# Dirichlet boundary conditions

The Dirichlet boundary conditions are:

where is the unit normal vector to . Let be an open set with Lipschitz boundary , and

where we denote Clearly, is a closed, convex set of a Hilbert space equipped with the norm . The function set incorporates the Dirichlet boundary conditions.

Let us consider the energy functional

for . If then is -convex on . (see Lemma 1 of the RSG report for the proof)

Hence, for such we have the existence of a unique solution to the problem:

Find , s.t.

(this follows from -convexity of and convexity and closedness of - see Lemma 2 of the RSG report for the proof)

If , then this minimisation problem has a unique solution.

## The Euler-Lagrange equations

The Euler-Lagrange equations corresponding to this minimisation problem are

for all

which is the weak formulation of the equations

## What if ?

Suppose that and and consider the case . One can then show that is concave with respect to where . Hence such a is not convex on . In fact, is not bounded below on this space, so the minimiser does not exist. As for the function set , it is not clear how analyse it convexity (and hence the existence of the minimiser) for such .

On the other hand suppose .

The approach applied in Turner [1] uses modified Bessel functions to write the explicit form of the solution of the Euler-Lagrange equations. In the stability region (i.e. when ) one can use the approximation when to obtain the asymptotic behaviour (i.e. for large ) of

where depends only on the parameters and are given by the real and imaginary parts of

(the numbers are the solutions of the characteristic equation

One can observe that becomes negative when crosses the stability threshold , from where the asymptotic approximation suggests a blowup of as . This is an unphysical behaviour and the region is called *Leiber unstable regime*.