. We also note the following lemma.

Lemma 7.6. $\left(L^{p}(\Omega)^{N} \right)^{\ast}\simeq (L^{q}(\Omega))^{N}$, i.e. if $G\in \left((L^{p}(\Omega))^{N}\right)^{\ast}$, then there exist unique $g_{1},\dots,g_{N}\in L^{q}(\Omega)$ such that for all $u=(u_{1},\dots,u_{N})\in (L^{p}(\Omega))^{N}$,

and $||G||=||(g_{1},\dots,g_{N})||_{q}$.

Proof. Let $G\in \left(L^{p}(\Omega)^{N}\right)^{\ast}$ and note that for any $u\in L^{p}(\Omega)^{N}$,

$G(u)=G(u_{1},0,\dots,0) + G(0,u_{2},0,\dots,0)+\dots + G(0,\dots,0,u_{N}).$

Note that for each $i\in\{1,\dots,n\}$, $L^{p}(\Omega)\ni u_{i}\mapsto G(\dots,u_{i},\dots)\in\mathbb{K}$ is continuous so that by the canonical isomorphism $L^{p}(\Omega)^{\ast}\simeq L^{q}(\Omega)$, there exists a unique $g_{i}\in L^{q}(\Omega)$ such that for all $u_{i}\in L^{p}(\Omega)$,

$G(\dots,u_{i},\dots)=\int_{\Omega}g_{i}u_{i},$

i.e. $G(u)=\sum_{i=1}^{N}\int_{\Omega}g_{i}u_{i}$. By the same computation employed in the discussion above, we see that $||G||\leq ||(g_{1},\dots,g_{N})||_{q}$. To see that equality holds, choose $u$ such that

$u_{i}(x)=\begin{cases}0,& x\in g_{i}^{-1}(\{0\})\\ ||g||_{q}^{-q/p} \bar{g}_{i}\cdot |g_{i}|^{q-2},&\textup{otherwise} \end{cases}.$

Since this function is measurable and $(\int_{\Omega}|u_{i}|^{p})^{1/p}=||g||^{-q/p}_{q}(\int_{\Omega}|g_{i}|^{p(q-1)})^{1/p} =||g||_{q}^{-q/p}||g_{i}||_{q}^{q/p}< \infty$, $u_{i}\in L^{p}(\Omega)$ and we compute that

$||u||_{p}=\left(\sum_{i=1}^{N}||u_{i}||_{p}^{p} \right)^{1/p}=||g||_{q}^{-q/p}\cdot\left(\sum_{i=1}^{N}||g_{i}||_{q}^{q} \right)^{1/p}=1$

and we have that

$G(u)=||g||_{q}^{-q/p}\sum_{i=1}^{N}\int_{\Omega}|g_{i}|^{q}=||g||_{q}^{q(1-1/p)}=||g||_{q},$

whence $||G||=||g||_{q}$

Proof of Theorem 7.4. Consider the linear mapping $\iota:H_{0}^{k,p}(\Omega)\rightarrow L^{p}(\Omega)^{N}$ defined by

$\iota(u)=(\D^{\alpha}u)_{|\alpha|\leq k},$

where we assume that the set $\{\alpha\leq k\}$ is ordered in some manner. Note that $\iota$ is injective and $||\iota(u)||_{p}=||u||_{k,p}$, i.e. it is an isometry.

Now suppose $F\in H_{0}^{k,p}(\Omega)^{\ast}$, set $U:=\textup{im}\ \iota$ and define $F':U\rightarrow\mathbb{K}$ by $F'(\iota(u))=F(u)$. $F'$ is clearly an element of $U^{\ast}$, since

$||F'||_{U^{\ast}}=\sup\{|F'(\iota(u))|: ||\iota(u)||_{p}=1\}=\sup\{|F(u)|:||u||_{k,p}=1\}=||F||_{H_{0}^{k,p}(\Omega)^{\ast}}.$

By the Hahn-Banach theorem, $F'$ may be extended to all of $L^{p}(\Omega)^{N}$ in such a way that $||F'||_{(L^{p}(\Omega)^{N})^{\ast}}=||F||_{H_{0}^{k,p}(\Omega)^{\ast}}$, and by Lemma , there exists an $f=(f_{\alpha})_{|\alpha|\leq k}\in L^{q}(\Omega)^{N}$ such that $||F'||_{(L^{p}(\Omega)^{N})^{\ast}}=||f||_{q}$ and for all $v=(v_{\alpha})_{|\alpha|\leq k}\in L^{p}(\Omega)^{N}$,

$F'(v)=\sum_{|\alpha|\leq k}\int_{\Omega}f_{\alpha}v_{\alpha};$

in particular,

$F(u)=F'(\iota(u))=\sum_{|\alpha|\leq k}\int_{\Omega}f_{\alpha}\cdot\D^{\alpha}u.$

Remark 7.7. An inspection of the above considerations shows that we have not made essential use of the fact that we are dealing with $H_{0}^{k,p}(\Omega)$ rather than $W^{k,p}(\Omega)$; in particular, we have the same characterisation of $W^{k,p}(\Omega)^{\ast}$ in terms of $L^{q}(\Omega)^{N}$.

