Week 8: Existence of weak solutions

We now turn our attention to the notion of a weak solution to a (divergence-form) elliptic PDE. We will assume for simplicity that all functions considered are real-valued.

Weak solutions to divergence-form elliptic equations via Lax-Milgram

Our goal will be to establish the existence of solutions to the following Dirichlet problem for $u:\overline\Omega\rightarrow\mathbb{R}$:

We will first consider the so-called weak formulation of this problem; to this end, assume the following setup throughout this lecture:

• $\Omega\subset\mathbb{R}^{n}$ is an open bounded set.

• For each $i,j\in\{1,\dots,n\}$, $a_{ij}\in L^{\infty}(\Omega)$ and for all $v\in\mathbb{R}^{n}\backslash\{0\}$ and almost all $x\in\Omega$, the bounds

  $\lambda_{0}|v|^{2}\leq \sum_{i,j=1}^{n}a_{ij}(x) v^{i}v^{j}$

  and $|(a_{ij}(x))_{i,j=1}^{n}|\leq \Lambda$ hold.

• For each $i\in\{1,\dots,n\}$, $b^{i}\in L^{\infty}(\Omega)$ with $|b|\leq \beta$.

• $c\in L^{\infty}(\Omega)$ and the bound $|c|\leq \gamma$ holds.

• $f\in L^{2}(\Omega)$.

Note that if $u$ is twice differentiable and satisfies the equation $Lu=f$, then multiplying by a function $\varphi\in C_{0}^{\infty}(\Omega)$ and integrating the leading order term by parts, we end up with the equation

Definition 8.1. A function $u\in H^{1,2}_{0}(\Omega)$ is said to be a weak solution to the boundary-value problem (*) if for all $\varphi\in C_{0}^{\infty}(\Omega)$, the equation (**) holds.

Remark 8.2. We may more generally consider the boundary-value problem

where $\Omega$ is a bounded domain of class $C^{1}$ and $g\in T_{\partial\Omega}(H^{1,2}(\Omega))$, i.e. the equation (**) holds for all $\varphi\in C_{0}^{\infty}(\Omega)$ and $g=T_{\partial\Omega}g'$ for some $g'\in H^{1,2}(\Omega)$. By considering $u':=u-g'$ so that $u'\in H_{0}^{1,2}(\Omega)$, we then see that

If $g'$ may be taken to be in $C^{2}(\overline\Omega)$, then the right-hand side may be written in the form of the right-hand side of (**) and we have reduced the case of general boundary values to that of zero boundary values. More generally, the right-hand side is actually an element of $H_{0}^{-1,2}(\Omega)$, considered as a function of $\varphi$. We will discuss this version of (*) in our investigations.

Note that if (**) holds for all $\varphi\in C_{0}^{\infty}(\Omega)$, then by approximation it also holds for $\varphi\in H_{0}^{1,2}(\Omega)$. Moreover, if we let

and $F(\varphi):=\int_{\Omega}f\cdot \varphi$, then we may abbreviate (**) as finding a $u\in H_{0}^{1,2}(\Omega)$ such that for all $\varphi\in H_{0}^{1,2}(\Omega)$,

The mapping $B:H_{0}^{1,2}(\Omega)\times H_{0}^{1,2}(\Omega)\rightarrow\mathbb{R}$ is called the bilinear form associated to $L$. As it turns out, there is a general functional-analytic principle that allows us to treat equations of the form (**)' under suitable assumptions on $B$.

Theorem 8.3 (Lax-Milgram). Suppose $X$ is a Hilbert space, $F\in X^{\ast}$ and $B:X\times X\rightarrow\mathbb{K}$ is a bilinear form with the following properties:

• $B$ is bounded, i.e. there exists a $C_{0}>0$ such that for all $x,y\in X$, $|B(x,y)|\leq C_{0}\cdot ||x||\cdot ||y||$; and
• $B$ is coërcive, i.e. there exists a $C_{1}>0$ such that for all $x\in X$, $\Re B(x,x)\geq C_{1}\cdot ||x||^{2}$.

Then there exists a unique $x_{0}\in X$ satisfying the equation

for all $y\in X$.

