Navier boundary conditions

Here we consider Navier boundary conditions:

$\left. u\right|_{\partial{\Omega}}=g, \left. \Delta{u}\right|_{\partial{\Omega}}=f, \left. \phi\right|_{\partial{\Omega}}=p,$

where $g,f,p\in{H^{1/2}}(\partial{\Omega})$ . As the second of the above boundary conditions is a natural boundary condition (rather than an essential boundary condition) we have to modify the energy functional

$\mathcal{F}(u,\phi) := \frac{1}{2} \kappa\int_\Omega |\Delta u |^2 +\frac{1}{2} \sigma\int_\Omega |\nabla u |^2 +\frac{1}{2} a\int_\Omega \phi^2 +\frac{1}{2} b\int_\Omega |\nabla \phi |^2 + c\int_\Omega \phi \Delta u$

by adding the respective boundary terms

$-\int_{\partial\Omega }(\kappa{f}+cp) \frac{\partial{u}}{\partial\nu} dS(x)$ .

These terms correspond to the energy required to impose the boundary conditions on the solution. Hence we will consider the energy functional of the form

$\overline{\mathcal{F}}(u,\phi):=\mathcal{F}(u,\phi)-\int_{\partial\Omega}(\kappa{f}+cp) \frac{\partial{u}}{\partial\nu} dS(x) .$

We will consider the function set

$W:=\{v\in{H^2}(\Omega)\,|\,\left.v\right|_{\partial\Omega}=g\}\times{H^1_p}(\Omega),$

which is a convex, closed subset of the Hilbert space $H^2(\Omega)\times{H^1}(\Omega)$ . The respective energy minimisation problem is to find $(u, \phi)\in{W}$ such that

$\overline{\mathcal{F}}(u,\phi)=\inf_{(v,\psi)\in{W}}\overline{\mathcal{F}}(v,\psi). \qquad (1)$

Our aim is to find a solution to the above minimisation problem. We have, similarly as in the case of Dirichlet boundary conditions, that if $c<\sqrt{\kappa{a}}+\sqrt{\sigma{b}}$ then the functional $\overline{\mathcal{F}}$ is $\alpha$ -convex $W$ and the minimisation problem has a unique solution (see Lemma 3 of the RSG report).

The Euler-Lagrange equations

The respective Euler-Lagrange equations are

$\overline{a} ((u,\phi),(v,\psi))=0 \qquad \forall (v,\psi)\in{W_0}, \qquad (2)$

where $(u,\phi)\in{W}$ is the minimiser, $W_0:=H^1_0(\Omega)\cap{H^2}(\Omega)\times{H^1_0}(\Omega)$ and

$\overline{a}((u,\phi),(v,\psi)):=\kappa\int_\Omega\Delta{u}\,\Delta{v}+\sigma\int_\Omega\nabla{u}\cdot\nabla{v}+a\int_\Omega\phi\psi+b\int_\Omega\nabla\phi\cdot\nabla\psi\nonumber$
$+c\int_\Omega\phi\Delta{v}+c\int_\Omega\psi\Delta{u}-\int_{\partial\Omega}(\kappa{f}+cp)\frac{\partial{v}}{\partial\nu}dS(x) \qquad (3)$

We note that if $(u,\phi)\in{W}$ is a solution to (2) then it is also a solution to (1) (because $\overline{\mathcal{F}}$ is convex on $W$ ).

We shall now focus on finding the solution to (1). We will find the solution by formal integration by parts some of the terms appearing in (3) and considering the substitution $w=\Delta{u}$ .

Precisely, let $w\in{H^1_f}(\Omega)$ , $\phi\in{H^1_p}(\Omega)$ be the solution of

$0=\kappa\int_\Omega\nabla{w}\cdot\nabla{v}\,dx+\sigma\int_\Omega{w}\,v \,dx+a\int_\Omega\phi\,\psi\,dx+b\int_\Omega\nabla\phi\cdot\nabla\psi\,dx\nonumber$
$+c\int_\Omega\nabla\phi\cdot\nabla{v}\,dx+c\int_\Omega\psi\,w\,dx,\qquad\forall v,\psi\in{H^1_0}(\Omega).$

We note that this equation comes from integrating by parts the Euler-Lagrange equations and formally substituting $w:=\Delta{u}$ .
From here we define $u\in{H^1_g}$ to be the solution of

$\int_\Omega\nabla{u}\cdot\nabla{v}\,dx=\int_{\Omega}w\,v\,dx,\qquad\forall{v}\in{H^1_0}(\Omega).$

One can show that if $c^2<2\min\{\kappa{b}},\sigma{a}\}$ , then there exists a unique solution $(w,\phi)\in{H^1_f}(\Omega)\times{H^1_p}(\Omega)$ and that, given $w$ , there exists a unique solution $u\in{H^1_g}$ . In particular, this gives us the pair of functions $u\in{H^1_g}(\Omega)$ , $\phi\in{H^1_p}(\Omega)$ . One can then show that this pair of functions is the solution to the Euler-Lagrange equations. For the full proof we refer to Section 4.2.2 of RSG report.

MASDOC - Mathematics and Statistics Doctoral Training Centre

Navier boundary conditions

The Euler-Lagrange equations

The discretised Euler-Lagrange equations and the convergence rate of the approximation