Elliptic equations

We now turn our attention to elliptic partial differential equations.

Definition 7.8. A linear second order elliptic operator is an expression of the form

where $(a_{ij})_{i,j=1}^{n}$ is a symmetric positive-definite matrix, and $a_{ij},b^{i},c:\Omega\rightarrow \mathbb{K}$. We say that $L$ is uniformly elliptic if there exists a constant $\theta>0$ such that for all $v=(v^{1},\dots,v^{n})\in\mathbb{R}^{n}\backslash\{0\}$ and $x\in\Omega$,

Remark 7.9. We say that $L$ is in divergence form if for all $u\in C^{2}(\Omega)$, $Lu$ may be written in the form

where $(a_{ij})_{i,j=1}^{n}$ has the same properties as before.

Given a second-order elliptic operator $L$ as above and functions $f:\Omega\rightarrow\mathbb{K}$ and $g:\partial\Omega\rightarrow\mathbb{K}$, we will be interested in the question of whether there is a solution $u:\Omega\rightarrow\mathbb{K}$ to the elliptic boundary value problem

and how smooth $u$ is expected to be given appropriate smoothness conditions on $f$ and $g$. We call such a boundary-value problem the Dirichlet problem associated with $L$.

Solutions to the Dirichlet problem associated with an elliptic operator in divergence form typically arise as minimisiers or critical points of energy functionals defined on suitable spaces of functions. The following lemma illustrates this. We assume that all functions are real-valued.

Lemma 7.10. Suppose $(a_{ij})_{i,j=1}^{n}$ is a positive-definite, symmetric matrix, $a_{ij}\in C^{1}(\Omega)$, $c,f \in C^{0}(\Omega)$ and $g\in C^{0}(\partial\Omega)$. Define the energy functional

If $u\in C^{2}(\overline\Omega)$ is a minimiser of $E$ amongst all functions $v\in C^{2}(\overline{\Omega})$ with $\left.v\right|_{\partial\Omega}\equiv g$, i.e. $\left.u\right|_{\partial\Omega}\equiv g$ and $E(u)\leq E(v)$ for all such $v$, then $u$ solves the following Dirichlet problem:

Proof. Let $\varphi\in C_{0}^{\infty}(\Omega)$. The minimisation condition implies that for all $t\in\mathbb{R}$, $E(u+t\varphi)\leq E(u)$, i.e. $t\mapsto E(u+t\varphi)$ attains its minimum at $t=0$. In particular,

$E(u+t\varphi) = E(u) +t\left(\int_{\Omega}\sum_{i,j=1}^{n}a_{ij}\partial_{i}u\partial_{j}\varphi + cu\varphi - f\varphi\right) + t^{2}\cdot \left(\frac{1}{2}\int_{\Omega}\sum_{i,j=1}^{n}a_{ij}\partial_{i}\varphi\partial_{j}\varphi + c\varphi^{2} \right).$

Therefore, $t\mapsto E(u+t\varphi)$ is smooth and we may set its first derivative at $t=0$ to $0$:

$0=\left.\frac{d}{d t}\right|_{t=0}E(u+t\varphi) \Leftrightarrow \int_{\Omega}\sum_{i,j=1}^{n}a_{ij}\partial_{i}u\partial_{j}\varphi + cu\varphi= \int_{\Omega}f\varphi.\tag{***}$

Integrating by parts yields the equation

$\int_{\Omega}\left(\sum_{i,j=1}^{n}\partial_{i}\left( \partial_{j}u\right) + cu\right)\varphi = \int_{\Omega} f\varphi,$

which holds for all $\varphi\in C_{0}^{\infty}(\Omega)$, immediately implying

Remark 7.11. Note that if $a_{ij},c\in L^{\infty}(\Omega)$, then $E(u)$ is also defined for $u\in W^{1,2}(\Omega)$, as is the intermediate equation (***). We say that (***) is the weak formulation of (**) and call the minimisation posed in Lemma the variational formulation of (**).