Weak formulation of partial differential equations

In this chapter, we briefly discuss how the functional analysis and function space apparatus can be employed to analyse the well-posedness of certain class of PDEs when given in a so-called “weak” formulation. We start by considering the Poisson problem

(11)\[\nabla \cdot \nabla u = -\Delta u = f \quad \text{in } \Omega\]

supplemented with some suitable boundary conditions which \(u\) should satisfy on the boundary \(\Gamma = \partial \Omega\) of \(\Omega\).

The PDE (11) is the prototype example of a 2nd order elliptic operator. More generally and without any significant complications, we can consider a more general PDE of the form

\[\mcA u := - \nabla \cdot ( A \nabla u) = f\]

where \(A = (a_{ij}(x))_{i,j=1}^n\) is a pointwise defined matrix. Note that

(12)\[\nabla \cdot ( A(x) \nabla u(x)) = -\sum_{i,j=1}^n \partial_{i} (a_{ij}(x) \partial_{j} u(x))\]

For most part of the remaining lectures, we will require \(A(x)\) to satisfy the following defintion.

Definition 4 (Ellipticity of \(\mcA\) )

The partial differential operator \(\mcA\) given by (12) with coefficients \(A = (a_{ij})_{i,j=1}^N \in (L^{\infty}(\Omega))^{n\times n}\) is called elliptic constant \(\alpha > 0\) such that

• \( \lambda \cdot A(x) \lambda \geqslant \alpha |\lambda|^2\)

for any \(\lambda \in \RR^n\).

Remark 4

Note that \(A \in (L^{\infty}(\Omega))^{n\times n}\) also implies that that there exists an \(\beta \geqslant 0 \) such that also

• \(|A(x) \lambda| \leqslant \beta |\lambda|\)

holds for any \(\lambda \in \RR^n\), and by ellipticity, we can conclude that in fact \(\beta \geqslant \alpha > 0\).

Exercise 3

Prove the statements made in the previous remark

TODO

• Relate \(\mcA\) to classical Poisson problem

• Explain why general \(A(x)\) is useful, e.g. anisotropic heat conduction problems

We now prepared to investigate the well-posedness of a number of boundary value problems where we supplement the partial differential operator \(\mcA\) with one of the following boundary conditions

• Dirichlet boundary conditions Given function \(g_D: \Gamma \to \RR\), we require that

  \[u = u_D \quad \text{on } \Gamma\]

• Neumann boundary conditions Given function \(g_N: \Gamma \to \RR\), we require that

  \[\bfn \cdot A \nabla u = g_N \quad \text{on } \Gamma \]

• Robin boundary conditions Given functions \(g_R, \sigma: \Gamma \to \RR\), we require that

  \[\bfn \cdot A \nabla u = \sigma(g_R - u) \quad \text{on } \Gamma \]

These boundary conditions are called homogeneous if \(g_D\) (respectively \(g_N\), \(g_R\)) is zero, otherwise we deal with inhomogeneous boundary data. We start by looking at the Poisson supplemented with Neumann boundary conditions

TODO

Later, mention possible impact on test and trial spaces.

Neumann problems

Let us consider the homogenous Neumann problem

\begin{equation} \left\{ \begin{alignedat}{2} - \Delta u + u &= f & &\quad \text{in } \Omega \\ \partial_n u &= 0 & &\quad \text{on } \Gamma \end{alignedat} \right. \end{equation}

Here, we used the slightly simplified notation \(\partial_n u = \bfn \cdot \nabla u\). The idea to derive a so-called weak formulation of an PDE is very similar to the idea behind the introduction of weak derivatives: We multiply with a suitable test function \(v\), integrate over \(\Omega\) and perform integration by parts to transfer a number of derivatives to the test function \(v\).

What kind of test function space we choose is often dictated by 2 considerations:

1. What kind of smoothness do we require to make the derived formulation work?

2. How do we take into account the boundary conditions?

For the Neumann boundary problem, let’s assume for the moment that \(u\), our boundary \(\Gamma\) and our test functions \(v\) are smooth enough so that we can use Green’s theorem, e.g, \(u \in C^2(\overline{\Omega})\), \(\Gamma\) is a \(C^1\) boundary, and \(v \in C^{\infty}(\overline{\Omega})\). Then multiplying the PDE in (13) with \(v\) and integrating over \(\Omega\) and applying Green’s theorem leads to

(14)\[\begin{split}\int_{\Omega} f v \dx &= - \int_{\Omega} \nabla \cdot (\nabla u) v \dx +\int_{\Omega} uv \dx \\ &= - \int_{\Gamma} \underbrace{(\bfn \cdot \nabla u)}_{=0} v \dx + \int_{\Omega} \nabla u \cdot \nabla v \dx +\int_{\Omega} uv \dx\end{split}\]

Note that the Neumann boundary condition \(\bfn \cdot \nabla u = 0\) makes the boundary integrals vanish. Also observe that the right-hand side of (14) can be interpreted as taking the inner product associated with \(H^1(\Omega)\) between \(u\) and \(v\). In fact, the expression makes perfectly sense even if we assume only assume that both \(u, v \in H^1(\Omega) =:V\)!. With this assumption, we can define the bilinear form

(15)\[a(v, w) := \int_{\Omega} \nabla v \cdot \nabla w \dx +\int_{\Omega} vw \dx\]

on \(V\times V\), and it is straightforward to show that \(a(\cdot, \cdot)\) (being the \(H^1\) inner product itself) satisfies the required assumptions of the Lax-Milgram theorem:

\begin{align} \text{Boundedness: } a(v,w) &:= \int_{\Omega} \nabla v \cdot \nabla w \dx +\int_{\Omega} vw \dx = (v, w)_{H^1(\Omega)} \leqslant \|v\|_{H^1(\Omega)} \|w\|_{H^1(\Omega)} \\ \text{Coercivity: } a(v,v) &= \int_{\Omega} |\nabla v|^2 \dx + \int_{\Omega} |v|^2 \dx = (v, v)_{H^1(\Omega)} = \|v\|_{H^1(\Omega)}^2 \end{align}

