## Proceedings of the Centre for Mathematics and its Applications

### $H_{\infty}$ functional calculus of second order elliptic partial differntial operators on $L^p$ spaces

Xuan Thinh Duong

#### Abstract

Let L be a strongly elliptic partial differential operator of second order, with real coefficients on $L^p(\Omega), 1 \lt p \lt \infty$, with either Dirichlet, or Neumann, or "oblique" boundary conditions. Assume that $\Omega$ is an open, bounded domain with $C^2$ boundary. By adding a oonstant, if necessary, we then establish an $H_\infty$, functional calculus which associates an operator m(L) to each bounded holomorphic function m so that $\| m (L) \| \leq M \| m \|\infty$ where M is a constmt independent of m. Under suitable asumptions on L, we can also obtain a similar result in the case of Dirichlet boundary conditions where $\Omega$ is a non-smooth domain.

#### Article information

Dates
First available in Project Euclid: 18 November 2014

Duong, Xuan Thinh. $H_{\infty}$ functional calculus of second order elliptic partial differntial operators on $L^p$ spaces. Miniconference on Operators in Analysis, 91--102, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, 1990.https://projecteuclid.org/euclid.pcma/1416335063