Proof. Note that uniqueness immediately follows from coërcivity and bilinearity: If $x_{0}$ and $x_{0}'$ solve (**), then for all $y\in X$, $B(x_{0}-x_{0}',y)=0$ so that

$||x_{0}-x_{0}'||\leq C_{1}^{-1}\Re B(x_{0}-x_{0}',x_{0}-x_{0}')=0\Rightarrow x_{0}=x_{0}'.$

As for existence, let

$\underline{X}:=\{\underline{x}\in X:\exists\chi\in X\ \textup{s.t.}\ \forall y\in X\ \left<\underline{x},y\right>=B(\chi,y)\}$

We will show that $\underline{X}=X$. First note that $(\underline{X},\left<\cdot,\cdot\right>)$ is a Hilbert space:

• If $\underline{x}_{1},\underline{x}_{2}\in \underline{X}$ so that $\exists\chi_{1},\chi_{2}\in X$ with $\left<\underline{x}_{i},y\right>=B(\chi_{i},y)$ for $i\in\{1,2\}$ and $y\in X$, then whenever $\alpha\in \mathbb{K}$,

  $\left<\underline{x}_{1}+\alpha\underline{x}_{2},y\right>=\left<\underline{x}_{1},y\right>+\alpha\left<\underline{x}_{2},y\right> = B(\chi_{1},y) + \alpha B(\chi_{2},y)=B(\chi_{1}+\alpha\chi_{2},y)$

  so that $\underline{x}_{1}+\alpha\underline{x}_{2}\in\underline{X}$.

• If $\{\underline{x}_{i}\}_{i=1}^{\infty}\subset\underline{X}$ is a Cauchy sequence, then it is clearly a Cauchy sequence in $X$ and therefore converges to some $x\in X$. On the other hand, the associates sequence of $\{\chi_{i}\}_{i=1}^{\infty}\subset X$ satisfying the equation

  $B(\chi_{i},\cdot)= \left<\underline{x}_{i},\cdot\right>$

  is also Cauchy, since

  $||\chi_{i}-\chi_{j}||^{2}\leq C_{1}^{-1}\Re B(\chi_{i}-\chi_{j},\chi_{i}-\chi_{j})=C_{1}^{-1}||\underline{x}_{i}-\underline{x}_{j}||^{2}\xrightarrow{i,j\rightarrow\infty}0$

  so that $\chi_{i}\xrightarrow{i\rightarrow\infty}\chi$ in $X$, which in turn implies by the boundedness of $B$ that for all $y\in X$,

  $\left<x,y\right>=\lim_{i\rightarrow\infty}\left<\underline{x}_{i},y\right>=\lim_{i\rightarrow\infty}B(\chi_{i},y)=B(\chi,y),$

  i.e. $x\in \underline{X}$ so that $\underline{X}$ is complete.

Now suppose $x\in X$, which implies that $\left(\underline{X}\ni\underline{z}\mapsto\left<x,\underline{z}\right>\right)\in (\underline{X})^{\ast}$, whence we may find an $\underline{x}\in \underline{X}$ such that for all $\underline{z}\in\underline{X}$,

$\left<\underline{x},\underline{z}\right>=\left<x,\underline{z}\right>\Leftrightarrow\left<x-\underline{x},\underline{z}\right>=0.$

Since $y\mapsto B(x-\underline{x},y)$ is in $X^{\ast}$, we may find an $x_{0}\in X$ such that $\left<x_{0},y\right>=B(x-\underline{x},y)$ for all $y\in X$, i.e. $x_{0}\in\underline{X}$, which implies that

$0=\Re\left<x_{0},x-\underline{x}\right>=\Re B(x-\underline{x},x-\underline{x})\geq C_{1}||x-\underline{x}||^{2},$

i.e. $x=\underline{x}$ so that $x\in \underline{X}\Rightarrow \underline{X}=X$. Since $X$ is a Hilbert space, the Riesz representation theorem yields that every element $F\in X^{\ast}$ is of the form $\left<\underline{x},\cdot\right>$ for some $\underline{x}\in X$, which establishes the claim.

