1
$\begingroup$

Consider the following partial differential equation in the bounded domain $\Omega\subset \mathbb{R}^2$ $$\Delta^2 - \nabla\cdot(k(x,y)\nabla u) + \lambda u = f,\text{ in }\Omega$$ where $\lambda >0$ is a constant and $k(x,y) >0$ is a given function of the position. Further, $f(x,y)$ is a given function. The boundary conditions are given by $$u =0,\,\Delta u = 0,\text{ on }\partial\Omega.$$

I need to find the minimisation problem corresponding to this boundary value problem and explain whether the number of boundary conditions is sufficient for a unique solution. I have the following theorem:

Let $L:\sum(\Omega)\to\sum'(\Omega)$, where $\sum(\Omega)$ is a linear space, and suppose that $L$ is linear, self-adjoint, positive and coercive, and let $u_0\in\sum:Lu_0 = f$. Then $u_0\in\sum(\Omega)$ minimises $$F(u) = \int_\Omega\dfrac{1}{2}uLu - ufd\Omega.$$

Hence, after applying integration by parts twice on the first term and once on the second term and using Gauss' theorem, the minimisation problem corresponding to the boundary value problem is $$F(u) = \int_\Omega \dfrac{1}{2}\nabla\cdot\|\nabla u\|^2 + \dfrac{1}{2}k(x,y)\|\nabla u\|^2 + \lambda u^2 - uf\,d\Omega.$$

My Question is: How do I show whether or not the number of boundary conditions is sufficient for a unique solution? I don't have a clue how I should show this.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.