scholarly journals The Maxwell Compactness Property in Bounded Weak Lipschitz Domains with Mixed Boundary Conditions

2016 ◽  
Vol 48 (4) ◽  
pp. 2912-2943 ◽  
Author(s):  
Sebastian Bauer ◽  
Dirk Pauly ◽  
Michael Schomburg
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Lipschitz Domains ◽  
Compactness Property

scholarly journals A global div-curl-lemma for mixed boundary conditions in weak Lipschitz domains and a corresponding generalized A 0 * \mathrm{A}_{0}^{*} - A 1 \mathrm{A}_{1} -lemma in Hilbert spaces

Analysis ◽  
2019 ◽  
Vol 39 (2) ◽  
pp. 33-58 ◽  
Author(s):  
Dirk Pauly
Keyword(s):  
Boundary Conditions ◽  
Hilbert Spaces ◽  
Differential Operators ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Homogenization Theory ◽  
Lipschitz Domains ◽  
Compact Embeddings ◽  
Arbitrary Dimensions ◽  
Global And Local

Abstract We prove global and local versions of the so-called {\operatorname{div}} - {\operatorname{curl}} -lemma, a crucial result in the homogenization theory of partial differential equations, for mixed boundary conditions on bounded weak Lipschitz domains in 3D with weak Lipschitz interfaces. We will generalize our results using an abstract Hilbert space setting, which shows corresponding results to hold in arbitrary dimensions as well as for various differential operators. The crucial tools and the core of our arguments are Hilbert complexes and related compact embeddings.

scholarly journals General solutions in Chern-Simons gravity and $$ T\overline{T} $$-deformations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Eva Llabrés
Keyword(s):  
Variational Principle ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Spin Connection ◽  
Departure Point ◽  
Boundary Stress ◽  
Chern Simons

Abstract We find the most general solution to Chern-Simons AdS3 gravity in Fefferman-Graham gauge. The connections are equivalent to geometries that have a non-trivial curved boundary, characterized by a 2-dimensional vielbein and a spin connection. We define a variational principle for Dirichlet boundary conditions and find the boundary stress tensor in the Chern-Simons formalism. Using this variational principle as the departure point, we show how to treat other choices of boundary conditions in this formalism, such as, including the mixed boundary conditions corresponding to a $$ T\overline{T} $$ T T ¯ -deformation.

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