We now establish some estimates on the bilinear form associated to $L$ with the goal of applying the Lax-Milgram theorem.

Lemma 8.4 (energy estimates). Suppose $B_{L}$ is the bilinear form associated to $L$. There exist constants $C_{0},C_{1}>0$ and $C_{2}>0$ such that for all $u,v\in H_{0}^{1,2}(\Omega)$,

and

Proof. First note that by the triangle inequality and Hölder's inequality,

$\begin{aligned}|B_{L}(u,v)|=\left|\int_{\Omega}a_{ij}\partial^{i}u\cdot \partial^{j}v + b^{i}\partial_{i}u\cdot v + cu\cdot v \right|&\leq \int_{\Omega}|a_{ij}\partial^{i}u\cdot \partial^{j}v| + |b^{i}\cdot \partial_{i}u|\cdot |v| + |c|\cdot |u|\cdot |v|\\&\leq \Lambda\int_{\Omega}|\D u|\cdot |\D v| + \beta \int_{\Omega}|\D u|\cdot |v| + \gamma\int_{\Omega}|u|\cdot |v|\\ &\leq(\Lambda + \beta + \gamma)||u||_{1,2}\cdot ||v||_{1,2},\end{aligned}$

As for the other estimate, we again make use of the bounds on the various coëfficients of $L$ to deduce that

$B_{L}(u,u)\geq \lambda_{0} \int_{\Omega} |\D u|^{2} - \beta \int_{\Omega}|\D u|\cdot |u| - \gamma\int_{\Omega}|u|^{2},$

or written more suggestively,

$B_{L}(u,u) + \beta \int_{\Omega}|\D u|\cdot |u| + \gamma\int_{\Omega}|u|^{2} \geq \lambda_{0} \int_{\Omega}|\D u|^{2}.$

We now apply the Peter-Paul inequality to the second integrand on the left-hand side, choosing $\eps=\frac{\lambda_{0}}{2\beta}$ so that

$B_{L}(u,u)+ \left(\frac{\beta^{2}}{2\lambda_{0}} + \gamma\right)\int_{\Omega}|u|^{2}\geq \frac{\lambda_{0}}{2}\int_{\Omega}|\D u|^{2} \geq \frac{\lambda_{0}}{4}\min\{1,C_{\textup{Poinc}}^{-1}\}||u||_{1,2}^{2},$

where $C_{\textup{Poinc}}$ is the constant appearing in the Poincaré inequality for $u\in H_{0}^{1,2}(\Omega)$. This establishes the claim.

As may be gleaned from this lemma, $B_{L}$ is almost coërcive, save for the constant $C_{2}$ getting in the way. However, we can still use Lax-Milgram to solve a slightly modified version of our problem.

Theorem 8.5 (first existence theorem). There exists a constant $C\geq 0$ such that whenever $\mu\geq C$, there exists a unique weak solution to the Dirichlet problem

Proof. Let $B_{L+\mu I}$ be the bilinear form corresponding to the elliptic operator $L+\mu I$ and $B_{L}$ that corresponding to $L$. For all $u,v\in H_{0}^{1,2}(\Omega)$, we have that

$B_{L+\mu I}(u,v)=B_{L}(u,v) + \mu\int_{\Omega}u\cdot v.$

This bilinear form is bounded, since by Hölder's inequality and Lemma 8.4,

$|B_{L+\mu I}(u,v)|\leq C_{0}||u||_{1,2}||v||_{1,2} + \mu ||u||_{2}||v||_{2}\leq (C_{0}+\mu)||u||_{1,2}||v||_{1,2}.$

On the other hand, we compute using Lemma 8.4 that

$B_{L+\mu I}(u,u)=B_{L}(u,u) + \mu ||u||_{2}^{2}\geq C_{1}||u||_{1,2}^{2} + (\mu-C_{2})||u||_{2}^{2}\geq C_{1}||u||_{1,2}^{2}$

provided $\mu\geq C:=C_{2}$. Since the linear mapping $H_{0}^{1,2}(\Omega)\ni v\mapsto \int_{\Omega}f\cdot v$ is continuous, Lax-Milgram implies that there exists a unique $u\in H_{0}^{1,2}(\Omega)$ such that for all $v\in H_{0}^{1,2}(\Omega)$,

$B_{L+\mu I}(u,v)=\int_{\Omega}f\cdot v,$

i.e. there exists a unique weak solution to the aforementioned Dirichlet problem.

