Let $\Omega \subset {\Bbb R}^2$ be a bounded domain with a regular boundary $\partial\Omega$, and let $\Gamma$ be a part of $\partial \Omega$ with nonempty relative interior.
Assume that a subdomain $\omega \subset \subset \Omega$ is given such that its boundary $\gamma := \partial\omega$ is a Jordan curve and let us denote by ${\bf n}$ the exterior normal to the boundary of $\Omega \setminus \overline{\omega}$. The question we address in this talk is the following: given a target function $h$ defined on $\gamma$, can one find a control function $v$ defined on $\partial\Omega$ having its support ${\rm supp}(v)\subset \Gamma$, and such that the solution $\Psi$ of
$\Delta\Psi = 0\quad\mbox{in }\, \Omega, \qquad {\partial\Psi \over \partial{\bf n}} = v\quad\mbox{on }\, \partial\Omega,$
satisfies
${\partial\Psi \over \partial{\bf n} } = h\quad \mbox{on }\, \gamma\qquad\mbox{or}\qquad \left\|{\partial\Psi \over \partial{\bf n} } - h\right\|_{H^{-1/2}(\gamma)} \leq \epsilon\mbox{ ?}$
(Here $H^{-1/2}(\gamma)$ denotes the Sobolev space of order $-1/2$ on $\gamma$). The motivation of this question lies in its application to the approximate Lagrangian control of Euler equation. The same question is studied when $\Omega \subset {\Bbb R}^N$ is a sooth bounded domain and $N \geq 2$.
The talk will also address the following ill-posed problem: let $\Omega$ be the rectangular domain $\Omega := (0,\pi )\times (0,\ell) \subset {\Bbb R}^2$ for some $\ell >0$ and set $\Gamma_{0} := [0,\pi]\times\{0\}$ and $\Gamma := \{0\}\times[0,\ell] \cup \{\pi\}\times[0,\ell]$. We give necessary and sufficient conditions on the Cauchy data $f_{0},g_{0}$ so that there exists $u \in H^1(\Omega)$ satisfying $\Delta u = 0$ in $\Omega$ and
${\partial u\over \partial{\bf n} } = 0 \quad \mbox{on }\, \Gamma,\qquad u = f_{0} \quad\mbox{on }\, \Gamma_{0}\quad \mbox{and }\, {\partial u \over\partial{\bf n} } = g_{0}\quad \mbox{on }\, \Gamma_{0}.$