Elliptic differential operators on Lipschitz domains and abstract boundary value problems

This paper consists of two parts. In the first part, which is of more abstract nature, the notion of quasi boundary triples and associated Weyl functions is developed further in such a way that it can be applied to elliptic boundary value problems on non-smooth domains. A key feature is the extension of the boundary maps by continuity to the duals of certain range spaces, which directly leads to a description of all self-adjoint extensions of the underlying symmetric operator with the help of abstract boundary values. In the second part of the paper a complete description is obtained of all self-adjoint realizations of the Laplacian on bounded Lipschitz domains, as well as Kre\u{\i}n type resolvent formulas and a spectral characterization in terms of energy dependent Dirichlet-to-Neumann maps. These results can be viewed as the natural generalization of recent results from Gesztesy and Mitrea for quasi-convex domains. In this connection we also characterize the maximal range spaces of the Dirichlet and Neumann trace operators on a bounded Lipschitz domain in terms of the Dirichlet-to-Neumann map. The general results from the first part of the paper are also applied to higher order elliptic operators on smooth domains, and particular attention is paid to the second order case which is illustrated with various examples

Extension theory for symmetric operators and elliptic boundary value problems on Lipschitz domains

Diese Dissertationsschrift besteht aus zwei wesentlichen Teilen. In dem ersten eher abstrakten Teil wird die Theorie der Quasirandtripel und deren Weylfunktionen weiterentwickelt, so dass diese auf elliptische Randwertprobleme auf nichtglatten Gebieten angewendet werden kann. Ein Hauptresultat ist die stetige Fortsetzung der Randabbildungen auf die Anti-Dualräume spezieller Bildräume. Dies führt zu einer Beschreibung aller selbstadjungierter Erweiterungen des zugrunde liegenden symmetrischen Operators durch abstrakte Randbedingungen. Im zweiten Teil wird eine komplette Beschreibung aller selbstadjungierter Realisierungen des Laplaceoperators auf beschränkten Lipschitzgebieten durch Randparameter sowie Kreinsche Resolventenformeln angegeben. Diese Resultate können als natürliche Generalisierung der kürzlich erschienenen Resultate von F. Gesztesy und M. Mitrea über quasi-konvexen Gebieten gesehen werden. In diesem Zusammenhang werden auch die maximalen Bildräume des Dirichlet- und des Neumann-Spuroperators auf beschränkten Lipschitzgebieten charakterisiert und durch die energieabhängige Dirichlet-zu-Neumann-Abbildung topologisiert. Die allgemeinen Aussagen des ersten Teils der Dissertation werden auch auf elliptische Differentialoperatoren höherer Ordnung auf glatten Gebieten angewendet und durch Beispiele illustriert.This dissertation consists of two main parts. In the first part, which is of more abstract nature, the notion of quasi boundary triples and associated Weyl functions is developed further in such a way that it can be applied to elliptic boundary value problems on non-smooth domains. A key feature is the extension of the boundary maps by continuity to the anti-duals of certain range spaces, which directly leads to a description of all self-adjoint extensions of the underlying symmetric operator with the help of abstract boundary values. In the second part a complete description of all self-adjoint realizations of the Laplacian on bounded Lipschitz domains, as well as Krein type resolvent formulas and a spectral characterization in terms of energy dependent Dirichlet-to-Neumann maps is obtained. These results can be viewed as the natural generalization of recent results from F. Gesztesy and M. Mitrea for quasi-convex domains. In this connection we also characterize the maximal range spaces of the Dirichlet and Neumann trace operators on a bounded Lipschitz domain in terms of the Dirichlet-to-Neumann map. The general results from the first part of the paper are also applied to higher order elliptic operators on smooth domains, and particular attention is paid to the second order which is illustrated with many examples

Acknowledgements

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