scholarly journals Positivity and anti-maximum principles for elliptic operators with mixed boundary conditions

2008 ◽  
pp. 73-104 ◽  
Author(s):  
Catherine Bandle ◽  
Joachim von Below ◽  
Wolfgang Reichel
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Elliptic Operators ◽  
Maximum Principles

scholarly journals Strict Positivity for the Principal Eigenfunction of Elliptic Operators with Various Boundary Conditions

2020 ◽  
Vol 20 (3) ◽  
pp. 633-650
Author(s):  
Wolfgang Arendt ◽  
A. F. M. ter Elst ◽  
Jochen Glück
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Elliptic Operators ◽  
Maximum Principles ◽  
Lipschitz Continuous ◽  
Various Boundary Conditions ◽  
Measurable Coefficients ◽  
Strict Positivity ◽  
Robin Boundary

AbstractWe consider elliptic operators with measurable coefficients and Robin boundary conditions on a bounded domain {\Omega\subset\mathbb{R}^{d}} and show that the first eigenfunction v satisfies {v(x)\geq\delta>0} for all {x\in\overline{\Omega}}, even if the boundary {\partial\Omega} is only Lipschitz continuous. Under such weak regularity assumptions the Hopf–Oleĭnik boundary lemma is not available; instead we use a new approach based on an abstract positivity improving condition for semigroups that map {L_{p}(\Omega)} into {C(\overline{\Omega})}. The same tool also yields corresponding results for Dirichlet or mixed boundary conditions. Finally, we show that our results can be used to derive strong minimum and maximum principles for parabolic and elliptic equations.

scholarly journals Strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions

2019 ◽  
Vol 150 (1) ◽  
pp. 475-495 ◽  
Author(s):  
Begoña Barrios ◽  
Maria Medina
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Classical Case ◽  
Parabolic Problems ◽  
Laplacian Operator ◽  
Maximum Principles ◽  
Local Version ◽  
Comparison Results ◽  
Non Local

AbstractWe present some comparison results for solutions to certain non-local elliptic and parabolic problems that involve the fractional Laplacian operator and mixed boundary conditions, given by a zero Dirichlet datum on part of the complementary of the domain and zero Neumann data on the rest. These results represent a non-local generalization of a Hopf's lemma for elliptic and parabolic problems with mixed conditions. In particular we prove the non-local version of the results obtained by Dávila and Dávila and Dupaigne for the classical cases= 1 in [23, 24] respectively.

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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