# Ngô Quốc Anh

## October 29, 2007

### Dirichlet-to-Neumann operator

Filed under: Nghiên Cứu Khoa Học — Ngô Quốc Anh @ 23:29

You have a domain $\Omega\subset \mathbb R^n$ and a partial differential equation (such as $\Delta u=0$) in that domain. Take a function $\phi$ defined on $\partial\Omega$, solve the boundary value problem with $u=\phi$ on the boundary, and compute $\psi=\frac{\partial u}{\partial n}$, the normal derivative of solution on the boundary. The map $\phi\mapsto\psi$ is the D-n-N operator.

A typical problem (motivated by tomography) is to recover the PDE from the D-n-N operator.

### Mountain Pass Theorem

Filed under: Nghiên Cứu Khoa Học, PDEs — Ngô Quốc Anh @ 23:00

Suppose $F \in C^1(V)$ satisfies (PS) condition with $F(0)=0$. There exist $\rho >0$, $\alpha >0$ and $e \in V$ such that

$\displaystyle\mathop {\inf }\limits_{\left\| u \right\| = \rho } F\left( u \right) \geqslant \alpha, \left\| e \right\| \geqslant \rho$

and

$F\left( e \right) < \alpha$.

Then

$\displaystyle \beta = \mathop {\inf }\limits_{\Sigma \in \Gamma } \mathop {\sup }\limits_{u \in \Sigma } F\left( u \right)$

is a critical value.