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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.

The discretised equations