Next, we define the linear form \(l : V \to \RR\)

(16)\[l(v) := \int_{\Omega} f v \dx = (f, v)_{L^2(\Omega)}\]

If we assume that \(f \in L^2(\Omega)\), then thanks to the Cauchy-Schwarz inequality,

\[|l(v)| = |(f, v)_{L^2(\Omega)}| \leqslant f\|_{L^2(\Omega)} \|v\|_{L^2(\Omega)} \leqslant f\|_{L^2(\Omega)} \|v\|_{H^1(\Omega)},\]

we can immediately conclude that \(l\) is a continuous bilinear form with \(C_l = \|f\|_{L^2(\Omega)}\). Thus the the Lax-Milgram theorem let us conclude that the problem: find \(u \in H^1(\Omega)=: V\) such that \(\forall v \in V\)

\[a(u,v) = l(v)\]

has a unique solution for every \(f \in L^2(\Omega)\) with \(\|u\|_{H^1(\Omega)} \leqslant \|f\|_{L^2(\Omega)}\).

TODO (variants of Neumann problems)

Discuss Neumann problems when a) lower term is not present b) \(g_N \neq 0\):

\begin{equation} \left\{ \begin{alignedat}{2} - \Delta u &= f & &\quad \text{in } \Omega \\ \partial_n u &= 0 & &\quad \text{on } \Gamma \end{alignedat} \right. \end{equation}

and

\begin{equation} \left\{ \begin{alignedat}{2} - \Delta u &= f & &\quad \text{in } \Omega \\ \partial_n u &= g_N & &\quad \text{on } \Gamma \end{alignedat} \right. \end{equation}

Robin problems

\begin{equation} \left\{ \begin{alignedat}{2} - \Delta u &= f & &\quad \text{in } \Omega \\ \partial_n u &= 0 & &\quad \text{on } \Gamma \end{alignedat} \right. \end{equation}

Dirichlet problems

Homogeneous Dirichlet problem for \(-\Delta + \mrm{Id}\) operator

Next, we consider

\begin{equation} \left\{ \begin{alignedat}{2} - \Delta u + u &= f & &\quad \text{in } \Omega \\ u &= 0 & &\quad \text{on } \Gamma \end{alignedat} \right. \end{equation}

We proceed as for the Neumann problem: we multiply with suitable test functions \(v\) and integrate by part, but this time, the boundary integral does not vanish since we don’t have natural boundary conditions to incorporate. To compensate, we only consider test functions \(v \in C^{\infty}_c(\Omega)\) which vanish at the boundary. Then again, we obtain

(18)\[\begin{split}\int_{\Omega} f v \dx &= - \int_{\Omega} \nabla \cdot (\nabla u) v \dx +\int_{\Omega} uv \dx \\ &= - \int_{\Gamma} (\bfn \cdot \nabla u)\underbrace{v}_{=0} \dx + \int_{\Omega} \nabla u \cdot \nabla v \dx +\int_{\Omega} uv \dx.\end{split}\]

Intuitively speaking we know how the solution \(u\) is going to look like on the boundary, namely \(u=0\), so we don’t need test functions which test for how the equation “behaves” at the boundary. Also, we now require that our function \(u\) comes from a function space where the boundary condition \(u=0\) is already incorporated. This is exactly what the \(H^1_0(\Omega)\) space is made for! So the weak formulation for (17) is:

Find \(u \in V := H^1_0(\Omega)\) such that

(19)\[a(u,v) = l(v) \quad \forall v \in V,\]

where \(a(\cdot, \cdot)\) and \(l(\cdot)\) are defined as in (15) and (16), respectively. As in the case for the homogeneous Neumann problem (13), we can show that \(a\) and \(l\) satisfy the assumption of the the Lax-Milgram theorem, and therefore we can conclude there there is a unique solution \(u\) to the weak formulation of the homogeneous Poisson problem which depends continuously on the data \(f\).

Important

The only but very important difference between the weak formulation of the homogeneous Neumann problem (13) and the homogeneous Dirichlet problem (17) is the Hilbert space on which they are posed on.

Homogeneous Dirichlet problem for \(-\Delta\) operator

Now, we consider a slightly modified problem Poisson problem where the low order term \(u\) is left out:

\begin{equation} \left\{ \begin{alignedat}{2} - \Delta u &= f & &\quad \text{in } \Omega \\ u &= 0 & &\quad \text{on } \Gamma \end{alignedat} \right. \end{equation}

Repeating the steps from the previous section, we arrive at the problem: Find \(u \in V := H^1_0(\Omega)\) such that

(21)\[a(u,v) = l(v) \quad \forall v \in V,\]

with the only distinction that \(a(\cdot, \cdot)\) is now given by

\[a(v, w) = \int_{\Omega} \nabla v \cdot \nabla w dx.\]

The boundedness of \(a(\cdot, \cdot)\) and \(l(\cdot)\) can be shown (almost) exactly as before. But let’s have a look at the coercivity/ellipticity: Setting \(u=v\), we obtain

\[a(v,v) = \int_{\Omega} |\nabla v|^2\]

But thanks to the Poincaré inequalty and <cor:poincare>

TODO (variants of Dirichlet problems)

Next, consider

\begin{equation} \left\{ \begin{alignedat}{2} - \Delta u &= f & &\quad \text{in } \Omega \\ u &= 0 & &\quad \text{on } \Gamma \end{alignedat} \right. \end{equation}