Existence and uniqueness theorems for variational problems

This article summarizes a list of existence and uniqueness theorems for variational problems from (Monk 2003), which are organized from simple to complex.

Theorem 1 (Riesz representation) Let \(\mathcal{X}\) be a Hilbert space. For each \(g \in \mathcal{X}'\) there exists a unique \(u\in\mathcal{X}\) such that \[\label{eq:riesz-representation} (u, v)_{\mathcal{X}} = g(v) \quad \forall v\in\mathcal{X},\] and \(\norm{u}_{\mathcal{X}}=\norm{g}_{\mathcal{X}'}\).

N.B. This is the simplest variational formulation where the left hand side is just an inner product.

Theorem 2 (Lax-Milgram) Let \(a(\cdot,\cdot):\mathcal{X}\times\mathcal{X}\rightarrow\mathbb{C}\) be a bounded and coercive sesquilinear form, i.e. for boundedness \[\label{eq:boundedness-in-lax-milgram} \abs{a(u,\phi)} \leq C\norm{u}_{\mathcal{X}} \norm{\phi}_{\mathcal{X}} \quad \forall u\in\mathcal{X},\phi\in\mathcal{X}\] and for coercivity, there is exists a constant \(\alpha>0\) independent of \(u\in\mathcal{X}\) such that \[\label{eq:coercivity} \abs{a(u,u)} \geq \alpha\norm{u}_{\mathcal{X}}^2 \quad \forall u\in\mathcal{X}.\] Then for all \(f\in\mathcal{X}'\) there exists a unique solution \(u\in\mathcal{X}\) to the following variational problem \[\label{eq:variational-problem} a(u,\phi)=f(\phi) \quad \forall \phi\in\mathcal{X}.\] Furthermore, the norm of \(u\) is controlled by the norm of \(f\) as \[\label{eq:lax-milgram-error-estimate} \norm{u}_{\mathcal{X}} \leq \frac{C}{\alpha}\norm{f}_{\mathcal{X}'}.\]

Theorem 3 (Generalized Lax-Milgram) Let \(\mathcal{X}\) and \(\mathcal{Y}\) be Hilbert spaces. Let \(a(\cdot,\cdot):\mathcal{X}\times\mathcal{Y}\rightarrow\mathbb{C}\) be a bounded sesquilinear form which satisfies the following properties:

1. For all \(u\in\mathcal{X}\) and \(v\in\mathcal{Y}\), \[\label{eq:boundedness-in-general-lax-milgram} \abs{a(u,v)} \leq C\norm{u}_{\mathcal{X}} \norm{v}_{\mathcal{Y}}.\]

2. There is a constant \(\alpha\) such that \[\label{eq:inf-sup-condition} \inf_{u\in\mathcal{X},\norm{u}_{\mathcal{X}}=1} \sup_{v\in\mathcal{Y},\norm{v}_{\mathcal{Y}} \leq 1} \abs{a(u,v)} \geq \alpha > 0.\]

3. For every \(v\in\mathcal{Y}\), \(v \neq 0\) \[\sup_{u\in\mathcal{X}} \abs{a(u,v)} > 0.\]

If \(g\in\mathcal{Y}'\), there exists a unique solution \(u\in\mathcal{X}\) to the following variational problem \[\label{eq:general-variational-problem} a(u,\phi) = g(\phi) \quad \forall \phi\in\mathcal{Y}.\] Furthermore, the norm of \(u\) is controlled by the norm of \(g\) as \[\label{eq:general-lax-milgram-error-estimate} \norm{u}_{\mathcal{X}} \leq \frac{C}{\alpha} \norm{g}_{\mathcal{Y}'}.\]

Theorem 4 (Existence and uniqueness theorem for mixed variational problem) Let \(\mathcal{X}\) and \(S\) be Hilbert spaces. Let \(a(\cdot,\cdot): \mathcal{X}\times\mathcal{X}\rightarrow\mathbb{C}\) and \(b(\cdot,\cdot): \mathcal{X}\times\mathcal{S}\rightarrow\mathbb{C}\) be bounded sesquilinear forms: \[\label{eq:boundedness-in-mixed-variational-problem} \begin{aligned} \abs{a(u,\phi)} & \leq C\norm{u}_{\mathcal{X}}\norm{\phi}_{\mathcal{X}} \quad \forall u,\phi\in\mathcal{X}, \\ \abs{b(u,\xi)} &\leq C\norm{u}_{\mathcal{X}}\norm{\xi}_{\mathcal{S}} \quad \forall u\in\mathcal{X}, \xi\in\mathcal{S}. \end{aligned}\]

\(a(\cdot,\cdot)\) satisfies the \(\mathcal{Z}\)-coercive condition, i.e. there exists a constant \(\alpha > 0\) independent of \(u\) such that \[\abs{a(u,u)} \geq \alpha\norm{u}_{\mathcal{X}}^2 \quad \forall u \in \mathcal{Z},\] where \[\label{eq:z-coercivity} \mathcal{Z} = \{ u\in\mathcal{X} \vert b(u,\xi)=0 \; \forall \xi\in\mathcal{S} \},\] i.e. \(\mathcal{Z}\) is the annihilator of \(\mathcal{S}\).

\(b(\cdot,\cdot)\) satisfies the Babuška-Brezzi condition, i.e. there exists a constant \(\beta>0\) independent of \(p\) such that \[\label{eq:babuska-brezzi-condition} \sup_{w\in\mathcal{X}} \frac{\abs{b(w,p)}}{\norm{w}_{\mathcal{X}}} \geq \beta\norm{p}_{\mathcal{S}} \quad \forall p\in\mathcal{S}.\]

If \(f\in\mathcal{X}'\) and \(g\in\mathcal{S}'\), there exists a unique solution \((u,p)\in\mathcal{X}\times\mathcal{S}\) to the following variational problem \[\begin{aligned} \label{eq:mixed-variational-problem-a} a(u,\phi) + b(\phi,p) &= f(\phi) \quad \forall \phi\in\mathcal{X} \\ \label{eq:mixed-variational-problem-b} b(u,\xi) &= g(\xi) \quad \forall \xi\in\mathcal{S}. \end{aligned}\] Furthermore, the norms of \(u\) and \(p\) can be controlled by the norms of \(f\) and \(g\) as \[\label{eq:mixed-variational-problem-error-estimate} \norm{u}_{\mathcal{X}} + \norm{p}_{\mathcal{S}} \leq C(\norm{f}_{\mathcal{X}'} + \norm{g}_{\mathcal{S}'})\]

N.B. By fixing an arbitrary \(p\) in \(\mathcal{S}\), the left hand side of Babuška-Brezzi condition is the norm of the functional \(b(\cdot,p): \mathcal{X}\rightarrow\mathcal{C}\), which has a lower bound determined by \(\norm{p}_{\mathcal{S}}\).

References

Monk, Peter. 2003. Finite Element Methods for Maxwell’s Equations.