scholarly journals A class of degenerate diffusion equations with mixed boundary conditions

2004 ◽  
Vol 298 (2) ◽  
pp. 589-603 ◽  
Author(s):  
Jing Wang ◽  
Zejia Wang ◽  
Jingxue Yin
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Diffusion Equations ◽  
Degenerate Diffusion ◽  
Degenerate Diffusion Equations

Existence theorems for the quantum drift–diffusion equations with mixed boundary conditions

2016 ◽  
Vol 18 (04) ◽  
pp. 1550048 ◽  
Author(s):  
Xiangsheng Xu
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Existence Theorems ◽  
Neumann Boundary ◽  
Mixed Boundary Conditions ◽  
Approximate Solutions ◽  
Diffusion Equations ◽  
Drift Diffusion ◽  
High Regularity ◽  
Recent Theorem

In this paper, we construct a weak solution to the unipolar quantum drift–diffusion equations coupled with initial and mixed boundary conditions in up to four space dimensions. We only assume that the boundary of the domain is Lipschitz and the interface between the Dirichlet boundary and the Neumann boundary is a Lipschitzian hypersurface, and thus the lack of high regularity for solutions of such problems is an issue. The convergence of a sequence of approximate solutions is established via an application of a recent theorem by the author on the logarithmic upper bound for solutions of elliptic equations with the same type of boundary conditions [Logarithmic up bounds for solutions of elliptic partial differential equations, Proc. Amer. Math. Soc. 139 (2011) 3485–3490].

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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