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.

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