L 2 existence theorems for the $$\bar \partial _b - Neumann$$ problem on strongly pseudoconvex CR manifoldsproblem on strongly pseudoconvex CR manifolds

1991 ◽  
Vol 1 (2) ◽  
pp. 139-163 ◽  
Author(s):  
Mei-Chi Shaw
Keyword(s):  
Neumann Problem ◽  
Existence Theorems ◽  
Cr Manifolds

The Neumann problem in plane deformations of a micropolar elastic solid with micropolar surface effects

2020 ◽  
Vol 26 (1) ◽  
pp. 30-44
Author(s):  
Alireza Gharahi ◽  
Peter Schiavone
Keyword(s):  
Neumann Problem ◽  
Existence Theorems ◽  
Surface Elasticity ◽  
Elastic Solid ◽  
Singular Integral Equations ◽  
Surface Effects ◽  
Integral Representations ◽  
Matrix Functions ◽  
Micropolar Elasticity ◽  
The Neumann Problem

We consider the Neumann problem in a theory of plane micropolar elasticity incorporating micropolar surface effects. The incorporation of surface elasticity utilizes the Eremeyev–Lebedev–Altenbach shell model, leading to a set of second-order boundary conditions describing the separate micropolar elasticity of the surface. The Neumann problem is of particular interest, since the question of solvability is complicated by the fact that the corresponding systems of homogeneous singular integral equations admit nontrivial solutions that affect the solvability of both the interior and exterior Neumann boundary value problems. We overcome this difficulty by constructing integral representations of the solutions based on specifically constructed auxiliary matrix functions leading to uniqueness and existence theorems in appropriate classes of smooth matrix functions.

Teichmuller Geometry

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Uniqueness Theorem ◽  
Existence Theorems ◽  
Quadratic Differentials ◽  
Complex Structures ◽  
Metric Geometry ◽  
Quasiconformal Maps ◽  
Hyperbolic Structures ◽  
Teichmüller Metric ◽  
Uniqueness And Existence ◽  
Teichmüller Mapping

This chapter focuses on the metric geometry of Teichmüller space. It first explains how one can think of Teich(Sɡ) as the space of complex structures on Sɡ. To this end, the chapter defines quasiconformal maps between surfaces and presents a solution to the resulting Teichmüller's extremal problem. It also considers the correspondence between complex structures and hyperbolic structures, along with the Teichmüller mapping, Teichmüller metric, and the proof of Teichmüller's uniqueness and existence theorems. The fundamental connection between Teichmüller's theorems, holomorphic quadratic differentials, and measured foliations is discussed as well. Finally, the chapter describes the Grötzsch's problem, whose solution is tied to the proof of Teichmüller's uniqueness theorem.

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