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# Dirichlet boundary conditions

The Dirichlet boundary conditions are:

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

where $\nu$ is the unit normal vector to $\partial\Omega$. Let $\Omega\subset\mathbb{R}^3$ be an open set with Lipschitz boundary $\partial\Omega$, $g,p\in{H^{1/2}}(\partial\Omega)$ and

$V:= H^2_{g,f}(\Omega)\times{H^1_p}(\Omega),$

where we denote $H^2_{g,f}(\Omega):=\{ u\in{H^2}(\Omega)\,|\,u=g\text{ on }\partial\Omega\,,\,\frac{\partial{u}}{\partial\nu}=f \text{ on }\partial\Omega\}.$ Clearly, $V$ is a closed, convex set of a Hilbert space $H:=H^2(\Omega)\times{H^1}(\Omega)$ equipped with the norm $||(v,\psi)||^2_H :=||v||_{H^2(\Omega)}^2+||\psi||^2_{H^1(\Omega)}$. The function set $V$ incorporates the Dirichlet boundary conditions.

Let us consider the energy functional

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

for $(u,\phi)\in{V}$. If $c<\sqrt{\kappa{a}}+\sqrt{\sigma{b}}$ then $\mathcal{F}$ is $\alpha$-convex on $V$. (see Lemma 1 of the RSG report for the proof)

Hence, for such $c$ we have the existence of a unique solution to the problem:
Find $(u,\phi)\in{V}$, s.t.

$\mathcal{F}(u,\phi)=\min_{(v,\psi)\in{V}}\mathcal{F}(v,\psi)$

(this follows from $\alpha$-convexity of $\mathcal{F}$ and convexity and closedness of $V$ - see Lemma 2 of the RSG report for the proof)

If $c<\sqrt{\kappa{a}}+\sqrt{\sigma{b}}$, then this minimisation problem has a unique solution.

## The Euler-Lagrange equations

The Euler-Lagrange equations corresponding to this minimisation problem are

$0= \kappa \int_\Omega \Delta u \, \Delta v + \sigma \int_\Omega \nabla u \cdot \nabla v + a \int_\Omega \phi \psi + b \int_\Omega \nabla \phi \cdot \nabla \psi + c \int_\Omega \phi \Delta v + c \int_\Omega \psi \Delta u$

for all $(v,\psi)\in{V_0}:=H^2_{0,0}(\Omega)\times{H^1_0}(\Omega)$

which is the weak formulation of the equations

$\begin{cases}\kappa\Delta^2{u}-\sigma\Delta{u}+c\Delta\phi=0,\\a\phi-b\Delta\phi+c\Deltau=0\end{cases}$

## What if $c\geq\sqrt{\kappa{a}}+\sqrt{\sigma{b}}$?

Suppose that $c\geq\sqrt{\kappa{a}}+\sqrt{\sigma{b}}$ and $\Omega:=(0,2\pi)^2$ and consider the case $\kappa=a, \sigma=b$. One can then show that $\mathcal{F}(\alpha(u_0,u_0))$ is concave with respect to $\alpha$ where $u_0(x,y)=\sin{x}\sin{y}$. Hence such a $\mathcal{F}$ is not convex on $\{u\in{H^2}(\Omega)\,:\,\left.u\right|_{\partial\Omega}=0\}\times{H^1_0}(\Omega)$. In fact, $\mathcal{F}$ is not bounded below on this space, so the minimiser does not exist. As for the function set $V$, it is not clear how analyse it convexity (and hence the existence of the minimiser) for such $c$.

On the other hand suppose $\Omega'=\mathbb{R}^2\setminus\overline{B(0,1)}$.

The approach applied in Turner [1] uses modified Bessel functions $K_n(\cdot)$ to write the explicit form of the solution $(u,\phi)$ of the Euler-Lagrange equations. In the stability region (i.e. when $c<\sqrt{\kappa{a}} + \sqrt{\sigma{b}}$) one can use the approximation $K_n(\rho)\approx\rm{e}^{-\rho}(\pi\rho/2)^{-1/2}$ when $\rho\gg{n}$ to obtain the asymptotic behaviour (i.e. for large $|x|$) of $u$

$u(x)\sim\frac{1}{\sqrt{r}}\rm{e}^{-\lambda(r-1)}\cos(\omega{r}+\vartheta),$

where $\vartheta$ depends only on the parameters $\sigma,\kappa,a,b,c$ and $\lambda,\omega$ are given by the real and imaginary parts of

$k_\pm=\frac{1}{2\sqrt{\kappa{b}}}\left(\sqrt{(\sqrt{\kappa{a}}+\sqrt{\sigma{b}})^2-c^2}\pm\sqrt{(\sqrt{\kappa{a}}-\sqrt{\sigma{b}})^2-c^2}\right).$

(the numbers $k_\pm^2$ are the solutions of the characteristic equation

$r^2(\kappa b)+r(c^2-\sigma{b}-a\kappa)+\sigma{a}=0)$

One can observe that $\lambda$ becomes negative when $c$ crosses the stability threshold $\sqrt{\kappa{a}}+\sqrt{\sigma{b}}$, from where the asymptotic approximation suggests a blowup of $u$ as $|x| \to \infty$. This is an unphysical behaviour and the region $c>\sqrt{\kappa{a}}+\sqrt{\sigma{b}}$ is called Leiber unstable regime.