Calderón projector

Definition

Let $\Omega \subset \R^n$ be a bounded open set with smooth boundary $\Gamma = \partial\Omega$ and outward unit normal $\nu$. Let $P$ be an elliptic linear differential operator of order $2m$ with smooth coefficients on a neighbourhood $U$ of $\overline{\Omega}$. Assume $P$ has a fundamental solution $E$ on $U$, that is, $P E = \delta$ in the sense of distributions. For $u$ smooth up to $\Gamma$, let $\gamma u = (u|_\Gamma, \partial_\nu u|_\Gamma, \ldots, \partial_\nu^{2m-1} u|_\Gamma)$ be its Cauchy data, given by the trace operator. Green's formula for $P$ gives a potential operator $K$, built from $E$, such that $u = K(\gamma u)$ in $\Omega$ whenever $Pu = 0$ in $\Omega$. Let $\gamma_+$ take the Cauchy data of a function on $\Omega$ as limits from inside. The Calderón projector of $P$ and $\Omega$ is the operator $C = \gamma_+ K$ on $2m$-tuples of functions on $\Gamma$.