Remark 8.6. The proof of the first existence theorem actually gives us a bit more: Given any $F\in H_{0}^{-1,2}(\Omega)$, there exists a unique solution $u$ to the equation $B_{L+\mu I}(u,v)=F(v)$ for all $v\in H_{0}^{1,2}(\Omega)$. We formally say that such a $u$ solves the equation $Lu +\mu u=F$ weakly. Using coërcivity, we also note that such a solution satisfies an estimate of the form

Remark 8.7. In light of the first existence theorem and the preceding remark, we see that the mapping $H_{0}^{1,2}(\Omega)\ni u\mapsto B(u,\cdot)\in H_{0}^{-1,2}(\Omega)$ is an isomorphism. We formally express this by saying that $L+\mu I:H_{0}^{1,2}(\Omega)\rightarrow H_{0}^{-1,2}(\Omega)$ is an isomorphism.

Remark 8.8. From the above proofs, we immediately see that if $b^{i}\equiv 0$ and $c\geq 0$, then the Dirichlet problem corresponding to $L:=-\sum_{i,j=1}^{n}a_{ij}\partial_{i}\partial_{j} + c$ admits a unique weak solution, since in this case $C_{2}=0$. This includes in particular the operator $L=-\Delta$.

Existence via Fredholm Alternative

To get a clearer picture of what gets in the way of the existence and uniqueness of solutions to $Lu=f$, we employ the so-called Fredholm alternative from functional analysis.

Recall that if $X$ is a Hilbert space, $D\subset X$ is a dense subspace, and $L:D\rightarrow X$ is a linear mapping, the adjoint of $L$ is the unique mapping $L^{\ast}:D'\rightarrow X$ such that

• for all $u\in D$ and $v\in D'$, $\left<Lu,v \right>=\left<u, L^{\ast}v\right>$; and
• if $L^{\ast}_{0}:D_{0}'\rightarrow X$ is any other linear mapping satisfying the relation $\left<Lu,v\right>=\left<u,L_{0}^{\ast}v\right>$ for all $u\in D$ and $v\in D_{0}'$, then $D_{0}'\subset D'$ and $L_{0}^{\ast}=\left.L\right|_{D_{0}'}$.

Note that $L$ is not necessarily continuous (if it were, then $L$ could be uniquely extended to all of $X$).

Theorem 8.9 (Fredholm alternative). Suppose $X$ is a Hilbert space and $K:X\rightarrow X$ is compact, i.e. if $\{x_{i}\}_{i=1}^{\infty}$ is a bounded sequence, then $\{Kx_{i}\}_{i=1}^{\infty}$ has a convergent subsequence. Either one of the following alternatives holds:

• For all $g\in X$ and $G\in X^{\ast}$, there exist unique $f\in X$ and $F\in X^{\ast}$ such that the equations

  $(I-K)f=g\tag{\#}$

  and

  $(I-K^{\ast})F=G\tag*{(\#)'}$

  hold.

• The homogeneous equations

  $(I-K)\varphi=0\tag{\#\#}$

  and

  $(I-K^{\ast})\Phi=0\tag*{(\#\#)'}$

  each have (the same finite number of) non-trivial solutions $\varphi\in X$ and $\Phi\in X^{\ast}$. In this case, a necessary and sufficient condition that (#) and (#)' possess solutions is that $g$ and $G$ satisfy the relations $\left<\Phi,g\right>=0$ and $\left<G,\varphi\right>=0$ whenever $\varphi$ and $\Phi$ solve (##) and (##)' respectively.

