St. Petersburg Mathematical Journal

Abstract:

Let $(M,g)$ be a compact (in general, nonorientable) surface with boundary $\partial M$ and let $\Gamma _0$, …, $\Gamma _{m-1}$ be connected components of $\partial M$. Let $u=u^{f}(x)$ be a solution to the problem $\Delta _{g}u=0$ in $M$, $u\big |_{\Gamma _0}=f$, $u\big |_{\Gamma _j}=0$, $j=1$, …, $m’$, $\partial _{\nu }u\big |_{\Gamma _j}=0$, $j=m’+1$, …, $m-1$, where $\nu$ is the outward normal. With this problem, one associates the DN map $\Lambda \colon f\mapsto \partial _{\nu }u^{f}\big |_{\Gamma _0}$. The purpose is to determine $M$ from $\Lambda$.

To this end, an algebraic version of the boundary control method is applied. The key instrument is the algebra $\mathfrak {A}$ of functions holomorphic on the appropriate orientable double cover of $M$. It is proved that $\mathfrak {A}$ is determined by $\Lambda$ up to isometric isomorphism. The spectrum of the algebra $\mathfrak {A}$ provides a relevant copy $M’$ of $M$. This copy is conformally equivalent to $M$ while its DN map coincides with $\Lambda$.