Elliptic differential operators on Lipschitz domains and abstract boundary value problems.

Area of Science:

• Mathematical Analysis
• Partial Differential Equations
• Operator Theory

Background:

• Elliptic boundary value problems are crucial in physics and engineering.
• Non-smooth domains present significant challenges in analysis.
• Self-adjoint operators are fundamental in quantum mechanics and spectral theory.

Purpose of the Study:

• To develop abstract methods for analyzing elliptic boundary value problems on non-smooth domains.
• To provide a complete description of self-adjoint realizations of the Laplacian on Lipschitz domains.
• To generalize existing results for smooth and quasi-convex domains.

Main Methods:

• Development of quasi-boundary triples and associated Weyl functions.
• Extension of boundary maps by continuity to dual spaces.
• Application of abstract boundary values to characterize self-adjoint extensions.
• Analysis of Dirichlet-to-Neumann maps and trace operators.

Main Results:

• A framework for self-adjoint extensions of symmetric operators using abstract boundary values.
• Complete characterization of self-adjoint realizations of the Laplacian on bounded Lipschitz domains.
• Derivation of Kreĭn type resolvent formulas and spectral characterizations.
• Identification of maximal range spaces for Dirichlet and Neumann trace operators.

Conclusions:

• The developed theory provides a unified approach for elliptic boundary value problems on non-smooth domains.
• The results generalize and extend previous work on self-adjoint realizations and spectral properties.
• The study offers new insights into the spectral characterization of operators via boundary maps.