We now reformulate the Dirichlet problem so as to make use of the Fredholm alternative. Suppose $u\in H_{0}^{1,2}(\Omega)$ solves $Lu=f$ on $\Omega$ (in the weak sense), where $f\in L^{2}(\Omega)$, assume $\mu\geq 0$ is as in the first existence theorem and set $L_{\mu}:=L+\mu I$, which by the first existence theorem and subsequent remarks is an isomorphism $H_{0}^{1,2}(\Omega)\rightarrow H_{0}^{-1,2}(\Omega)$. Noting that $L=L_{\mu}-\mu I$, we see that

Setting $K:=\mu L_{\mu}^{-1}$, we also note that if $u\in L^{2}(\Omega)$ satisfies $(I-K)u=\mu^{-1}Kf$, then $u= Ku + \mu^{-1}Kf\in H_{0}^{1,2}(\Omega)$ so that by reversing the steps above, $Lu=f$ in the weak sense. Moreover, the mapping $K:L^{2}(\Omega)\rightarrow H_{0}^{1,2}(\Omega)\subset L^{2}(\Omega)$ is compact, since by Remark 8.6, if $\{f_{i}\}\subset L^{2}(\Omega)$ is a bounded sequence, then $\{Kf_{i} \}$ is bounded by the continuity of $K$ (as a mapping into $H_{0}^{1,2}(\Omega)$), but then Rellich-Kondrakov implies that $\{K f_{i}\}$ has a subsequence converging in $L^{2}(\Omega)$, which establishes the claim. Finally, to fully leverage the Fredholm alternative, we will need a more explicit description of $K^{\ast}$. We first define some useful notions.

Definition 8.10. If $b^{i}\in C^{1}(\overline{\Omega})$, the elliptic operator $L^{\ast}$ defined by

is called the formal adjoint of $L$. More generally, we define the adjoint bilinear form $B_{L^{\ast}}:H_{0}^{1,2}(\Omega)\times H_{0}^{1,2}(\Omega)\rightarrow \mathbb{R}$ such that

Finally, if $F\in H_{0}^{-1,2}(\Omega)$, we say that $v\in H_{0}^{1,2}(\Omega)$ is a weak solution of the adjoint problem

if for all $u\in H_{0}^{1,2}(\Omega)$, $B_{L^{\ast}}(v,u) = F(u)$.

Remark 8.11. If $a_{ij},b\in C^{1}(\overline\Omega)$ and $u,v\in C_{0}^{\infty}(\Omega)$, we see that $B_{L}(u,v)=\left<Lu,v\right>=\left<u,L^{\ast}v\right>=B_{L^{\ast}}(v,u)$.

As before, we formally write $L^{\ast}$ for the mapping $H_{0}^{1,2}(\Omega)\rightarrow H_{0}^{-1,2}(\Omega)$ such that $v\mapsto (u\mapsto B_{L^{\ast}}(v,u))$. Since $B_{L^{\ast}}$ satisfies the same estimates as $B_{L}$, we see that $L_{\mu}^{\ast}=(L^{\ast}+\mu I):H_{0}^{1,2}(\Omega)\rightarrow H_{0}^{-1,2}(\Omega)$ is also an isomorphism, and a quick calculation shows that the adjoint of $K$ defined before is given by

In particular, $\Phi\in L^{2}(\Omega)$ solves (##)' iff

i.e. iff $\Phi$ solves the adjoint problem.

Finally, if $\Phi$ solves (##)', then for $f\in L^{2}(\Omega)$,

Altogether, the Fredholm alternative yields the following existence theorem:

Theorem 8.12 (second existence theorem). One of the following statements is true:

• For any $f\in L^{2}(\Omega)$, the Dirichlet problem (*) possesses a unique solution $u\in H_{0}^{1,2}(\Omega)$.

• There exist a finite number of nontrivial solutions to the homogeneous problem

  $\begin{aligned}Lu&=0\ \textup{on }\Omega\\\left.u\right|_{\partial\Omega}&\equiv 0\end{aligned}$

  and the same number of nontrivial solutions $\Phi\in H_{0}^{1,2}(\Omega)$ to the adjoint homogeneous problem

  $\begin{aligned}L^{\ast}\Phi&=0\ \textup{on }\Omega\\\left.\Phi\right|_{\partial\Omega}&\equiv 0.\end{aligned}$

  A solution to (*) exists iff for any solution $\Phi$ to the adjoint homogeneous problem, $\left<\Phi,f\right>=0$.