Queridos compañeros.

Este próximo miércoles 19 de junio a las 12h. en el Seminario del IUMPA
se impartirá la siguiente conferencia:

Conferenciante: Thomas Kalmes

Título: “An approximation theorem of Runge type for kernels of certain non-elliptic partial differential operators”

From Runge’s classical theorem on rational approximation it follows that for open subsets $X_1\subseteq X_2$ of the complex plane $\mathbb{C}$ every function holomorphic in $X_1$ can be approximated uniformly on compact subsets of $X_1$ by functions which are holomorphic in $X_2$ if and only if $\mathbb{C}\backslash X_1$ has no compact connected component which is contained in $X_2$. This approximation theorem has been generalized independently by Lax and Malgrange from holomorphic functions, i.e.\ functions in the kernel of the Cauchy-Riemann operator, to kernels of elliptic constant coefficient partial differential operators $P(D)$.
We report on a recent generalization of the above approximation theorem to constant coefficient partial differential operator $P(D)$ with a single characteristic direction such as the time-dependent free Schr\”odinger operator as well as non-degenerate parabolic differential operators like the heat operator. Under the additional hypothesis that $P(D)$ is surjective on both $C^\infty (X_1)$ and $C^\infty(X_2)$, we characterize when solutions of the equation $P(D)u=0$ in $X_1$ can be approximated by solutions of the same equation in $X_2$. As part of our result we prove that for a large class of non-elliptic operators $P(D)$ there are non-trivial smooth solutions $u$ to the equation $P(D)u=0$ in $\mathbb{R}^d$ with support contained in an arbitarily narrow slab bounded by two parallel characteristic hyperplanes for $P(D)$.