Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, first online, pp. 1-16

The Dirichlet-to-Neumann operator on rough domains with finite volume

A. F. M. ter Elst, Varun Narula

Received August 8, 2023. Published online March 5, 2025. OPEN ACCESS

Abstract: Using a variational formulation we consider the Dirichlet-to-Neumann operator on a connected open set $\Omega\subset\mathbb{R}^d$ of finite volume, assuming only that the surface measure is locally finite on the boundary. Then the boundary may have infinite measure and trace properties become delicate. We show that this has consequences for the kernel of the Dirichlet-to-Neumann operator and characterise the situation in which a trace on $\Omega$ both exists and is unique.

Keywords: Dirichlet-to-Neumann operator; trace; form method; rough boundary

Classification MSC: 46E35, 47A07

DOI: 10.21136/CMJ.2025.0362-23

PDF available at: Springer Institute of Mathematics CAS

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Affiliations: A. F. M. ter Elst (corresponding author), Varun Narula, University of Auckland, 38 Princes Street, Auckland Central, Auckland 1010, New Zealand e-mail: terelst@math.auckland.ac.nz, vnar603@aucklanduni.ac.nz

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