Vol. 13, No. 6, 2020

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On uniqueness results for Dirichlet problems of elliptic systems without de Giorgi–Nash–Moser regularity

Pascal Auscher and Moritz Egert

Vol. 13 (2020), No. 6, 1605–1632
Abstract

We study uniqueness of Dirichlet problems of second-order divergence-form elliptic systems with transversally independent coefficients on the upper half-space in the absence of regularity of solutions. To this end, we develop a substitute for the fundamental solution used to invert elliptic operators on the whole space by means of a representation via abstract single-layer potentials. We also show that such layer potentials are uniquely determined.

Keywords
Dirichlet problems, uniqueness of solutions, elliptic systems, single-layer operators
Mathematical Subject Classification 2010
Primary: 35J57, 35A02
Secondary: 35J50, 42B25, 35C15
Milestones
Received: 28 March 2017
Accepted: 13 August 2019
Published: 12 September 2020
Authors
Pascal Auscher
Université Paris-Saclay
CNRS
Laboratoire de Mathématiques d’Orsay
Orsay
France
Laboratoire Amiénois de Mathématique Fondamentale et Appliquée
CNRS-UMR 7352
Université de Picardie-Jules Verne
Amiens
France
Moritz Egert
Université Paris-Saclay
CNRS
Laboratoire de Mathématiques d’Orsay
Orsay
France