# Seminario: “On the existence of classical solutions of the Helmholtz equation”

## Lunedì 14 Novembre 2016, ore 15:00 - Aula 2BC30 - Chia-Chi Tung

ARGOMENTI: Seminari

Lunedì 14 Novembre 2016 alle ore 15:00 in Aula 2BC30 Chia-Chi Tung (Minnesota State University) terrà un seminario dal titolo “On the existence of classical solutions of the Helmholtz equation”.

Abstract
In this talk the Helmholtz equation is defined on a domain in a complex space and its solvability in the classical sense will be considered. Given a Riemann subdomain $(D, p)$ of a complex space and a Helmholtz operator $\mathscr{H}_{p,\mu}(\psi) := \Delta_p\psi + \mu\psi$, with a real parameter $\mu$, there exists a mapping $\mathscr{G}_{D,\mu,0} : \mathscr{C}^\infty(\overline{D}) \rightarrow \mathscr{C}^0(\overline{D}) \cap \mathscr{C}^\infty(D^*)$ such that $\mathscr{H}_{p,\mu} \circ \mathscr{G}_{D,\mu,0} =$ Id. in $D^*$.
Moreover, in the case $D$ is a pseudoball, an explicit integral representation of the operator $\mathscr{G}_{D,\mu,0}$ can be given. This integral formula extends corresponding results of potential theory on an Euclidean space to a (possibly) singular domain. Such results are useful in, for instance, finding conditions under which a $\mathscr{C}^2$-function satisfies an (inhomogeneous) Schrödinger equation with a (possibly) complex parameter and/or a nonzero potential on a complex space.