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

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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

Source
Miniconference on Operators in Analysis. Ian Doust, Brian Jefferies, Chun Li, and Alan McIntosh, eds. Proceedings of the Centre for Mathematical Analysis, v. 24. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 1990), 91-102

Dates
First available in Project Euclid: 18 November 2014

Permanent link to this document
https://projecteuclid.org/ euclid.pcma/1416335063

Citation

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


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